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Research Article Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation J. F. Gómez Aguilar, 1 T. Córdova-Fraga, 2 J. Tórres-Jiménez, 3 R. F. Escobar-Jiménez, 4 V. H. Olivares-Peregrino, 4 and G. V. Guerrero-Ramírez 4 1 CONACYT-Centro Nacional de Investigaci´ on y Desarrollo Tecnol´ ogico, Tecnol´ ogico Nacional de M´ exico, Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico 2 Departamento de Ingenier´ ıa F´ ısica, Divisi´ on de Ciencias e Ingenier´ ıas Campus Le´ on, Universidad de Guanajuato, 37150 Le´ on, GTO, Mexico 3 Departamento de Electromec´ anica, Instituto Tecnol´ ogico Superior de Irapuato, 36821 Irapuato, GTO, Mexico 4 Centro Nacional de Investigaci´ on y Desarrollo Tecnol´ ogico, Tecnol´ ogico Nacional de M´ exico, Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico Correspondence should be addressed to J. F. G´ omez Aguilar; [email protected] Received 19 February 2016; Accepted 17 April 2016 Academic Editor: Juan J. Trujillo Copyright © 2016 J. F. G´ omez Aguilar et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. e anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range (0, 2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. e advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. e classical cases are recovered when the fractional derivative exponents are equal to 1. 1. Introduction Fourier’s law satisfies the heat conduction induced by a small temperature gradient in steady state. In steady state, the heat transfer through a material is proportional to the negative gradient of the temperature and to the area. However, there are some cases in which the Fourier equation is not adequate to describe the heat conduction process. More precisely, Fourier law is diffusive and cannot predict the finite temperature propagation speed in transient situations, in this context, the Cattaneo-Vernotte equation corrects the nonphysical property of infinite propagation of the Fourier and Fickian theory of the diffusion of heat, and this equation also known as the telegraph equation for the temperature is a generalization of the heat diffusion (Fourier’s law) and particle diffusion (Fick’s laws) equations. Processes where the traditional Fourier heat equation leads to inaccurate temperature and heat flux profiles are known as non- Fourier type processes [1]; these processes can be Markovian or non-Markovian [2]. In the Markovian processes case, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 7845874, 15 pages http://dx.doi.org/10.1155/2016/7845874
Transcript of Research Article Nonlocal Transport Processes and...
Research ArticleNonlocal Transport Processes and the FractionalCattaneo-Vernotte Equation
J F Goacutemez Aguilar1 T Coacuterdova-Fraga2 J Toacuterres-Jimeacutenez3 R F Escobar-Jimeacutenez4
V H Olivares-Peregrino4 and G V Guerrero-Ramiacuterez4
1CONACYT-Centro Nacional de Investigacion y Desarrollo Tecnologico Tecnologico Nacional de MexicoInterior Internado Palmira SN Colonia Palmira 62490 Cuernavaca MOR Mexico2Departamento de Ingenierıa Fısica Division de Ciencias e Ingenierıas Campus Leon Universidad de Guanajuato37150 Leon GTO Mexico3Departamento de Electromecanica Instituto Tecnologico Superior de Irapuato 36821 Irapuato GTO Mexico4Centro Nacional de Investigacion y Desarrollo Tecnologico Tecnologico Nacional de Mexico Interior Internado Palmira SNColonia Palmira 62490 Cuernavaca MOR Mexico
Correspondence should be addressed to J F Gomez Aguilar jgomezcenidetedumx
Received 19 February 2016 Accepted 17 April 2016
Academic Editor Juan J Trujillo
Copyright copy 2016 J F Gomez Aguilar et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations this mathematical model combineswaves and diffusion with a finite velocity of propagation In disordered systems the diffusion can be anomalous In these kinds ofsystems the mean-square displacement is proportional to a fractional power of time not equal to one The anomalous diffusionconcept is naturally obtained fromdiffusion equations using the fractional calculus approach In this paper we present an alternativerepresentation of the Cattaneo-Vernotte equation using the fractional calculus approach the spatial-time derivatives of fractionalorder are approximated using the Caputo-type derivative in the range (0 2] In this alternative representation we introduce theappropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivativesinto the fractional Cattaneo-Vernotte equation Finally consider the Dirichlet conditions the Fourier method was used to findthe full solution of the fractional Cattaneo-Vernotte equation in analytic way and Caputo and Riesz fractional derivatives areconsidered The advantage of our representation appears according to the comparison between our model and models presentedin the literature which are not acceptable physically due to the dimensional incompatibility of the solutions The classical cases arerecovered when the fractional derivative exponents are equal to 1
1 Introduction
Fourierrsquos law satisfies the heat conduction induced by asmall temperature gradient in steady state In steady statethe heat transfer through a material is proportional tothe negative gradient of the temperature and to the areaHowever there are some cases in which the Fourier equationis not adequate to describe the heat conduction processMore precisely Fourier law is diffusive and cannot predict thefinite temperature propagation speed in transient situations
in this context the Cattaneo-Vernotte equation corrects thenonphysical property of infinite propagation of the Fourierand Fickian theory of the diffusion of heat and this equationalso known as the telegraph equation for the temperatureis a generalization of the heat diffusion (Fourierrsquos law) andparticle diffusion (Fickrsquos laws) equations Processes wherethe traditional Fourier heat equation leads to inaccuratetemperature and heat flux profiles are known as non-Fourier type processes [1] these processes can be Markovianor non-Markovian [2] In the Markovian processes case
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7845874 15 pageshttpdxdoiorg10115520167845874
2 Mathematical Problems in Engineering
the mean-square displacement of the diffusing particle isproportional to time while in the disordered systems ornon-Markovian process the diffusion can be anomalousin this case the mean-square displacement is proportionalto a fractional power of time not equal to one whenanomalous diffusion occurs the probability density for adiffusing particle is not the usual Gaussian distributionThe mechanism of diffusion is Brownian motion and thismotion is the simplest continuous-time stochastic processContinuous-time random walks can be coupled with Brow-nian motion and fractional calculus (FC) to provide animproved estimator in the modeling of anomalous diffusionA random walk is a mathematical formalization of a paththat consists of a succession of random steps [3] A Levyflight also referred to as Levy motion is a random walkin which the step-lengths have a heavy-tailed probabilitydistribution When a random walk is defined as a walkin a space of dimension greater than one the steps aredefined in terms of a probability distribution and steps movewith isotropic random directions [4] and continuous-timerandom walk schemes are considered in the derivation oftime-fractional differential equations Recently the subjectof FC has attracted interest of researches this mathematicalconcept involves nonlocal operators which can be applied inphysical systems yielding new information about its behaviorfractional derivatives with respect to coordinates describepower-law nonlocal properties of the distributed systemand there are several papers about the recent history ofthe FC see [5ndash7] Several approaches have been used forinvestigating anomalous diffusion Langevin equations [8 9]random walks [10 11] or fractional derivatives based on FCseveral works connected to anomalous diffusion processeswhich may be found in [12ndash30] Using phenomenologicalarguments Compte andMetzler [31] generalize theCattaneo-Vernotte equation by introducing fractional derivatives witha continuous-time random walk scheme The authors of theworks presented in [32ndash34] studied the generalizedCattaneo-Vernotte equation with fractional space-time derivatives andthe order of the spatial and temporal fractional derivativesare 120573 120574 isin (0 2] Lewandowska and Kosztołowicz in [35]investigate the subdiffusive impedance phenomena of aspatially limited sample for large pulsation of electric fieldTarasov in [36] based on the Liouville equation obtainedthe fractional analogues of the classical kinetic and transportequations Qi and Jiang in [37] derived the exact solution ofthe Cattaneo-Vernotte equation by joint Laplace and Fouriertransforms Other applications of FC to Cattaneo-Vernotteequation are given in [38ndash41]
The aim of this work is to contribute to the developmentof a new version of fractional fundamental Cattaneo-Vernotteequation applying the idea proposed in the work [42] theorder considered is (0 2] for the fractional equation in space-time domain this representation preserves the dimensional-ity of the equation for any value taken by the exponent of thefractional derivative
The paper is structured as follows in Section 2 we explainthe basic concepts of the fractional calculus in Section 3 wepresent the fractional Cattaneo-Vernotte equation and giveconclusions in Section 4
2 Basic Definitions of Fractional Calculus
The most commonly used definitions in FC are Riemann-Liouville (RL) Grunwald-Letnikov (GL) Caputo fractionalderivative (CFD) and Riesz fractional derivative (119877) [43ndash46]
The RL definition of the fractional derivative for (120593 gt 0)
is
RL119886119863120593
119905119891 (119905) =
1
Γ (119898 minus 120593)
119889119898
119889119905119898int119905
119886
119891 (120578)
(119905 minus 120578)120593minus119898+1
119889120578
119898 minus 1 lt 120593 lt 119898
(1)
For function 119891(119905) the CFD is given by
119862
119886119863120593
119905119891 (119905) =
1
Γ (119899 minus 120593)int119905
119886
119891(119899)
(120578)
(minus120578 + 119905)120593minus119899+1
119889120578
119899 minus 1 lt 120593 le 119899
(2)
where 119889120593119889119905120593 = 119862119886119863120593
119905is a CFD with respect to 119905 120593 isin 119877 is the
order of the fractional derivative 119899 = 1 2 isin 119873 and Γ(sdot)
represents Eulerrsquos gamma functionIn the present paper we would use the CFD definition
since the former is more popular in real applications Forthe CFD definition we need to specify additional conditionsin order to produce a unique solution these additionalconditions are expressed in terms of integer-order derivatives[46] and this definition is used mainly for the problemwith memories In the case of the RL definition there existphysically unacceptable initial conditions [47]
The Laplace transform of Caputorsquos derivative (2) has theform [48]
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus
119899minus1
sum119896=0
119904120593minus119896minus1
119891(119896)
(0) (3)
where 119865(119904) is the Laplace transform of the function 119891(119905) and119899 = [R(120593)] + 1 From this expression we have two particularcases
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus 119904120593minus1
119891 (0) 0 lt 120593 le 1 (4)
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus 119904120593minus1
119891 (0) minus 119904120593minus2
1198911015840
(0)
1 lt 120593 le 2
(5)
The Mittag-Leffler function has gained extensive interestamong physicists due its vast potential of applications in thesolution of fractional differential equations [48]
119864120572120579
(119905) =
infin
sum119898=0
119905119898
Γ (120572119898 + 120579) (120572 gt 0) (120579 gt 0) (6)
when 120572 = 1 and 120579 = 1 from (6) we obtain the exponentialfunction
erfc(120572) denotes the complementary error function [48]and it is defined as
erfc (120572) = 2
radic120587int120572
0
exp (minus1199052
) 119889119905 (7)
Mathematical Problems in Engineering 3
Some common Mittag-Leffler functions are described in[48]
119864121
(plusmn120572) = exp (1205722
) [1 plusmn erfc (120572)] (8)
11986411
(plusmn120572) = exp (plusmn120572) (9)
11986421
(minus1205722
) = cos (120572) (10)
11986431
(120572) =1
2[exp (120572
13
)
+ 2 exp(minus(1
2) 12057213
) cos(radic3
212057213
)]
(11)
11986441
(120572) =1
2(cos (12057214) + cosh (120572
14
)) (12)
The Mittag-Leffler function 1198641199042119903
is defined by Miller in[49]
1198641199042119903
(119911) =1199112120581(1minus119903)
119904
119904minus1
sum119895=0
1205721minus(1199042+119903)
119895(exp (120572
1198951199112120581
))
sdot (1205721199042
119895+ erfc (12057212
119895119911120581
))
minus 119911minus2119899
2119899minus1
sum119896=0
119911119896
Γ (1199041198962 + 120583)
(13)
where 120581 = 1119904 119903 = 119899119904 + 120583 119899 = 0 1 2 3 120583 = 1 2 3 For the calculation of generalized Mittag-Leffler functions atarbitrary precision see [50 51]
The Riesz fractional derivative for (120593 gt 0) is [43ndash46]
119877
119863120593
119909119891 (119909) = minus
1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]
119899 minus 1 lt 120593 le 119899
(14)
where 119889120593119889119909120593 = 119877119863120593
119909is a Riesz fractional derivative with
respect to 119909 120593 isin 119877 is the order of the fractional derivative119899 = 1 2 isin 119873 and Γ(sdot) represents Eulerrsquos gamma function
3 Fractional Cattaneo-Vernotte Equation
In previous studies of the fractional Cattaneo-Vernotte equa-tion the authors did not consider the physical dimensionalityof the solutions The authors of the work [42] proposed asystematic way to construct fractional differential equationsfor the physical systems To keep the dimensionality of
the fractional differential equations a new parameter 120590 wasintroduced in the following way
120597
120597119909997888rarr
1
1205901minus120593
119909
sdot120597120593
120597119909120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(15)
120597
120597119905997888rarr
1
1205901minus120593
119905
sdot120597120593
120597119905120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(16)
where 120593 is an arbitrary parameter which represents theorder of the derivative 120590
119909has dimension of length and 120590
119905
has the dimension of time These new parameters maintainthe dimensionality of the equation invariant and character-izes the fractional space or fractional temporal structures(components that show an intermediate behavior between aconservative system and dissipative one) [42] when120593 = 1 theexpressions (15) and (16) reduce to the ordinary derivative Inthe following we will apply this idea to generalize the case ofthe fractional Cattaneo-Vernotte equation
In this work we consider generalized Cattaneo-Vernotteequation in the 119909 direction of the form [32ndash35]
nabla2
119879 minus1
119863 minus
120591
119863 = 0 (17)
where 120591 is a characteristic relaxation time constant (orthe non-Fourier character of the material) and 119863 is thegeneralized thermal diffusivity (17) is a hyperbolic diffusionequation when the parameter 120591 = 0 (17) recovers a parabolicform in this limit one has to replace Cattaneo-Vernotteequation by Fourierrsquos heat transfer equation
Considering the CFD (2) and (15) and (16) the fractionalrepresentation of (17) is
1
1205902(1minus120593)
119909
119862
01198632120593
119909119879 (119909 119905) minus
1
119863sdot
1
1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905) minus
120591
119863
sdot1
1205902(1minus120593)
119905
119862
01198632120593
119905119879 (119909 119905) = 0
(18)
The order of the derivative considered is 120593 isin (0 2] for thefractional Cattaneo-Vernotte equation in space-time domain
31 Fractional Space Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the space derivative is fractionalequation (15) and the time derivative is ordinary the spatialfractional equation is
119862
01198632120593
119909119879 (119909 119905) minus
1
1198631205902(1minus120593)
119909sdot120597119879 (119909 119905)
120597119905minus
120591
1198631205902(1minus120593)
119909
sdot1205972119879 (119909 119905)
1205971199052= 0
(19)
Suppose the solution
119879 (119909 119905) = 1198790sdot exp (119894120596119905) 119906 (119909) (20)
4 Mathematical Problems in Engineering
substituting (20) into (19) we obtain
1198892120593119906 (119909)
1198891199092120593+ (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909119906 (119909) = 0 (21)
where
1198962
119909= (
120591
1198631205962
minus 1198941
119863120596) (22)
is the dispersion relation in the 119909 direction and
2
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909= 1198962
1199091205902(1minus120593)
119909 (23)
is the fractional dispersion relation from the fractionaldispersion relation (23) we can expect the fractional wavenumber in the 119909 direction to have real and imaginary parts120575119909and 120573
119909 respectively Let us write
119909= 120575119909minus 119894120573119909 (24)
substituting (24) into (23) we have
(120575119909minus 119894120573119909)2
= 1205752
119909minus 2119894120575119909120573119909minus 1205732
119909 (25)
where
1205752
119909minus 2119894120575119909120573119909minus 1205732
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909 (26)
solving for 120573119909we obtain
120573119909=
120596
119863sdot
1
2120575119909
1205902(1minus120593)
119909 (27)
and for 120575119909
120575119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909 (28)
substituting (28) into (27) we have
120573119909=
1
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909 (29)
Now the fractional wave number is 119909= 120575119909minus 119894120573119909 where
120575119909and 120573
119909are given by (28) and (29) respectively
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909
(30)
equation (30) describes the real and imaginary part of thefractional wave number in terms of the frequency 120596 therelaxation time 120591 and the generalized thermal diffusivity 119863in presence of fractional space components 120590
119909
Considering (23) (21) gives
1198892120593119906 (119909)
1198891199092120593+ 2
119909119906 (119909) = 0 (31)
the solution of (31) can be obtained applying direct andinverse Laplace transform [47] and the solution of the aboveequation is given by
119906 (119909) = 11986421205931
(minus2
1199091199092120593
) (32)
where 11986421205931
(minus2
1199091199092120593) is the Mittag-Leffler function
Therefore the general solution of (21) is given by
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21205931(minus2
1199091199092120593
) (33)
Next we will analyze the case when 120593 takes differentvalues
Case 1 When 120593 = 2 we have
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus1
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus1
119909
(34)
and equation (34) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
41(minus2
1199091199094
) (35)
where 11986441
is given by (12) and solution (35) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905)
sdot [cos(minus12
119909119909) + cosh (minus
12
119909119909)]
(36)
Case 2 When 120593 = 32 we have
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus12
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus12
119909
(37)
and equation (37) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
31(minus2
1199091199093
) (38)
Mathematical Problems in Engineering 5
where 11986431
is given by (11) and solution (38) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
(39)
Case 3 When 120593 = 1 we have 119909= 119896119909
119896119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
(40)
and equation (40) represents the classical wave number 119896119909
From (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21(minus2
1199091199092
) (41)
where
119879 (119909 119905) = R [1198790sdot exp (119894120596119905) sdot exp (minus119894
119909119909)] (42)
and in (42) 119909
= 119896119909 R indicates the real part and 119896
119909=
120575119909minus 119894120573119909is wave number (40) substituting 119896
119909in (42) we have
119879 (119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)] (43)
Equation (43) represents the classical case for the spaceCattaneo-Vernotte equation The first exponential exp(119894(120596119905 minus120575119909119909)) gives the usual plane-wave variation of the thermal
field with position 119909 and time 119905 The second exponentialexp(minus120573
119909119909) gives and exponential decay in the amplitude of
the thermal wave
Case 4 When 120593 = 12 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
11(minus2
119909119909) (44)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059012
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059012
119909
(45)
and equation (45) represents the fractional wave number inpresence of fractional space components 120590
119909
The solution for (44) is
119879 (119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909))
sdot exp (minus120591
1198631205962
120590119909119909)]
(46)
whereR indicates the real part
Case 5 When 120593 = 14 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (47)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059034
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059034
119909
(48)
and equation (48) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (49)
where 119864121
is given by (8) and solution (49) is
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909)
sdot [1 minus erfc (minus2
11990912
)]
(50)
erfc(120572) denotes the error function defined in (8) Equation(50) represents the space evolution of the temperature andthe amplitude exhibits an algebraic decay for 119909 rarr infin
For this case there exists a physical relation between theauxiliary parameter 120590
119909and the wave number 119896
119909given by the
order 120593 of the fractional differential equation
120593 = 119896119909120590119909=
120590119909
120582 0 lt 120590
119909le 120582 (51)
where 120582 is the wavelength We can use this relation in orderto write (33) as
119879 ( 119905) = 1198790sdot exp (119894120596119905) sdot 119864
2120593(minus1205932(1minus120593)
2120593
) (52)
where = 119909120582 is a dimensionless parameter Figures 1(a) and1(b) show the simulation of (52) for 120593 values 07 lt 120593 le 1 and17 lt 120593 le 2 respectively
Table 1 shows the different solutions of (52) The order ofthe fractional differential equation is 120593 = 2 120593 = 32 120593 = 1120593 = 12 and 120593 = 14
32 Fractional Time Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the time derivative is fractionalequation (16) and the space derivative is ordinary the tempo-ral fractional equation is
119862
01198632120593
119905119879 (119909 119905) +
1
120591sdot 1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905)
minus119863
1205911205902(1minus120593)
119905
1205972119879 (119909 119905)
1205971199092= 0
(53)
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
the mean-square displacement of the diffusing particle isproportional to time while in the disordered systems ornon-Markovian process the diffusion can be anomalousin this case the mean-square displacement is proportionalto a fractional power of time not equal to one whenanomalous diffusion occurs the probability density for adiffusing particle is not the usual Gaussian distributionThe mechanism of diffusion is Brownian motion and thismotion is the simplest continuous-time stochastic processContinuous-time random walks can be coupled with Brow-nian motion and fractional calculus (FC) to provide animproved estimator in the modeling of anomalous diffusionA random walk is a mathematical formalization of a paththat consists of a succession of random steps [3] A Levyflight also referred to as Levy motion is a random walkin which the step-lengths have a heavy-tailed probabilitydistribution When a random walk is defined as a walkin a space of dimension greater than one the steps aredefined in terms of a probability distribution and steps movewith isotropic random directions [4] and continuous-timerandom walk schemes are considered in the derivation oftime-fractional differential equations Recently the subjectof FC has attracted interest of researches this mathematicalconcept involves nonlocal operators which can be applied inphysical systems yielding new information about its behaviorfractional derivatives with respect to coordinates describepower-law nonlocal properties of the distributed systemand there are several papers about the recent history ofthe FC see [5ndash7] Several approaches have been used forinvestigating anomalous diffusion Langevin equations [8 9]random walks [10 11] or fractional derivatives based on FCseveral works connected to anomalous diffusion processeswhich may be found in [12ndash30] Using phenomenologicalarguments Compte andMetzler [31] generalize theCattaneo-Vernotte equation by introducing fractional derivatives witha continuous-time random walk scheme The authors of theworks presented in [32ndash34] studied the generalizedCattaneo-Vernotte equation with fractional space-time derivatives andthe order of the spatial and temporal fractional derivativesare 120573 120574 isin (0 2] Lewandowska and Kosztołowicz in [35]investigate the subdiffusive impedance phenomena of aspatially limited sample for large pulsation of electric fieldTarasov in [36] based on the Liouville equation obtainedthe fractional analogues of the classical kinetic and transportequations Qi and Jiang in [37] derived the exact solution ofthe Cattaneo-Vernotte equation by joint Laplace and Fouriertransforms Other applications of FC to Cattaneo-Vernotteequation are given in [38ndash41]
The aim of this work is to contribute to the developmentof a new version of fractional fundamental Cattaneo-Vernotteequation applying the idea proposed in the work [42] theorder considered is (0 2] for the fractional equation in space-time domain this representation preserves the dimensional-ity of the equation for any value taken by the exponent of thefractional derivative
The paper is structured as follows in Section 2 we explainthe basic concepts of the fractional calculus in Section 3 wepresent the fractional Cattaneo-Vernotte equation and giveconclusions in Section 4
2 Basic Definitions of Fractional Calculus
The most commonly used definitions in FC are Riemann-Liouville (RL) Grunwald-Letnikov (GL) Caputo fractionalderivative (CFD) and Riesz fractional derivative (119877) [43ndash46]
The RL definition of the fractional derivative for (120593 gt 0)
is
RL119886119863120593
119905119891 (119905) =
1
Γ (119898 minus 120593)
119889119898
119889119905119898int119905
119886
119891 (120578)
(119905 minus 120578)120593minus119898+1
119889120578
119898 minus 1 lt 120593 lt 119898
(1)
For function 119891(119905) the CFD is given by
119862
119886119863120593
119905119891 (119905) =
1
Γ (119899 minus 120593)int119905
119886
119891(119899)
(120578)
(minus120578 + 119905)120593minus119899+1
119889120578
119899 minus 1 lt 120593 le 119899
(2)
where 119889120593119889119905120593 = 119862119886119863120593
119905is a CFD with respect to 119905 120593 isin 119877 is the
order of the fractional derivative 119899 = 1 2 isin 119873 and Γ(sdot)
represents Eulerrsquos gamma functionIn the present paper we would use the CFD definition
since the former is more popular in real applications Forthe CFD definition we need to specify additional conditionsin order to produce a unique solution these additionalconditions are expressed in terms of integer-order derivatives[46] and this definition is used mainly for the problemwith memories In the case of the RL definition there existphysically unacceptable initial conditions [47]
The Laplace transform of Caputorsquos derivative (2) has theform [48]
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus
119899minus1
sum119896=0
119904120593minus119896minus1
119891(119896)
(0) (3)
where 119865(119904) is the Laplace transform of the function 119891(119905) and119899 = [R(120593)] + 1 From this expression we have two particularcases
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus 119904120593minus1
119891 (0) 0 lt 120593 le 1 (4)
119871 [119862
0119863120593
119905119891 (119905)] = 119904
120593
119865 (119904) minus 119904120593minus1
119891 (0) minus 119904120593minus2
1198911015840
(0)
1 lt 120593 le 2
(5)
The Mittag-Leffler function has gained extensive interestamong physicists due its vast potential of applications in thesolution of fractional differential equations [48]
119864120572120579
(119905) =
infin
sum119898=0
119905119898
Γ (120572119898 + 120579) (120572 gt 0) (120579 gt 0) (6)
when 120572 = 1 and 120579 = 1 from (6) we obtain the exponentialfunction
erfc(120572) denotes the complementary error function [48]and it is defined as
erfc (120572) = 2
radic120587int120572
0
exp (minus1199052
) 119889119905 (7)
Mathematical Problems in Engineering 3
Some common Mittag-Leffler functions are described in[48]
119864121
(plusmn120572) = exp (1205722
) [1 plusmn erfc (120572)] (8)
11986411
(plusmn120572) = exp (plusmn120572) (9)
11986421
(minus1205722
) = cos (120572) (10)
11986431
(120572) =1
2[exp (120572
13
)
+ 2 exp(minus(1
2) 12057213
) cos(radic3
212057213
)]
(11)
11986441
(120572) =1
2(cos (12057214) + cosh (120572
14
)) (12)
The Mittag-Leffler function 1198641199042119903
is defined by Miller in[49]
1198641199042119903
(119911) =1199112120581(1minus119903)
119904
119904minus1
sum119895=0
1205721minus(1199042+119903)
119895(exp (120572
1198951199112120581
))
sdot (1205721199042
119895+ erfc (12057212
119895119911120581
))
minus 119911minus2119899
2119899minus1
sum119896=0
119911119896
Γ (1199041198962 + 120583)
(13)
where 120581 = 1119904 119903 = 119899119904 + 120583 119899 = 0 1 2 3 120583 = 1 2 3 For the calculation of generalized Mittag-Leffler functions atarbitrary precision see [50 51]
The Riesz fractional derivative for (120593 gt 0) is [43ndash46]
119877
119863120593
119909119891 (119909) = minus
1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]
119899 minus 1 lt 120593 le 119899
(14)
where 119889120593119889119909120593 = 119877119863120593
119909is a Riesz fractional derivative with
respect to 119909 120593 isin 119877 is the order of the fractional derivative119899 = 1 2 isin 119873 and Γ(sdot) represents Eulerrsquos gamma function
3 Fractional Cattaneo-Vernotte Equation
In previous studies of the fractional Cattaneo-Vernotte equa-tion the authors did not consider the physical dimensionalityof the solutions The authors of the work [42] proposed asystematic way to construct fractional differential equationsfor the physical systems To keep the dimensionality of
the fractional differential equations a new parameter 120590 wasintroduced in the following way
120597
120597119909997888rarr
1
1205901minus120593
119909
sdot120597120593
120597119909120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(15)
120597
120597119905997888rarr
1
1205901minus120593
119905
sdot120597120593
120597119905120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(16)
where 120593 is an arbitrary parameter which represents theorder of the derivative 120590
119909has dimension of length and 120590
119905
has the dimension of time These new parameters maintainthe dimensionality of the equation invariant and character-izes the fractional space or fractional temporal structures(components that show an intermediate behavior between aconservative system and dissipative one) [42] when120593 = 1 theexpressions (15) and (16) reduce to the ordinary derivative Inthe following we will apply this idea to generalize the case ofthe fractional Cattaneo-Vernotte equation
In this work we consider generalized Cattaneo-Vernotteequation in the 119909 direction of the form [32ndash35]
nabla2
119879 minus1
119863 minus
120591
119863 = 0 (17)
where 120591 is a characteristic relaxation time constant (orthe non-Fourier character of the material) and 119863 is thegeneralized thermal diffusivity (17) is a hyperbolic diffusionequation when the parameter 120591 = 0 (17) recovers a parabolicform in this limit one has to replace Cattaneo-Vernotteequation by Fourierrsquos heat transfer equation
Considering the CFD (2) and (15) and (16) the fractionalrepresentation of (17) is
1
1205902(1minus120593)
119909
119862
01198632120593
119909119879 (119909 119905) minus
1
119863sdot
1
1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905) minus
120591
119863
sdot1
1205902(1minus120593)
119905
119862
01198632120593
119905119879 (119909 119905) = 0
(18)
The order of the derivative considered is 120593 isin (0 2] for thefractional Cattaneo-Vernotte equation in space-time domain
31 Fractional Space Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the space derivative is fractionalequation (15) and the time derivative is ordinary the spatialfractional equation is
119862
01198632120593
119909119879 (119909 119905) minus
1
1198631205902(1minus120593)
119909sdot120597119879 (119909 119905)
120597119905minus
120591
1198631205902(1minus120593)
119909
sdot1205972119879 (119909 119905)
1205971199052= 0
(19)
Suppose the solution
119879 (119909 119905) = 1198790sdot exp (119894120596119905) 119906 (119909) (20)
4 Mathematical Problems in Engineering
substituting (20) into (19) we obtain
1198892120593119906 (119909)
1198891199092120593+ (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909119906 (119909) = 0 (21)
where
1198962
119909= (
120591
1198631205962
minus 1198941
119863120596) (22)
is the dispersion relation in the 119909 direction and
2
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909= 1198962
1199091205902(1minus120593)
119909 (23)
is the fractional dispersion relation from the fractionaldispersion relation (23) we can expect the fractional wavenumber in the 119909 direction to have real and imaginary parts120575119909and 120573
119909 respectively Let us write
119909= 120575119909minus 119894120573119909 (24)
substituting (24) into (23) we have
(120575119909minus 119894120573119909)2
= 1205752
119909minus 2119894120575119909120573119909minus 1205732
119909 (25)
where
1205752
119909minus 2119894120575119909120573119909minus 1205732
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909 (26)
solving for 120573119909we obtain
120573119909=
120596
119863sdot
1
2120575119909
1205902(1minus120593)
119909 (27)
and for 120575119909
120575119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909 (28)
substituting (28) into (27) we have
120573119909=
1
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909 (29)
Now the fractional wave number is 119909= 120575119909minus 119894120573119909 where
120575119909and 120573
119909are given by (28) and (29) respectively
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909
(30)
equation (30) describes the real and imaginary part of thefractional wave number in terms of the frequency 120596 therelaxation time 120591 and the generalized thermal diffusivity 119863in presence of fractional space components 120590
119909
Considering (23) (21) gives
1198892120593119906 (119909)
1198891199092120593+ 2
119909119906 (119909) = 0 (31)
the solution of (31) can be obtained applying direct andinverse Laplace transform [47] and the solution of the aboveequation is given by
119906 (119909) = 11986421205931
(minus2
1199091199092120593
) (32)
where 11986421205931
(minus2
1199091199092120593) is the Mittag-Leffler function
Therefore the general solution of (21) is given by
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21205931(minus2
1199091199092120593
) (33)
Next we will analyze the case when 120593 takes differentvalues
Case 1 When 120593 = 2 we have
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus1
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus1
119909
(34)
and equation (34) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
41(minus2
1199091199094
) (35)
where 11986441
is given by (12) and solution (35) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905)
sdot [cos(minus12
119909119909) + cosh (minus
12
119909119909)]
(36)
Case 2 When 120593 = 32 we have
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus12
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus12
119909
(37)
and equation (37) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
31(minus2
1199091199093
) (38)
Mathematical Problems in Engineering 5
where 11986431
is given by (11) and solution (38) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
(39)
Case 3 When 120593 = 1 we have 119909= 119896119909
119896119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
(40)
and equation (40) represents the classical wave number 119896119909
From (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21(minus2
1199091199092
) (41)
where
119879 (119909 119905) = R [1198790sdot exp (119894120596119905) sdot exp (minus119894
119909119909)] (42)
and in (42) 119909
= 119896119909 R indicates the real part and 119896
119909=
120575119909minus 119894120573119909is wave number (40) substituting 119896
119909in (42) we have
119879 (119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)] (43)
Equation (43) represents the classical case for the spaceCattaneo-Vernotte equation The first exponential exp(119894(120596119905 minus120575119909119909)) gives the usual plane-wave variation of the thermal
field with position 119909 and time 119905 The second exponentialexp(minus120573
119909119909) gives and exponential decay in the amplitude of
the thermal wave
Case 4 When 120593 = 12 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
11(minus2
119909119909) (44)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059012
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059012
119909
(45)
and equation (45) represents the fractional wave number inpresence of fractional space components 120590
119909
The solution for (44) is
119879 (119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909))
sdot exp (minus120591
1198631205962
120590119909119909)]
(46)
whereR indicates the real part
Case 5 When 120593 = 14 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (47)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059034
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059034
119909
(48)
and equation (48) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (49)
where 119864121
is given by (8) and solution (49) is
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909)
sdot [1 minus erfc (minus2
11990912
)]
(50)
erfc(120572) denotes the error function defined in (8) Equation(50) represents the space evolution of the temperature andthe amplitude exhibits an algebraic decay for 119909 rarr infin
For this case there exists a physical relation between theauxiliary parameter 120590
119909and the wave number 119896
119909given by the
order 120593 of the fractional differential equation
120593 = 119896119909120590119909=
120590119909
120582 0 lt 120590
119909le 120582 (51)
where 120582 is the wavelength We can use this relation in orderto write (33) as
119879 ( 119905) = 1198790sdot exp (119894120596119905) sdot 119864
2120593(minus1205932(1minus120593)
2120593
) (52)
where = 119909120582 is a dimensionless parameter Figures 1(a) and1(b) show the simulation of (52) for 120593 values 07 lt 120593 le 1 and17 lt 120593 le 2 respectively
Table 1 shows the different solutions of (52) The order ofthe fractional differential equation is 120593 = 2 120593 = 32 120593 = 1120593 = 12 and 120593 = 14
32 Fractional Time Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the time derivative is fractionalequation (16) and the space derivative is ordinary the tempo-ral fractional equation is
119862
01198632120593
119905119879 (119909 119905) +
1
120591sdot 1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905)
minus119863
1205911205902(1minus120593)
119905
1205972119879 (119909 119905)
1205971199092= 0
(53)
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Some common Mittag-Leffler functions are described in[48]
119864121
(plusmn120572) = exp (1205722
) [1 plusmn erfc (120572)] (8)
11986411
(plusmn120572) = exp (plusmn120572) (9)
11986421
(minus1205722
) = cos (120572) (10)
11986431
(120572) =1
2[exp (120572
13
)
+ 2 exp(minus(1
2) 12057213
) cos(radic3
212057213
)]
(11)
11986441
(120572) =1
2(cos (12057214) + cosh (120572
14
)) (12)
The Mittag-Leffler function 1198641199042119903
is defined by Miller in[49]
1198641199042119903
(119911) =1199112120581(1minus119903)
119904
119904minus1
sum119895=0
1205721minus(1199042+119903)
119895(exp (120572
1198951199112120581
))
sdot (1205721199042
119895+ erfc (12057212
119895119911120581
))
minus 119911minus2119899
2119899minus1
sum119896=0
119911119896
Γ (1199041198962 + 120583)
(13)
where 120581 = 1119904 119903 = 119899119904 + 120583 119899 = 0 1 2 3 120583 = 1 2 3 For the calculation of generalized Mittag-Leffler functions atarbitrary precision see [50 51]
The Riesz fractional derivative for (120593 gt 0) is [43ndash46]
119877
119863120593
119909119891 (119909) = minus
1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]
119899 minus 1 lt 120593 le 119899
(14)
where 119889120593119889119909120593 = 119877119863120593
119909is a Riesz fractional derivative with
respect to 119909 120593 isin 119877 is the order of the fractional derivative119899 = 1 2 isin 119873 and Γ(sdot) represents Eulerrsquos gamma function
3 Fractional Cattaneo-Vernotte Equation
In previous studies of the fractional Cattaneo-Vernotte equa-tion the authors did not consider the physical dimensionalityof the solutions The authors of the work [42] proposed asystematic way to construct fractional differential equationsfor the physical systems To keep the dimensionality of
the fractional differential equations a new parameter 120590 wasintroduced in the following way
120597
120597119909997888rarr
1
1205901minus120593
119909
sdot120597120593
120597119909120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(15)
120597
120597119905997888rarr
1
1205901minus120593
119905
sdot120597120593
120597119905120593
119899 minus 1 lt 120593 le 119899 119899 isin 119873 = 1 2 3
(16)
where 120593 is an arbitrary parameter which represents theorder of the derivative 120590
119909has dimension of length and 120590
119905
has the dimension of time These new parameters maintainthe dimensionality of the equation invariant and character-izes the fractional space or fractional temporal structures(components that show an intermediate behavior between aconservative system and dissipative one) [42] when120593 = 1 theexpressions (15) and (16) reduce to the ordinary derivative Inthe following we will apply this idea to generalize the case ofthe fractional Cattaneo-Vernotte equation
In this work we consider generalized Cattaneo-Vernotteequation in the 119909 direction of the form [32ndash35]
nabla2
119879 minus1
119863 minus
120591
119863 = 0 (17)
where 120591 is a characteristic relaxation time constant (orthe non-Fourier character of the material) and 119863 is thegeneralized thermal diffusivity (17) is a hyperbolic diffusionequation when the parameter 120591 = 0 (17) recovers a parabolicform in this limit one has to replace Cattaneo-Vernotteequation by Fourierrsquos heat transfer equation
Considering the CFD (2) and (15) and (16) the fractionalrepresentation of (17) is
1
1205902(1minus120593)
119909
119862
01198632120593
119909119879 (119909 119905) minus
1
119863sdot
1
1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905) minus
120591
119863
sdot1
1205902(1minus120593)
119905
119862
01198632120593
119905119879 (119909 119905) = 0
(18)
The order of the derivative considered is 120593 isin (0 2] for thefractional Cattaneo-Vernotte equation in space-time domain
31 Fractional Space Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the space derivative is fractionalequation (15) and the time derivative is ordinary the spatialfractional equation is
119862
01198632120593
119909119879 (119909 119905) minus
1
1198631205902(1minus120593)
119909sdot120597119879 (119909 119905)
120597119905minus
120591
1198631205902(1minus120593)
119909
sdot1205972119879 (119909 119905)
1205971199052= 0
(19)
Suppose the solution
119879 (119909 119905) = 1198790sdot exp (119894120596119905) 119906 (119909) (20)
4 Mathematical Problems in Engineering
substituting (20) into (19) we obtain
1198892120593119906 (119909)
1198891199092120593+ (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909119906 (119909) = 0 (21)
where
1198962
119909= (
120591
1198631205962
minus 1198941
119863120596) (22)
is the dispersion relation in the 119909 direction and
2
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909= 1198962
1199091205902(1minus120593)
119909 (23)
is the fractional dispersion relation from the fractionaldispersion relation (23) we can expect the fractional wavenumber in the 119909 direction to have real and imaginary parts120575119909and 120573
119909 respectively Let us write
119909= 120575119909minus 119894120573119909 (24)
substituting (24) into (23) we have
(120575119909minus 119894120573119909)2
= 1205752
119909minus 2119894120575119909120573119909minus 1205732
119909 (25)
where
1205752
119909minus 2119894120575119909120573119909minus 1205732
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909 (26)
solving for 120573119909we obtain
120573119909=
120596
119863sdot
1
2120575119909
1205902(1minus120593)
119909 (27)
and for 120575119909
120575119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909 (28)
substituting (28) into (27) we have
120573119909=
1
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909 (29)
Now the fractional wave number is 119909= 120575119909minus 119894120573119909 where
120575119909and 120573
119909are given by (28) and (29) respectively
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909
(30)
equation (30) describes the real and imaginary part of thefractional wave number in terms of the frequency 120596 therelaxation time 120591 and the generalized thermal diffusivity 119863in presence of fractional space components 120590
119909
Considering (23) (21) gives
1198892120593119906 (119909)
1198891199092120593+ 2
119909119906 (119909) = 0 (31)
the solution of (31) can be obtained applying direct andinverse Laplace transform [47] and the solution of the aboveequation is given by
119906 (119909) = 11986421205931
(minus2
1199091199092120593
) (32)
where 11986421205931
(minus2
1199091199092120593) is the Mittag-Leffler function
Therefore the general solution of (21) is given by
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21205931(minus2
1199091199092120593
) (33)
Next we will analyze the case when 120593 takes differentvalues
Case 1 When 120593 = 2 we have
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus1
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus1
119909
(34)
and equation (34) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
41(minus2
1199091199094
) (35)
where 11986441
is given by (12) and solution (35) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905)
sdot [cos(minus12
119909119909) + cosh (minus
12
119909119909)]
(36)
Case 2 When 120593 = 32 we have
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus12
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus12
119909
(37)
and equation (37) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
31(minus2
1199091199093
) (38)
Mathematical Problems in Engineering 5
where 11986431
is given by (11) and solution (38) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
(39)
Case 3 When 120593 = 1 we have 119909= 119896119909
119896119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
(40)
and equation (40) represents the classical wave number 119896119909
From (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21(minus2
1199091199092
) (41)
where
119879 (119909 119905) = R [1198790sdot exp (119894120596119905) sdot exp (minus119894
119909119909)] (42)
and in (42) 119909
= 119896119909 R indicates the real part and 119896
119909=
120575119909minus 119894120573119909is wave number (40) substituting 119896
119909in (42) we have
119879 (119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)] (43)
Equation (43) represents the classical case for the spaceCattaneo-Vernotte equation The first exponential exp(119894(120596119905 minus120575119909119909)) gives the usual plane-wave variation of the thermal
field with position 119909 and time 119905 The second exponentialexp(minus120573
119909119909) gives and exponential decay in the amplitude of
the thermal wave
Case 4 When 120593 = 12 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
11(minus2
119909119909) (44)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059012
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059012
119909
(45)
and equation (45) represents the fractional wave number inpresence of fractional space components 120590
119909
The solution for (44) is
119879 (119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909))
sdot exp (minus120591
1198631205962
120590119909119909)]
(46)
whereR indicates the real part
Case 5 When 120593 = 14 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (47)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059034
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059034
119909
(48)
and equation (48) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (49)
where 119864121
is given by (8) and solution (49) is
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909)
sdot [1 minus erfc (minus2
11990912
)]
(50)
erfc(120572) denotes the error function defined in (8) Equation(50) represents the space evolution of the temperature andthe amplitude exhibits an algebraic decay for 119909 rarr infin
For this case there exists a physical relation between theauxiliary parameter 120590
119909and the wave number 119896
119909given by the
order 120593 of the fractional differential equation
120593 = 119896119909120590119909=
120590119909
120582 0 lt 120590
119909le 120582 (51)
where 120582 is the wavelength We can use this relation in orderto write (33) as
119879 ( 119905) = 1198790sdot exp (119894120596119905) sdot 119864
2120593(minus1205932(1minus120593)
2120593
) (52)
where = 119909120582 is a dimensionless parameter Figures 1(a) and1(b) show the simulation of (52) for 120593 values 07 lt 120593 le 1 and17 lt 120593 le 2 respectively
Table 1 shows the different solutions of (52) The order ofthe fractional differential equation is 120593 = 2 120593 = 32 120593 = 1120593 = 12 and 120593 = 14
32 Fractional Time Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the time derivative is fractionalequation (16) and the space derivative is ordinary the tempo-ral fractional equation is
119862
01198632120593
119905119879 (119909 119905) +
1
120591sdot 1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905)
minus119863
1205911205902(1minus120593)
119905
1205972119879 (119909 119905)
1205971199092= 0
(53)
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
substituting (20) into (19) we obtain
1198892120593119906 (119909)
1198891199092120593+ (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909119906 (119909) = 0 (21)
where
1198962
119909= (
120591
1198631205962
minus 1198941
119863120596) (22)
is the dispersion relation in the 119909 direction and
2
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909= 1198962
1199091205902(1minus120593)
119909 (23)
is the fractional dispersion relation from the fractionaldispersion relation (23) we can expect the fractional wavenumber in the 119909 direction to have real and imaginary parts120575119909and 120573
119909 respectively Let us write
119909= 120575119909minus 119894120573119909 (24)
substituting (24) into (23) we have
(120575119909minus 119894120573119909)2
= 1205752
119909minus 2119894120575119909120573119909minus 1205732
119909 (25)
where
1205752
119909minus 2119894120575119909120573119909minus 1205732
119909= (
120591
1198631205962
minus 1198941
119863120596)1205902(1minus120593)
119909 (26)
solving for 120573119909we obtain
120573119909=
120596
119863sdot
1
2120575119909
1205902(1minus120593)
119909 (27)
and for 120575119909
120575119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909 (28)
substituting (28) into (27) we have
120573119909=
1
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909 (29)
Now the fractional wave number is 119909= 120575119909minus 119894120573119909 where
120575119909and 120573
119909are given by (28) and (29) respectively
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
1205901minus120593
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
1205901minus120593
119909
(30)
equation (30) describes the real and imaginary part of thefractional wave number in terms of the frequency 120596 therelaxation time 120591 and the generalized thermal diffusivity 119863in presence of fractional space components 120590
119909
Considering (23) (21) gives
1198892120593119906 (119909)
1198891199092120593+ 2
119909119906 (119909) = 0 (31)
the solution of (31) can be obtained applying direct andinverse Laplace transform [47] and the solution of the aboveequation is given by
119906 (119909) = 11986421205931
(minus2
1199091199092120593
) (32)
where 11986421205931
(minus2
1199091199092120593) is the Mittag-Leffler function
Therefore the general solution of (21) is given by
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21205931(minus2
1199091199092120593
) (33)
Next we will analyze the case when 120593 takes differentvalues
Case 1 When 120593 = 2 we have
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus1
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus1
119909
(34)
and equation (34) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
41(minus2
1199091199094
) (35)
where 11986441
is given by (12) and solution (35) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905)
sdot [cos(minus12
119909119909) + cosh (minus
12
119909119909)]
(36)
Case 2 When 120593 = 32 we have
119909
= 120596radic120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
120590minus12
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
120590minus12
119909
(37)
and equation (37) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
31(minus2
1199091199093
) (38)
Mathematical Problems in Engineering 5
where 11986431
is given by (11) and solution (38) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
(39)
Case 3 When 120593 = 1 we have 119909= 119896119909
119896119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
(40)
and equation (40) represents the classical wave number 119896119909
From (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21(minus2
1199091199092
) (41)
where
119879 (119909 119905) = R [1198790sdot exp (119894120596119905) sdot exp (minus119894
119909119909)] (42)
and in (42) 119909
= 119896119909 R indicates the real part and 119896
119909=
120575119909minus 119894120573119909is wave number (40) substituting 119896
119909in (42) we have
119879 (119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)] (43)
Equation (43) represents the classical case for the spaceCattaneo-Vernotte equation The first exponential exp(119894(120596119905 minus120575119909119909)) gives the usual plane-wave variation of the thermal
field with position 119909 and time 119905 The second exponentialexp(minus120573
119909119909) gives and exponential decay in the amplitude of
the thermal wave
Case 4 When 120593 = 12 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
11(minus2
119909119909) (44)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059012
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059012
119909
(45)
and equation (45) represents the fractional wave number inpresence of fractional space components 120590
119909
The solution for (44) is
119879 (119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909))
sdot exp (minus120591
1198631205962
120590119909119909)]
(46)
whereR indicates the real part
Case 5 When 120593 = 14 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (47)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059034
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059034
119909
(48)
and equation (48) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (49)
where 119864121
is given by (8) and solution (49) is
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909)
sdot [1 minus erfc (minus2
11990912
)]
(50)
erfc(120572) denotes the error function defined in (8) Equation(50) represents the space evolution of the temperature andthe amplitude exhibits an algebraic decay for 119909 rarr infin
For this case there exists a physical relation between theauxiliary parameter 120590
119909and the wave number 119896
119909given by the
order 120593 of the fractional differential equation
120593 = 119896119909120590119909=
120590119909
120582 0 lt 120590
119909le 120582 (51)
where 120582 is the wavelength We can use this relation in orderto write (33) as
119879 ( 119905) = 1198790sdot exp (119894120596119905) sdot 119864
2120593(minus1205932(1minus120593)
2120593
) (52)
where = 119909120582 is a dimensionless parameter Figures 1(a) and1(b) show the simulation of (52) for 120593 values 07 lt 120593 le 1 and17 lt 120593 le 2 respectively
Table 1 shows the different solutions of (52) The order ofthe fractional differential equation is 120593 = 2 120593 = 32 120593 = 1120593 = 12 and 120593 = 14
32 Fractional Time Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the time derivative is fractionalequation (16) and the space derivative is ordinary the tempo-ral fractional equation is
119862
01198632120593
119905119879 (119909 119905) +
1
120591sdot 1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905)
minus119863
1205911205902(1minus120593)
119905
1205972119879 (119909 119905)
1205971199092= 0
(53)
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
where 11986431
is given by (11) and solution (38) is
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
(39)
Case 3 When 120593 = 1 we have 119909= 119896119909
119896119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
(40)
and equation (40) represents the classical wave number 119896119909
From (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
21(minus2
1199091199092
) (41)
where
119879 (119909 119905) = R [1198790sdot exp (119894120596119905) sdot exp (minus119894
119909119909)] (42)
and in (42) 119909
= 119896119909 R indicates the real part and 119896
119909=
120575119909minus 119894120573119909is wave number (40) substituting 119896
119909in (42) we have
119879 (119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)] (43)
Equation (43) represents the classical case for the spaceCattaneo-Vernotte equation The first exponential exp(119894(120596119905 minus120575119909119909)) gives the usual plane-wave variation of the thermal
field with position 119909 and time 119905 The second exponentialexp(minus120573
119909119909) gives and exponential decay in the amplitude of
the thermal wave
Case 4 When 120593 = 12 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
11(minus2
119909119909) (44)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059012
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059012
119909
(45)
and equation (45) represents the fractional wave number inpresence of fractional space components 120590
119909
The solution for (44) is
119879 (119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909))
sdot exp (minus120591
1198631205962
120590119909119909)]
(46)
whereR indicates the real part
Case 5 When 120593 = 14 from (33) we have
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (47)
where 119909is
119909= 120596radic
120591
119863[1
2plusmn
1
2radic1 +
1
12059121205962]
12
12059034
119909
minus 1198941
2119863radic120591119863 [12 plusmn (12)radic1 + 112059121205962]12
12059034
119909
(48)
and equation (48) represents the fractional wave number inpresence of fractional space components 120590
119909
In this case (33) is written as follows
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot 119864
121(minus2
11990911990912
) (49)
where 119864121
is given by (8) and solution (49) is
119879 (119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909)
sdot [1 minus erfc (minus2
11990912
)]
(50)
erfc(120572) denotes the error function defined in (8) Equation(50) represents the space evolution of the temperature andthe amplitude exhibits an algebraic decay for 119909 rarr infin
For this case there exists a physical relation between theauxiliary parameter 120590
119909and the wave number 119896
119909given by the
order 120593 of the fractional differential equation
120593 = 119896119909120590119909=
120590119909
120582 0 lt 120590
119909le 120582 (51)
where 120582 is the wavelength We can use this relation in orderto write (33) as
119879 ( 119905) = 1198790sdot exp (119894120596119905) sdot 119864
2120593(minus1205932(1minus120593)
2120593
) (52)
where = 119909120582 is a dimensionless parameter Figures 1(a) and1(b) show the simulation of (52) for 120593 values 07 lt 120593 le 1 and17 lt 120593 le 2 respectively
Table 1 shows the different solutions of (52) The order ofthe fractional differential equation is 120593 = 2 120593 = 32 120593 = 1120593 = 12 and 120593 = 14
32 Fractional Time Cattaneo-Vernotte Equation Consider-ing (18) and assuming that the time derivative is fractionalequation (16) and the space derivative is ordinary the tempo-ral fractional equation is
119862
01198632120593
119905119879 (119909 119905) +
1
120591sdot 1205901minus120593
119905
119862
0119863120593
119905119879 (119909 119905)
minus119863
1205911205902(1minus120593)
119905
1205972119879 (119909 119905)
1205971199092= 0
(53)
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9
Space diffusion of heat
0
02
04
06
08
1T(xt)
x
120593 = 1
120593 = 09
120593 = 08
120593 = 07
(a)
Space diffusion of heat
T(xt)
minus10
minus5
0
5
10
1 2 3 4 5 6 7 8 90x
120593 = 17
120593 = 18
120593 = 19
120593 = 2
(b)
Figure 1 Simulation of (52) for 07 lt 120593 le 2
Table 1 Solutions for fractional space Cattaneo-Vernotte equation(52) for different values of 120593 R indicates the real part and erfc(120572)denotes the error function defined in (8)
120593 Solution
2 119879(119909 119905) =1198790
2sdot exp (119894120596119905) sdot [cos (minus
12
119909119909) + cosh (minus
12
119909119909)]
32
119879 (119909 119905) =1198790
2sdot exp (119894120596119905) sdot [
[
exp(minus23
119909119909)
+ 2 exp(23
119909
2119909) sdot cos(minus
radic3
223
119909119909)]
]
1 119879(119909 119905) = R [1198790sdot exp (119894 (120596119905 minus 120575
119909119909)) sdot exp (minus120573
119909119909)]
12 119879(119909 119905) = R [1198790sdot exp(119894120596 (119905 +
1
119863120590119909119909)) sdot exp(minus
120591
1198631205962120590119909119909)]
14 119879(119909 119905) = 1198790sdot exp (119894120596119905) sdot exp (
4
119909) sdot [1 minus erfc (minus2
11990912)]
suppose the solution
119879 (119909 119905) = 1198790exp (119894
119909119909) 119906 (119905) (54)
where 119909is the wave number in the 119909 direction Substituting
(54) into (53) we obtain
1198892120593119906 (119905)
1198891199052120593+
1
1205911205901minus120593
119905
119889120593119906 (119905)
119889119905120593+
119863
1205912
1199091205902(1minus120593)
119905119906 (119905) = 0 (55)
the solution of (55) can be obtained applying direct andinverse Laplace transform [47] Taking solution (54) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205911205901minus120593
119905119905120593
)
sdot 11986421205931
(minus [119863
1205912
119909minus
1
41205912] 1205902(1minus120593)
1199051199052120593
)
(56)
and solution (56) represents a temporal nonlocal thermalequation interpreted as an existence of memory effectswhich correspond to intrinsic dissipation characterizedby the exponent of the fractional derivative 120593 in thesystem
For underdamped case we have ((119863120591)2
119909minus 141205912) = 0
1205960= radic119863120591
119909is the undamped natural frequency expressed
in radians per second and 120572 = radic12120591 is the damping factorexpressed inmeters per second Next we will analyze the casewhen 120593 takes different values
Case 1 When 120593 = 2 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
1
2120591120590119905
1199052
)
sdot 11986441
(minus [119863
1205912
119909minus
1
41205912] (
1
1205902119905
) 1199054
)
(57)
where11986421
is given by (10) and11986441
by (12) in this case solution(57) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot cos(radic
1
2120590119905120591119905)
sdot cos [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
+ cosh [minus(119863
1205912
119909minus
1
41205912)(
1
1205902119905
)]
14
119905
(58)
Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering 7
Case 2 When 120593 = 32 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
212059112059012
119905
11990532
)
sdot 11986431
(minus [119863
1205912
119909minus
1
41205912] (
1
120590119905
) 1199053
)
(59)
where 119864321
is given by (13) and 11986431
by (11) in this casesolution (59) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909)
sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot (12057232
119895+ erfc(120572
12
119895(minus
11990532
212059112059012
119905
)
120581
))
minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot[[
[
exp(minus[119863
1205912
119909minus
1
41205912]23
(1
12059023
119905
) 119905)
+ 2 exp([((119863120591)
2
119909minus 141205912) (1120590
119905)]23
2119905)
sdot cos(minusradic3
2[(
119863
1205912
119909minus
1
41205912)(
1
120590119905
)]
23
119905)]]
]
(60)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
Case 3 When 120593 = 1 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(minus
119905
2120591)
sdot cos(radic119863
1205912
119909minus
1
41205912119905)
(61)
and (61) represents the classic case and the well-known resultfrom (61) we see that there is a relation between120593 and120590
119905given
by
120593 = (119863
1205912
119909minus
1
41205912)12
120590119905
0 lt 120590119905le
1
((119863120591) 2
119909minus 141205832)
12
(62)
Then solution (56) for the underdamped case 120572 lt 1205960
takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic(119863120591) 2
119909minus 141205912
1205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
1199052120593
)
(63)
where = ((119863120591)2
119909minus141205912)
12
119905 is a dimensionless parameterDue to the condition 120572 lt 120596
0we can choose an example
1
2120591radic(119863120591) 2
119909minus 141205912
= 3
0 le1
2120591radic(119863120591) 2
119909minus 141205912
lt infin
(64)
So solution (56) takes its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus31205931minus120593
120593
)
sdot 11986421205931
(minus1205932(1minus120593)
2120593
)
(65)
Case 4 When 120593 = 12 from (56) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [minus (119863
1205912
119909minus
1
41205912)120590119905119905]
(66)
and erfc(120572) denotes the error function defined in (8) Equa-tion (66) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 2(a) and 2(b)
Table 2 shows the different solutions of (65) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
In the overdamped case 120572 gt 1205960or 120578 gt 2
119909radic120598120583 the
solution of (56) has the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1205901minus120593
119905
2120591119905120593
)
sdot 1198641205931
(minus[1
41205912minus
119863
1205912
119909]12
1205901minus120593
119905119905120593
)
(67)
Next we will analyze the case when 120593 takes differentvalues
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table2Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
underdam
pedcase
(65)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=
1198790 2sdotexp(119894119896119909119909)sdotcos(
radic1
2120590119905120591119905)
sdotcos[
minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905+cosh
[minus(119863 120591
1198962 119909minus
1 41205912
)(
1 1205902 119905
)]
14
119905
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot[ [ [
exp(minus[119863 120591
1198962 119909minus
1 41205912
]23
(1
12059023
119905
)119905)
+2exp(
[((119863
120591)1198962 119909minus14
1205912
)(1120590119905)]23
2119905)
sdotcos(
minusradic3 2[(119863 120591
1198962 119909minus
1 41205912
)(
1 120590119905
)]
23
119905)] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
119905 2120591)sdotcos(
radic119863 120591
1198962 119909minus
1 41205912
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905)sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[minus(119863 120591
1198962 119909minus
1 41205912
)120590119905119905]
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Temporal diffusion of heat
minus350
minus300
minus250
minus200
minus150
minus100
minus50
0
50
100
T
1 2 3 4 5 6 7 8 90t
120593 = 2
(a)
Temporal diffusion of heat
1 2 3 4 5 6 7 8 90t
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
T
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 2 Simulation of (65) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 1 When 120593 = 2 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
21(minus
120590minus1119905
21205911199052
)
sdot 11986421
(minus[1
41205912minus
119863
1205912
119909]12
120590minus1
1199051199052
)
(68)
where 11986421
is given by (10) in this case solution (69) is
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot cos(radic
1
2120591120590119905
)
sdot cos(radic(1
41205912minus
119863
1205911198962119909)12
(1
120590119905
)119905)
(69)
Case 2 When 120593 = 32 we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
321(minus
1
2120591120590minus12
119905
11990532
)
sdot 119864321
(minus[1
41205912minus
119863
1205912
119909]12
120590minus12
11990511990532
)
(70)
where 119864321
is given by (13) in this case solution (70) is
119879 (119909 119905) =1198790
2sdot exp (119894
119909119909) sdot [
[
(minus11990532212059112059012119905
)2120581(1minus119903)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895(exp(120572
119895(minus
11990532
212059112059012
119905
)
2120581
)) sdot (12057232
119895
+ erfc(12057212
119895(minus
11990532
212059112059012
119905
)
120581
)) minus (minus11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus11990532212059112059012119905
)119896
Γ (31198962 + 120583)]
]
sdot
[[[[
[
(minus(141205912 minus (119863120591)
2
119909)12
120590minus12119905
11990532)
3
sdot
2
sum119895=0
1205721minus(32+119903)
119895
sdot (exp(120572119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
))2119896
)
sdot (12057232
119895
+ erfc(12057212
119895(minus(
1
41205912minus
119863
1205912
119909)120590minus12
11990511990532
)119896
))
minus (minus(1
41205912minus
119863
1205912
119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum119896=0
(minus (141205912 minus (119863120591) 2
119909)12
120590minus12119905
11990532)
119896
Γ (31198962 + 120583)
]]]]
]
(71)
where 120581 = 13 119903 = 3119899 + 120583 119899 = 0 1 2 3 120583 = 1 2 3
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Temporal diffusion of heat
minus6
minus4
minus2
0
2
4
6
8
10T
1 2 3 4 5 6 7 8 90
120593 = 2
t
(a)
Temporal diffusion of heat
minus04
minus02
0
02
04
06
08
1
T
1 2 3 4 5 6 7 8 90t
120593 = 05
120593 = 1
120593 = 15
(b)
Figure 3 Simulation of (76) for 05 lt 120593 le 2 The crossover time occurs at = 10 independent of 120593 but the crossover time increases withdecreasing 120593
Case 3 When 120593 = 1 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
(72)
and solution (72) represents the classic caseTaking into account the relation between 120593 and 120590
119905is
120593 = (1
41205912minus
119863
1205912
119909)12
120590119905
0 lt 120590119905le
1
(141205912 minus (119863120591) 2
119909)12
(73)
Solution (67) takes the form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909)
sdot 1198641205931
(minus1
2120591radic141205912 minus (119863120591) 2
119909
1205931minus120593
119905120593
)
sdot 1198641205931
(minus1205931minus120593
119905120593
)
(74)
where = (141205912minus(119863120591)2
119909)12
119905 is a dimensionless parameter
Due to the condition 120572 gt 1205960we can choose an example
1
2120591radic141205912 minus (119863120591) 2
119909
=1
2
1 lt1
2120591radic141205912 minus (119863120591) 2
119909
lt infin
(75)
Then solution (67) can be written in its final form
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot 119864
1205931(minus
1
21205931minus120593
120593
)
sdot 1198641205931
(minus1205931minus120593
120593
)
(76)
Case 4 When 120593 = 12 from (67) we have
119879 (119909 119905) = 1198790sdot exp (119894
119909119909) sdot exp(
120590119905
41205912119905)
sdot [1 minus erfc(minus12059012119905
212059111990512
)]
sdot exp [(1
41205912minus
119863
1205912
119909)120590119905119905]
sdot [1 minus erfc( 1
41205912minus
119863
1205912
119909)12
12059012
11990511990512
]
(77)
and erfc(120572) denotes the error function defined in (8) Equa-tion (77) represents the time evolution of the temperature andthe amplitude exhibits an algebraic decay for 119905 rarr infin Plots fordifferent values of 120593 are shown in Figures 3(a) and 3(b)
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 3 shows the different solutions of (76) The orderof the fractional differential equation is 120593 = 2 120593 = 32120593 = 1 and 120593 = 12 The change of the order of thederivative describes the crossover from ballistic transport tothe diffusion behavior
33 Fractional Space-Time Cattaneo-Vernotte Equation Nowconsidering (18) and assuming that the space and time deriva-tive are fractional the order of the time-space fractionaldifferential equations is 120593 isin (0 1] for this example weconsider for 119905 gt 0 119909 = 0 and 119909 = 119871 with Dirichlet condition119879(0 119905) = 119879(119871 119905) = 0 and initial conditions 0 lt 119909 lt 119871 119905 = 0
119879(119905 0) = 1198790gt 0 and 0 lt 119909 lt 119871 119905 = 0 (120597119879120597119905)|
119905=0= 0
Applying the Fourier method of the variable separationthe full solution of (18) is
119879 (119909 119905) =1198790
120587120591sdot 119864120593(minus
1
21205911205901minus120593
119905119905120593
)
sdot
infin
sum119898=1
1
2119898 minus 1I [119864119894120593((2119898 minus 1) 120587
1198711205901minus120593
119909119909120593
)]
sdot [minus1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(minus1199112119898minus1
1205901minus120593
119905119905120593
)
+1 + 2120591119911
2119898minus1
1199112119898minus1
sdot 119864120593(1199112119898minus1
1205901minus120593
119905119905120593
)]
(78)
where I indicates the imaginary part and 119911119898
=
radic(1198712 minus 4119898120587120591119863)2119871120591 In the case when 120593 = 1 we havethe classical solution
119879 (119909 119905) =1198790
120587120591sdot exp(minus
1
2120591119905) sdot
infin
sum119898=1
1
2119898 minus 1
sdot sin((2119898 minus 1) 120587
119871119909) sdot [
minus1 + 21205911199112119898minus1
1199112119898minus1
sdot exp (minus1199112119898minus1
119905) +1 + 2120591119911
2119898minus1
1199112119898minus1
sdot exp (1199112119898minus1
119905)]
(79)
Now considering (18) with the Riesz space fractionalderivative the order of the space fractional differential equa-tions is 120593 isin (0 1] for this example we consider for 0 lt 119905 le 1198790 lt 119909 lt 119871 and Dirichlet condition 119879(0 119905) = 119879(119871 119905) = 0 andinitial conditions 119879(119909 0) = ℎ(119909)
1205972119879 (119909 119905)
1205971199052+
1
1205911205901minus120593
119905
120597119879 (119909 119905)
120597119905= minus
119863
1205912
1199091205902(1minus120593)
119905
sdot [1
2 cos (1205871205932)
sdot [1
Γ (119899 minus 120593)(
120597
120597119909)
119899
intminus
infin119909
119891 (120594 119905) 119889120594
(119909 minus 120594)120593+1minus119899
+(minus1)119899
Γ (119899 minus 120593)(
120597
120597119909)
119899
intinfin
119909
119891 (120594 119905) 119889120594
(120594 minus 119909)120593+1minus119899
]]
(80)
and the solution is given by
119879 (119909 119905) =
infin
sum119899=1
119860119899(119905) sin(
119899120587119909
119871) (81)
which satisfies the boundary condition substituting thiscondition into (81) we obtain
infin
sum119899=1
[1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899
+ [119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899] sin(
119899120587119909
119871) = 0
(82)
and the problem for 119860119899becomes
1198892119860119899
1198891199052+
1
1205911205901minus120593
119905
119889
119889119905119860119899+ [
119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
]119860119899= 0 (83)
which has the general solution
119879 (119909 119905) = 119860119899(0)
sdot exp(minus1
2120591(1 + radic1 minus
41205912
119909
119863)119905)
sdot exp(119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(84)
to obtain 119860119899(0) we use the initial condition
119879 (119909 119905) =
infin
sum119899=1
119879119899(0) sin(
119899120587119909
119871) = 119892 (119909) (85)
from which we deduce that
119879119899(0) =
2
119871int119871
0
(119892120594) sin(119899120587120594
119871)119889120594 = 119861
119899 (86)
Hence the solution is given by
119879 (119909 119905)
=
infin
sum119899=1
119861119899sin(
119899120587119909
119871)
sdot exp((minus1
21205911205901minus120593
119905minus (minus
1
41205912minus
119863
1205912
119909)12
1205901minus120593
119905) 119905)
sdot exp (minus119863
1205912
1199091205902(1minus120593)
119905(120582119899)1205932
119905)
(87)
where 120582119899= 119899212058721198712
4 Conclusions
In this paper we introduced an alternative representationof the fractional Cattaneo-Vernotte equation In particular
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table3Solutio
nsforthe
fractio
naltim
eCattaneo-Ve
rnotteequatio
nin
overdampedcase
(76)
ford
ifferentvalueso
f120593erfc(120572)deno
testhe
errorfun
ctiondefin
edin
(8)
120593Solutio
n
2119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotcos(
radic1
2120591120590119905
)sdotcos(
radic(
1 41205912
minus119863 120591
1198962 119909)12
(1 120590119905
)119905)
32
119879(119909
119905)=
1198790 2sdotexp
(119894119896119909119909)sdot[ [
(minus(11990532
212059112059012
119905))2120581(1minus119903)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp(120572119895(minus
11990532
212059112059012
119905
)
2120581
))
sdot(12057232
119895+erfc
(12057212
119895(minus
11990532
212059112059012
119905
)
120581
))
minus(minus
11990532
212059112059012
119905
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(11990532
212059112059012
119905))119896
Γ(31198962
+120583)
] ]
sdot
[ [ [ [ [
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
3
2 sum 119895=0
1205721minus(32+119903)
119895sdot(exp120572119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)2119896
)sdot(12057232
119895+erfc
(12057212
119895(minus(
1 41205912
minus119863 120591
1198962 119909)120590minus12
11990511990532
)119896
))
minus(minus(
1 41205912
minus119863 120591
1198962 119909)12
120590minus12
11990511990532
)
minus2119899
sdot
2119899minus1
sum 119896=0
(minus(14
1205912
minus(119863
120591)1198962 119909)12
120590minus12
11990511990532
)
119896
Γ(31198962
+120583)
] ] ] ] ]
1119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(minus
1 2120591(1+radic1minus
41205911198962 119909
119863)
119905)
12119879(119909
119905)=1198790sdotexp(119894119896119909119909)sdotexp(
120590119905
41205912
119905 )sdot[1minuserfc
(minus12059012
119905 2120591
11990512
)]sdotexp[(
1 41205912
minus119863 120591
1198962 119909)120590119905119905 ]
sdot[1minuserfc
minus(
1 41205912
minus119863 120591
1198962 119909)12
12059012
11990511990512
]
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
a one-dimensional model was considered in detail Weshowed that the fractional Cattaneo-Vernotte equation inher-its some crucial characteristics in particular the parameters120590119909and 120590
119905need to be introduced in order to characterize
the behavior of the physical system which is located intoan intermediate state between a conservative and dissipativesystem presenting anomalous relaxations This combinationof stored and dissipated energy is conveniently based onthe representation of linear thermoviscoelasticity theoryUsually this dissipation is known as internal friction Somespecial cases are also discussed Our results indicate thatthe fractional order 120593 has an important influence on thetemperature Considering the Dirichlet conditions in therange 0 lt 120593 le 1 for the spatial case we observe thenon-Markovian Levy flights and in the temporal case forthe range 0 lt 120593 lt 1 the subdiffusion phenomena Inthe spatial case when 120593 = 32 and 120593 = 2 the diffusionexhibits an increment in the amplitude and the behaviorpresents anomalous dispersion (the diffusion increases withincreasing order of 120593) in the range 1 lt 120593 le 2 we observe theMarkovian Levy flights Finally in the temporal case when120593 = 32 and 120593 lt 2 the diffusion exhibits an increment in theamplitude and presents the superdiffusion and the case120593 = 2
represents the ballistic diffusion A crossover from a power-law behavior for short times to an exponential decay for longtimes has been foundWhenmemory effects are incorporatedusing fractional time derivatives the crossover dynamicsis richer The alternative model and results in this paperprovide a new theoretical perspective of the non-Fourier heatconduction Furthermore since the solutions are given interms of the multivariate Mittag-Leffler functions dependingonly on a small number of parameters the universalityconcept (when the class of behavior does not depend on thedetails of the physical system) can be considered through thismethodology since the analytic solutions presented only needa few parameters to describe their behavior in all cases thesolutions preserve the physical units of the system studied
Among problems for further research we mention theproblem of thermal convection of non-Fourier fluids andthe non-Newtonian effects in thermal convection (somesituations exist where the non-Fourier and non-Newtonianeffects are simultaneously present such as rarefied gaseswith high Knudsen numbers [52ndash54] in this case it isimportant to describe the interaction between the thermalrelaxation and viscous (stress) relaxation) another problemis the two- and three-dimensional fractional wave equationsconsidering fractional variational calculus (see [54] and thereferences therein) with different initial orand boundaryconditions of course it would be interesting to consider thefractional thermal wave equations with fractional derivativesdefined in different ways Furthermore the methodologyproposed in this work can be applied in the critical phe-nomena theory self-similarity scale-invariance propagationof energy in dissipative systems theory of viscoelastic fluidsand solids relaxing gas dynamics irreversible thermody-namics theory of thermal stresses thermoelasticity cos-mological models finance modeling theory of diffusion incrystalline solids and the description of anomalous complexprocesses
Competing Interests
The authors declare no competing interests
Acknowledgments
The authors would like to thank Mayra Martınez for theinteresting discussions J F GomezAguilar acknowledges thesupport provided by CONACYT Catedras CONACYT paraJovenes Investigadores 2014
References
[1] D D Joseph and L Preziosi ldquoHeat wavesrdquo Reviews of ModernPhysics vol 61 no 1 pp 41ndash73 1989
[2] E K Lenzi L R Evangelista M K Lenzi H V Ribeiro andE C de Oliveira ldquoSolutions for a non-Markovian diffusionequationrdquo Physics Letters A vol 374 no 41 pp 4193ndash4198 2010
[3] D Ben-Avraham and S Havlin Diffusion and Reactions inFractals and Disordered Systems Cambridge University PressCambridge UK 2000
[4] R Metzler A V Chechkin and J Klafter ldquoLevy statisticsand anomalous transport levy flights and subdiffusionrdquo inEncyclopedia of Complexity and Systems Science pp 5218ndash52392009
[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[6] J A Tenreiro Machado A M Galhano and J J TrujilloldquoScience metrics on fractional calculus development since1966rdquo Fractional Calculus and Applied Analysis vol 16 no 2pp 479ndash500 2013
[7] J A Tenreiro Machado A M S F Galhano and J J TrujilloldquoOn development of fractional calculus during the last fiftyyearsrdquo Scientometrics vol 98 no 1 pp 577ndash582 2014
[8] T Sandev R Metzler and Z Tomovski ldquoVelocity and displace-ment correlation functions for fractional generalized Langevinequationsrdquo Fractional Calculus and Applied Analysis vol 15 no3 pp 426ndash450 2012
[9] C H Eab and S C Lim ldquoFractional Langevin equations ofdistributed orderrdquo Physical Review E vol 83 no 3 Article ID031136 10 pages 2011
[10] P De Anna T Le Borgne M Dentz A M TartakovskyD Bolster and P Davy ldquoFlow intermittency dispersion andcorrelated continuous time random walks in porous mediardquoPhysical Review Letters vol 110 no 18 Article ID 184502 2013
[11] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E-Statistical Nonlinear and Soft MatterPhysics vol 85 no 5 Article ID 051103 2012
[12] A Zeb G Zaman I H Jung andM Khan ldquoOptimal campaignstrategies in fractional-order smoking dynamicsrdquo Zeitschrift furNaturforschung A vol 69 no 5-6 pp 225ndash231 2014
[13] I S Jesus and J A Tenreiro Machado ldquoFractional control ofheat diffusion systemsrdquo Nonlinear Dynamics vol 54 no 3 pp263ndash282 2008
[14] M Zingales ldquoFractional-order theory of heat transport in rigidbodiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 19 no 11 pp 3938ndash3953 2014
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
[15] J F Gomez-Aguilar and D Baleanu ldquoFractional transmissionline with lossesrdquo Zeitschrift fur Naturforschung vol 69 no 10-11 pp 539ndash546 2014
[16] J F Gomez Aguilar and D Baleanu ldquoSolutions of the telegraphequations using a fractional calculus approachrdquo Proceedings ofthe Romanian Academy Series A vol 15 no 1 pp 27ndash34 2014
[17] V Mishra K Vishal S Das and S H Ong ldquoOn the solution ofthe nonlinear fractional diffusion-wave equation with absorp-tion a homotopy approachrdquo Zeitschrift fur Naturforschung Avol 69 no 3-4 pp 135ndash144 2014
[18] R T Sibatov and V V Uchaikin ldquoFractional differentialapproach to dispersive transport in semiconductorsrdquo Physics-Uspekhi vol 52 no 10 pp 1019ndash1043 2009
[19] X-J Yang D Baleanu and H M Srivastava ldquoLocal fractionalsimilarity solution for the diffusion equation defined on Cantorsetsrdquo Applied Mathematics Letters vol 47 pp 54ndash60 2015
[20] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[21] G-C Wu D Baleanu S-D Zeng and Z-G Deng ldquoDiscretefractional diffusion equationrdquo Nonlinear Dynamics vol 80 no1-2 pp 281ndash286 2015
[22] O Defterli M DrsquoElia Q Du M Gunzburger R Lehoucqand M M Meerschaert ldquoFractional diffusion on boundeddomainsrdquo Fractional Calculus and Applied Analysis vol 18 no2 pp 342ndash360 2015
[23] R Schumer M M Meerschaert and B Baeumer ldquoFractionaladvection-dispersion equations for modeling transport at theearth surfacerdquo Journal of Geophysical Research Earth Surfacevol 114 no 4 Article ID F00A07 2009
[24] M M Meerschaert D A Benson H-P Scheffler and BBaeumer ldquoStochastic solution of space-time fractional diffusionequationsrdquo Physical Review E vol 65 no 4 Article ID 0411034 pages 2002
[25] F Liu M M Meerschaert R J McGough P Zhuang andQ Liu ldquoNumerical methods for solving the multi-term time-fractional wave-diffusion equationrdquo Fractional Calculus andApplied Analysis vol 16 no 1 pp 9ndash25 2013
[26] A Kullberg D del-Castillo-Negrete G J Morales and JE Maggs ldquoIsotropic model of fractional transport in two-dimensional bounded domainsrdquo Physical Review E vol 87 no5 Article ID 052115 2013
[27] R Gorenflo Y Luchko and M Stojanovi ldquoFundamentalsolution of a distributed order time-fractional diffusion-waveequation as probability densityrdquo Fractional Calculus andAppliedAnalysis vol 16 no 2 pp 297ndash316 2013
[28] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus and Applied Analysis vol 4 no 2 pp 153ndash192 2001
[29] J F G Aguilar and M M Hernandez ldquoSpace-time fractionaldiffusion-advection equation with Caputo derivativerdquo Abstractand Applied Analysis vol 2014 Article ID 283019 8 pages 2014
[30] RGorenflo FMainardi DMoretti and P Paradisi ldquoTime frac-tional diffusion a discrete random walk approachrdquo NonlinearDynamics vol 29 no 1 pp 129ndash143 2002
[31] A Compte and R Metzler ldquoThe generalized cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997
[32] Y Postvenko ldquoTheories of termal stresses based on space-time-fractional telegraph equationsrdquo Computers and Mathematicswith Applications vol 64 no 10 pp 3321ndash3328 2012
[33] T Atanackovic S Konjik L Oparnica and D Zorica ldquoTheCattaneo type space-time fractional heat conduction equationrdquoContinuum Mechanics and Thermodynamics vol 24 no 4ndash6pp 293ndash311 2012
[34] T M Atanackovic S Pilipovic and D Zorica ldquoA diffusionwave equationwith two fractional derivatives of different orderrdquoJournal of Physics A Mathematical and Theoretical vol 40 no20 pp 5319ndash5333 2007
[35] K D Lewandowska and T Kosztołowicz ldquoApplication of gen-eralized cattaneo equation to model subdiffusion impedancerdquoActa Physica Polonica B vol 39 no 5 pp 1211ndash1220 2008
[36] V E Tarasov ldquoTransport equations from Liouville equations forfractional systemsrdquo International Journal of Modern Physics BCondensed Matter Physics Statistical Physics Applied Physicsvol 20 no 3 pp 341ndash353 2006
[37] H T Qi and X Y Jiang ldquoSolutions of the space-time fractionalCattaneo diffusion equationrdquo Physica A vol 390 no 11 pp1876ndash1883 2011
[38] H-Y Xu H-T Qi and X-Y Jiang ldquoFractional Cattaneo heatequation in a semi-infinite mediumrdquo Chinese Physics B vol 22no 1 Article ID 014401 6 pages 2013
[39] Y J Yu X G Tian and T J Lu ldquoFractional order general-ized electro- magneto-thermo-elasticityrdquo European Journal ofMechanics-ASolids vol 42 pp 188ndash202 2013
[40] S-W Vong H-K Pang and X-Q Jin ldquoA high-order differencescheme for the generalized Cattaneo equationrdquo East AsianJournal on Applied Mathematics vol 2 no 2 pp 170ndash184 2012
[41] X Zhao and Z-Z Sun ldquoCompact Crank-Nicolson schemesfor a class of fractional Cattaneo equation in inhomogeneousmediumrdquo Journal of Scientific Computing vol 62 no 3 pp 747ndash771 2015
[42] J F Gomez-Aguilar M Miranda-Hernandez M G Lopez-Lopez V M Alvarado-Martınez and D Baleanu ldquoModelingand simulation of the fractional space-time diffusion equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 30 no 1ndash3 pp 115ndash127 2016
[43] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993
[44] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[45] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[46] K Diethelm N J Ford A D Freed and Y Luchko ldquoAlgorithmsfor the fractional calculus a selection of numerical methodsrdquoComputer Methods in Applied Mechanics and Engineering vol194 no 6ndash8 pp 743ndash773 2005
[47] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type Springer Berlin Germany 2010
[48] H J Haubold A M Mathai and R K Saxena ldquoMittag-Leffler functions and their applicationsrdquo Journal of AppliedMathematics vol 2011 Article ID 298628 51 pages 2011
[49] K S Miller ldquoSome simple representations of the generalizedMittag-Leffler functionsrdquo Integral Transforms and Special Func-tions vol 11 no 1 pp 13ndash24 2001
[50] R Garrappa and M Popolizio ldquoEvaluation of generalizedMittag-Leffler functions on the real linerdquo Advances in Compu-tational Mathematics vol 39 no 1 pp 205ndash225 2013
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[51] H Struchtrup and P Taheri ldquoMacroscopic transport modelsfor rarefied gas flows a brief reviewrdquo IMA Journal of AppliedMathematics vol 76 no 5 pp 672ndash697 2011
[52] H Brenner ldquoSteady-state heat conduction in a gas undergoingrigid-body rotation Comparison of Navier-Stokes-Fourier andbivelocity paradigmsrdquo International Journal of Engineering Sci-ence vol 70 pp 29ndash45 2013
[53] H Struchtrup ldquoResonance in rarefied gasesrdquo ContinuumMechanics and Thermodynamics vol 24 no 4ndash6 pp 361ndash3762012
[54] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
