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SYMMETRY-ENHANCING FOR A THIN FILM EQUATION
TANYA L.M. WALKER
A thesis submitted in fulfilment
of the requirements for the degree of
Doctor of Philosophy - Science
University of Western Sydney
2008
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ABSTRACT
This thesis is concerned with the construction of new one-parameter symmetry groups
and similarity solutions for a generalisation of the one-dimensional thin film equation by
the method of symmetry-enhancing constraints involving judicious equation-splitting.
Firstly by Lie classical analysis we obtain symmetry groups and similarity solutions of
this thin film equation. Via the Bluman-Cole non-classical procedure, we then construct
non-classical symmetry groups of this thin film equation and compare them to the
classical symmetry groups we derive for this equation.
Next we apply the method of symmetry-enhancing constraints to this thin film equation,
obtaining new Lie symmetry groups for this equation. We construct similarity solutions
for this thin film equation in association with these new groups. Subsequently we
retrieve further new symmetry groups for this thin film equation by an approach
combining the method of symmetry-enhancing constraints and the Bluman-Cole non-
classical procedure. We derive similarity solutions for this thin film equation in
connection with these new groups.
Then we incorporate nontrivial functions into a partition (of this thin film equation)
which has previously led to new Lie symmetry groups. The resulting system admits newLie symmetry groups. We recover similarity solutions for this system and hence for the
thin film equation in question.
Finally we attempt to derive potential symmetries for this thin film equation but our
investigations reveal that none occur for this equation.
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PREFACE
In this thesis, the symmetry groups and similarity solutions obtained for the thin film
equation and the systems of equations under consideration form an original contribution.
Where the work of other authors has been used, this has always been specifically
acknowledged in the relevant sections of the text.
Tanya Walker
31stMarch 2008
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ACKNOWLEDGEMENTS
I would like to express my indebtedness to my supervisor Dr. Alec Lee whose
encouragement, enthusiasm, intellectual stimulation and unlimited reserves of patience
have guided my researches since the commencement of this degree.
I wish to thank Professor Broadbridge for discussions leading to the final form of the
generalised thin film equation (1.1) studied in this thesis.
Furthermore I would like to express my deep appreciation of my beloved husband David
for his constant love, tenderness, understanding and confidence in me throughout my
candidature.
Finally I would like to thank my closest friend Karen for the understanding and support
she has always shown me, especially in the undertaking of these studies.
All these factors have combined to make this thesis a reality.
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This thesis is dedicated with deepest love to my husband David.
O how vast the shores of learning,
There are still uncharted seas,
And they call to bold adventure,
Those who turn from sloth and ease
Excerpt from A Students Prayer
Author unknown
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CHAPTER 6: CLASSICAL SYMMETRY-ENHANCING
CONSTRAINTS FOR THE THIN FILM EQUATION
INVOLVING ARBITRARY FUNCTIONS 163
6.1 Introduction 163
6.2 Classical Symmetry-Enhancing Constraints 164
6.3 Tables Of Results 193
6.4 Concluding Remarks 197
CHAPTER 7: LOCATING POTENTIAL SYMMETRIES FOR THE
THIN FILM EQUATION 198
7.1 Introduction 198
7.2 The Method Of Obtaining Potential Symmetries 199
7.3 Concluding Remarks 200
CHAPTER 8: CONCLUSION 201
BIBLIOGRAPHY 204
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CHAPTER 1
INTRODUCTION
We construct new one-parameter symmetry groups and corresponding similarity
solutions for a generalised thin film equation via the method of symmetry-enhancing
constraints introduced and developed by Goard and Broadbridge [29]. This technique
involves systematic equation-splitting and is restricted to classical symmetries. In
conjunction with this method of symmetry-enhancing constraints, Saccomandi
considered special classes of non-classical symmetries [47]. By similarly augmenting
this method of symmetry-enhancing constraints with the non-classical symmetry method
of Bluman and Cole [16], we retrieve symmetry groups for the enlarged system resulting
from the partitioning of the generalised thin film equation in question.
By means of the symmetry groups obtained for this thin film equation via the method of
symmetry-enhancing constraints, we identify similarity solutions of the latter equation.
Computer techniques involving the Mathematica and Maple programs are instrumental
in the process of deriving these groups and solutions [46, 54].
Applying the method of symmetry-enhancing constraints to solve this generalised thin
film equation does not consistently prove successful in deriving solutions, as is clear
from Chapter 5 of this thesis. However, this method of solving differential equations is
successfully applicable to nonlinear differential equations such as cylindrical boundary-
layer equations, generating new similarity solutions [29].
Other treatments of recovering solutions include the approach developed by Burde to
derive explicit similarity solutions of partial differential equations (PDEs) [20]. His
approach is an extension of the Bluman-Cole non-classical group method [15]. Burdes
method involves directly substituting a similarity form of the solution into the given
PDE and was developed via a variation of the Clarkson-Kruskal technique [22]. Instead
of requiring this given PDE be reduced to an ordinary differential equation (ODE) as in
the Clarkson-Kruskal technique [22], a weaker condition is imposed, namely that this
PDE be reduced to an overdetermined system of ODEs solvable in closed form. The
viability of Burdes approach was justified as it enabled Burde to recover new, exact,
explicit, physically significant similarity solutions for the two-dimensional steady-state
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boundary layer problems. Although the solutions thus obtained extend beyond the
confines of those retrievable via classical Lie analysis and the Bluman-Cole non-
classical group method [15], they proved to be merely a special case of solutions derived
within the framework of the method of symmetry-enhancing constraints [29].
The equation under consideration in this thesis is a generalisation of the one-dimensional
thin film equation and is given by
[ ] ;0)()()( =++
txxxxx
hhhjhhghhfx
(1.1)
where h denotes the height of a thin viscous droplet (or film) as a function of time tand
the (one-dimensional) spatial coordinate x parallel to the solid surface. This thesis
assumes the y - independence of ,h namely that the film flows without developing any
structure in the transverse direction [44].
The term )(hf arises from surface tension (which tends to flatten the free surface [44])
between two liquids or between liquid and air and incorporates any slippage at the
liquid/solid interface. This term represents surface tension effects and the viscosity of the
liquid [45].
The term )(hg results from film destabilisation due to thermocapillarity or a density
mismatch between two liquids or physical effects such as evaporation, condensation, the
normal component of gravity to a solid surface and intermolecular forces [2]. This term
can indicate additional forces such as gravity, van der Waals interactions or
thermocapillary effects [45]. If ,0)( hg occurring with repulsive van der Waals
interactions, a long wave instability appears. If ,0)( hg the thin film equation (1.1)
lacks a long wave instability.
The convective term )(hj includes any directed driving forces (such as gravity or
Marangoni stress) corresponding to a dimensionless flux function [14]. In the case of
dominant Marangoni stress, the Burgers flux 2)( hhj = occurs while the compressive
3)( hhj = features in the case of gravitational stress [14]. The Marangoni effect
corresponds to tangential stresses at the gas-liquid interface due to surface tension
gradients while Marangoni flow refers to film flow induced by surface tension
gradients [2].
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The thin film equation (1.1) is a nonlinear degenerate fourth order diffusion equation
describing the flow of thin liquid films of height (or dimensionless thickness) h on an
inclined flat surface under the action of forces of gravity, viscosity and surface tension at
the air/liquid interface [14, 34]. This equation features 0>h in a one-dimensional
geometry so that h depends on one space variable x and time t[18].
The most common derivation of the thin film equation is as a lubrication approximation
(or limit) of the Navier-Stokes equations for incompressible fluids [2, 33, 44]. Thin films
are effectively described by lubrication approximation in which the equation of motion is
given by the thin film equation (1.1) with nhhf =)( and 0)()( == hjhg where 0>h is
a requirement [18].
Grun and Rumpf presented numerical experiments indicating the occurrence of a waiting
time phenomenon for fourth order degenerate parabolic equations [33]. Grun proved
such an occurrence in space dimensions 4n is a parameter [26, 34, 37, 38].
Hastings and Peletier regarded 0>n as a constant dependent on the type of flow
considered [34].
The above case of equation (1.1) with the critical value 3=n features in [6, 38, 43, 53]
and is pronounced most common in physical situations [38] while 4=n is noted as a
critical exponent for the large time behaviour of solutions.
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Bernis, Peletier and Williams considered the critical value2
3=n at which the nature of
the solution near the interface changes [8]. Hulshof studied similarity solutions of the
thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and ,0>n recovering one
such explicit solution via Maple 5 release 2; [37]. Bernis, Hulshof and Quiros studied the
limit of nonnegative, self-similar source-type solutions of this case of the thin film
equation (1.1) as ,0+n consequently obtaining a unique limiting function ,h a
solution of an obstacle-type free boundary problem with constraint ;0h [7].
The thin film equation (1.1) arises in fluid dynamics (hydrodynamics) and material
sciences (cf. the Cahn-Hilliard equation) [1, 31, 32]. The case of equation (1.1) with
nhhf =)( and 0)()( == hjhg (where 0>h is a requirement) occurs in certain fluid
dynamics problems in which inertia is negligible and the dynamics is governed by the
presence of viscosity and capillarity forces [18].
Upon assuming the lubrication approximation with the no-slip condition for the fluid at
the solid surface and the fact that the pressure is entirely due to surface tension, Beretta
and Bertsch derived the above case of thin film equation (1.1) with ;3=n [3]. This case
has great physical significance in lubrication theory in terms of governing the dynamicsof the spreading of a droplet over a solid surface under effects of viscosity and
capillarity. This case is depicted as the height ),( txh of a thin film of slowly flowing
viscous fluid over a horizontal substrate when surface tension is the dominating driving
force [3, 6, 12, 18, 38, 39, 43]. This case corresponding to no-slip boundary conditions
results in infinite viscous dissipation, generating variations on the same problem by
changing boundary conditions at the interface solid fluid [12, 18].
The case of the thin film equation (1.1) with 2)( hhf = and 0)()( == hjhg corresponds
to slip dominated spreading with a Navier slip law [43] and occurs in [4] and [18].
According to Laugesen and Pugh, the case of the thin film equation (1.1) with 0)( =hj
is used to model the dynamics of a thin film of viscous liquid where the air/liquid
interface is at height ),,( tyxhz = and the liquid/solid interface is at ;0=z [45]. These
authors also state that equation (1.1) with 0)( =hj applies if the liquid film is uniform in
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the y direction [45]. An application of equation (1.1) with 0)( =hj lies in its ability to
model the aggregation of aphids on a leaf where h represents population density [45].
The special case of the thin film equation (1.1) with hhf =)( and 0)()( == hjhg is
used to describe the evolution of the interface of a spreading droplet, modelling the
surface tension dominated motion of thin viscous films and spreading droplets,
according to Carrillo and Toscani [21]. This case describes the dynamics of the process
in the gravity-driven Hele-Shaw cell [6, 12, 18, 23, 25, 30, 38, 43, 45]. In this process,
liquid in a fluid droplet is sucked so as to produce a long thin bridge of thickness h
between two masses of fluids, the geometry of which problem being able to be
approximated as one-dimensional under appropriate conditions. This case emerges when
considering a drop on a porous surface [18].
Another of the varied applications of the thin film equation (1.1) is the modelling of
driven contact line experiments involving only one dominant driving force
(corresponding to a convex flux function )(hj ) [14]. In addition, equation (1.1) models
thin film slow viscous flows (viscosity driven flows) such as painting layers [37] and the
drying of a paint film in a specific parameter regime [52]. Equation (1.1) also plays a key
role in plasticity modelling where h represents the density of dislocations. This equation
occurs in the Cahn-Hilliard model of phase separation for binary mixtures where h
denotes the concentration of one component.
The case of the thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and )3,0(n
emerges as a lubrication theory model for the flow of thin viscous films (and spreading
droplets) driven by strong surface tension over a horizontal substrate with ),( txh
denoting the height of the free-surface of the film [7, 9, 26, 31, 32, 33]. The range 0
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profiles for this equation revealed that after initial transients, the flow develops a
travelling wave profile [44].
Via analysis methods (involving a Lyapunov function), Bertozzi and Shearer studied an
equation comparable to the thin film equation (1.1) with ,)()( 3hhghf == 23)( hhj =
and the size of the dimensionless parameter governing gravitational, viscous and surface
tension forces as well as the slope of the surface equalling 1; [14]. Experimental and
numerical studies of driven contact lines disclosed that travelling wave solutions of this
equation play a key role in the motion of the film [11, 13, 40, 51]. Travelling wave
solutions also arise in chapters 2 5 of this thesis.
Hulshof and Shishkov [39] examined compactly supported solutions of the case of the
thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and [ )3,2n on
( ) ( ]{ }TtRRxtxQT ,0,,:),( = with nonnegative initial data and lateral boundary
conditions respectively given by
)()0,( 0 xuxu = with ,00 u ( ) ( ) .0,, == tRutRu xxxx (1.2)
These authors regarded R as a finite positive number. It is also potentially considered as
=R for compactly supported solutions (the Cauchy problem). For the case of zerocontact angle boundary conditions on a finite domain, van den Berg et al. investigated
self-similar solutions of the above case of the thin film equation (1.1) where n is a real
parameter [53].
The outline of the thesis is as follows.
In chapter 2 we obtain the Lie classical symmetry groups of the thin film equation (1.1)
and derive its similarity solutions in association with each of these groups. We use the
one-parameter )( Lie group of general infinitesimal transformations in ,x t and ,h
namely
( ) ( )( ) ( )( ) ( ).,,
,,,
,,,
2
1
2
1
2
1
Ohtxhh
Ohtxtt
Ohtxxx
++=
++=
++=
(1.3)
In conjunction with Lie classical analysis discussed in [36], group transformations (1.3)
enable the recovery of the one-parameter Lie classical symmetry groups for the thin filmequation (1.1).
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In chapter 3 we construct non-classical symmetry groups for the thin film equation (1.1)
under the action of group transformations (1.3), using the non-classical symmetry
method of Bluman and Cole [16]. We compare these symmetry groups with those
obtained in chapter 2 and derive for equation (1.1) any similarity solutions not
retrievable by Lie classical analysis. Full details of these solutions occur in chapter 3.
In chapter 4 we apply the method of symmetry-enhancing constraints [29] to the thin
film equation (1.1) in association with group transformations (1.3) with a view to
obtaining new symmetry groups. In line with this method, we studied various partitions
of the thin film equation (1.1).
Two of these partitions lead to new Lie symmetry groups and generate the systems
( ) ,0)()( 2 =+ xxt hhghhjh [ ] ;0)()( =
xxxxx
hhghhfx
(1.4)
and
,0)()( =+ txxxxxx hhhghhf ( ) .0)()()(2
=+ xxxxxx hhjhhghhhf (1.5)
We construct similarity solutions for systems (1.4) and (1.5) and hence for the thin film
equation (1.1) in relation to each of these new groups. A full account of these solutions is
given in chapter 4.
In chapter 5 we derive symmetry groups for the thin film equation (1.1) in association
with group transformations (1.3) by a treatment combining the method of symmetry-
enhancing constraints [29] with the non-classical symmetry method of Bluman and Cole
[16]. Saccomandi considered the combination of these two techniques [47]. Investigating
systems (1.4) and (1.5) from the perspective of this combined approach generates new
symmetry groups for these systems. We retrieve the similarity solutions for systems (1.4)
and (1.5) and thus for the thin film equation (1.1) in connection with these groups.
In chapter 6 we augment system (1.4) with the arbitrary nontrivial functions )(xa and
),(tb obtaining the equations
( ) ,0)()()()( 2 =+ tbxahhghhjh xxt [ ] .0)()()()( =
tbxahhghhf
x xxxxx (1.6)
System (1.6) admits new Lie symmetry groups in association with transformations (1.3).
We derive similarity solutions for system (1.6) and hence for the thin film equation (1.1)
in relation to these groups. We give a full account of these groups and solutions in
chapter 6.
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In chapter 7 we seek potential symmetries for the thin film equation (1.1) by the method
introduced and developed by Bluman, Reid and Kumei [17].
At the end of each chapter, we tabulate all results obtained in the chapter concerned. This
thesis has been written largely in accordance with the guidelines in Higham [35],
Bluman and Kumei [55], Ibragimov [56], Olver [57] and Ovsiannikov [58].
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CHAPTER 2
LIE CLASSICAL SYMMETRIES FOR THE
THIN FILM EQUATION
2.1 INTRODUCTION
By the Lie classical procedure, we determine the Lie classical symmetry groups for the
thin film equation
[ ] [ ] ;0)()()( =++
txxxxx
hhhjhhgx
hhfx
(2.1)
where .0)( hf The restriction 0)( hf applies since the thin film equation (2.1)
generalises the fourth order nonlinear diffusion equation, a special case of equation (2.1)
with .0)()( == hjhg This case of the thin film equation (2.1) occurs in Bernoff and
Witelski [9] and King and Bowen [43]. The term )(hf in the thin film equation (2.1)
represents surface tension effects (Laugesen and Pugh [45]).
We consider the one-parameter )( Lie group of general infinitesimal transformations in
,x tand ,h namely
( ) ( )( ) ( )( ) ( );,,
,,,
,,,
2
1
2
1
2
1
Ohtxhh
Ohtxtt
Ohtxxx
++=
++=
++=
(2.2)
preserving the thin film equation (2.1).
Hence if ),,( txh = then from ),,( 111 txh = evaluating the expansion of
1h at 0=
gives the invariant surface condition
).,,(),,(),,( htxt
hhtx
x
hhtx =
+
(2.3)
Solutions of the invariant surface condition (2.3) are functional forms of similarity
solutions for the thin film equation (2.1).
The next section contains a brief outline of the Lie classical method, also described in.
Hill [36].
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2.2 THE CLASSICAL PROCEDURE
The classical method requires equating to zero the infinitesimal version of the thin film
equation (2.1) without using the invariant surface condition (2.3). In obtaining the
infinitesimal version of the thin film equation (2.1), we eliminate the highest order
derivative4
4
x
h
in equation (2.1) by expressing it with respect to all the remaining terms
of equation (2.1). Prolongation of the action of group transformations (2.2) on the thin
film equation (2.1) yields the invariance requirement, obtained by equating to zero the
coefficient of in the infinitesimal version of equation (2.1). Terms of order 2 are
neglected in these calculations since they involve relations between the group generators
, and already considered in the coefficient of , the left-hand side of the
invariance requirement.
The thin film equation (2.1) remains invariant under group transformations (2.2)
provided the group generators ),,,( htx ),,( htx and ),,( htx satisfy the determining
equations
,0=h ,0== xh ,0=hh ,0
)(
)(=
hf
hf
dh
d ( ) ,0)( = xxxhhf
,0)()()( =++ xxxxxxxt hfhghj ,0)(
)()(4 =
hf
hftx
[ ] xxxxxxtx hfhghgdh
d
hf
hj
dh
dhfhj )()()(2
)(
)()()(3 ++
+
( ) ,04)( =+ xxxxxxxhhf (2.4)
,064
)(
)(=+
xxxhx
hf
hf ,0
)(
)(
)(
)(246 =
hf
hg
dh
d
hf
hgxxxxxxh
( ) ( ) .0)(
)(2
)(
)(3
)(
)(=
+
hf
hg
dh
d
hf
hg
hf
hfxhxxxxxh
Equating to zero the coefficients of all derivatives of h and the sum of all remaining
terms not involving derivatives of h within the invariance requirement for the thin film
equation (2.1) produces system (2.4). All subscripts in system (2.4) denote partial
differentiation with ,x t and h as independent variables. Throughout this chapter,
primes represent differentiation with respect to the argument indicated.
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System (2.4) enables the recovery of all Lie classical symmetries and corresponding
conditions on ,0)( hf )(hg and )(hj for the thin film equation (2.1) under group
transformations (2.2).
We now partially solve the determining equations (2.4) to clarify derivations of sets of
conditions on ,0)( hf )(hg and )(hj associated with each Lie classical group we
obtain for the thin film equation (2.1). Subsequently we describe the functional forms of
,0)( hf )(hg and )(hj with the corresponding Lie classical group occurring for the
thin film equation (2.1). Eight such groups arise. Lastly we present the similarity
solutions of the thin film equation (2.1) in connection with each of these groups.
From equations (2.4)1 (2.4)3, it follows that),,(),,( txhtx = ),(),,( thtx = );,(),(),,( txbhtxahtx += (2.5)
where ),( txa and ),( txb are arbitrary functions of x and .t
By results (2.5)1and (2.5)3, equation (2.4)5gives ( ) ,0)( = xxxahf generating cases
(1) ),,(),( txtxa xxx = (2) .0)( =hf
We present the derivation of results for case (1) only.
Case (1) ),(),( txtxa xxx =
It follows that
);(),(),( ttxtxa x += (2.6)
where )(t is an arbitrary function of .t
Results (2.5)3 and (2.6) cause equation (2.4)9 to give [ ] ,)()(2)( xxx bhfhfhfh =
integrating which with respect to x implies
[ ] );,(),()()(2)( htctxbhfhfhfh x =+ (2.7)
where 0)( hf is an arbitrary function of h while ),( htc is an arbitrary function of t
and .h
By results (2.5)-(2.7), equation (2.4)7gives ,)(
)()(),()()(2
hf
thfhhtcttx
+== so
),(2
)()(),( tx
tttx
+
+= );()()()(),( thfhthfhtc = (2.8)
where ),(t )(t and )(t are arbitrary functions of .t
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Results (2.8) cause relations (2.5)3, (2.6) and (2.7) to give
),,(2
)(2)()(),,( txbh
ttthtx +
++=
(2.9)
[ ] ).(2
)(2)()()()(2)(),()( hfh
ttttthftxbhf
+++=
As equation (2.9)2gives ,0)( = xbhf we obtain the subcases
(a) ,0)( =hf (b) ).(),( tbtxb =
As case (2) includes subcase (a), we need consider only subcase (b).
Subcase (b) )(),( tbtxb =
Results (2.8) and (2.9) yield
),(2
)()(),( tx
tttx
+
+= ),(
2
)(2)()(),(),,( tbh
ttththtx +
++==
(2.10)
[ ] );(2
)(2)()()()(2)()()( hfh
ttttthftbhf
+++=
where )(tb is an arbitrary function of .t
Substituting result (2.10)2into equation (2.4)6gives
),()(2)( 1 ttdt = ;)( 2dtb = (2.11)
where 1d and 2d are arbitrary constants.
Results (2.10) and (2.11) give
[ ] ),()(),( 1 txtetx += ),()(2)( 1 ttdt = ,)(),( 21 dhehht +==
(2.12)( ) [ ] );()(2)()( 121 hfdtthfdhe +=+
where .2
11
de =
As equation (2.12)4has the form ),()( tmhk = giving ,0)()( == tmhk it follows that
,)(2)( 31 ddtt =+ ( ) );()( 321 hfdhfdhe =+ (2.13)
where 3d is an arbitrary constant.
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Results (2.12) and (2.13) yield
),(2
)(),( 3 tx
tdtx
+
= ),(2)( 3 tdt = ,)( 21 dheh +=
(2.14)
( ) ).()( 321 hfdhfdhe =+
By results (2.14)1, (2.14)3and (2.14)4, equation (2.4)8gives
( ) ).(2
)()(
2
)(3)( 321 tx
thj
tdhjdhe
=
+
++ (2.15)
Setting to zero the coefficient of x in equation (2.15) yields
;)( 4dt = (2.16)
where 4d is an arbitrary constant.
In view of result (2.16), equation (2.15) gives
;)( 65 dtdt += (2.17)
where 5d and 6d are arbitrary constants.
Redefining the constants, the determining equations (2.4) and the results for this case are
,),( 654 DtDxDtx ++= ,)( 73 DtDt += ,)( 21 DhDh +=
( ) ),()( 821 hfDhfDhD =+ ( ) ),()( 921 hgDhgDhD =+ (2.18)
( ) ;)()( 51021 DhjDhjDhD =++
whereiD is an arbitrary constant for all { }10,...,2,1i with ,4 348 DDD =
349 2 DDD = and .4310 DDD =
In view of equations (2.18)4and (2.18)6, we consider the cases
(1) ,021 =+DhD ,0108 ==DD ,05 =D (2) ,021 =+DhD ,0810 =DD
(3) ,021 +DhD ,01012 == DDD (4) ,021 +DhD ,01102 =DDD
(5) ,021 +DhD ,0101 =DD (6) ,021 +DhD .0101 DD
Rewriting cases (1)-(6) above with 348 4 DDD = and 4310 DDD = gives
(a) ,054321 ===== DDDDD (b) ,04 2143 === DDDD
(c) ,04312 == DDDD (d) ,012 =DD ,43 DD
(e) ,0431 = DDD (f) ,01 D .43 DD
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For each of the cases (a) (f), we describe ,0)( hf )(hg and )(hj (obtainable from
the defining equations (2.18)4 (2.18)6) with their associated Lie classical groups (I)
(VI). We also present ,0)( hf )(hg and )(hj with their corresponding Lie classical
groups (VII) (VIII) for case (2). As previously stated, we give the similarity solutions
of the thin film equation (2.1) in conjunction with each of these groups.
GROUP (I)
Subject to the conditions ,0)( hf )(hg and )(hj are arbitrary functions of ,h the thin
film equation (2.1) admits Lie classical group (I), namely
,),,( 6Dhtx = ,),,( 7Dhtx = ;0),,( =htx (2.19)
where 6D and 7D are arbitrary constants.
Similarity Solutions
Group (2.19), the invariant surface condition (2.3) and the thin film equation (2.1) give
,076 =+ tx hDhD [ ] ;0)()()( =++
txxxxx hhhjhhghhf
x (2.20)
where 6D and 7D are arbitrary constants while ,0)( hf )(hg and )(hj are arbitrary
functions of .h As 0=xh forces 0=th in equation (2.20)2 , giving =),( txh constant,
we require 0xh for system (2.20) to generate nonconstant similarity solutions.
As no similarity solutions are obtainable for the thin film equation (2.1) when
,076 ==DD we consider only the cases
(1) ,07 D (2) .076 =DD
Case (1) 07 D
By the method in [24], we solve equation (2.20)1and substitute its general solution into
equation (2.20)2 . Therefore under transformations (2.2) and with ,0)( hf )(hg and
)(hj arbitrary functions of ,h the similarity solution of the thin film equation (2.1) in
association with group (2.19) and the constraint 07 D is the travelling wave of
velocity ,11D namely
);(),( uytxh = (2.21)
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satisfying
( ) ( ) ( ) ( )[ ]2)4( )()()()()()()()()( uyuyguyuyguyuyuyfuyuyf +
( )[ ] .0)()( 11 =+ uyDuyj (2.22)
In relations (2.21)-(2.22), ,07 D 6D and7
611
D
DD = are arbitrary constants while
tDxu 11= and ( ) ,0)( uyf ( ))(uyg and ( ))(uyj are arbitrary functions of ).(uy We
require 0)( uy for solution (2.21) to be nonconstant.
When ,0116 ==DD the travelling wave (2.21) reduces to the steady state solution
satisfying the case of the ordinary differential equation (ODE) (2.22) with .011 =D
Case (2) 076 =DD
Since 06 D forces 0=xh in equation (2.20)1 , giving 0=th in equation (2.20)2 ,
system (2.20) yields only the constant solution. Hence under transformations (2.2) and
with ,0)( hf )(hg and )(hj arbitrary functions of ,h the similarity solution of the
thin film equation (2.1) in connection with group (2.19) and the constraints 076 =DD
is the constant solution.
GROUP (II)
Under the conditions 0)( hf is an arbitrary function of ,h 0)( =hg and ,)( 1jhj = the
thin film equation (2.1) yields Lie classical group (II), namely
( ) ,03),,( 614 ++= DtjxDhtx ,04),,( 74 += DtDhtx ;0),,( =htx (2.23)
where ,04 D ,6D 7D and 1j are arbitrary constants.
Similarity Solutions
Group (2.23), the invariant surface condition (2.3) and the thin film equation (2.1) imply
( )[ ] ( ) ,043 74614 =++++ tx hDtDhDtjxD [ ] ;0)( 1 =++
txxxx hhjhhf
x (2.24)
where ,04 D ,6D 7D and 1j are arbitrary constants while 0)( hf is an arbitrary
function of .h Since 0=xh causes 0=th in equation (2.24)2, giving =),( txh constant,
we require 0xh for system (2.24) to admit nonconstant solutions.
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Via the method in [24] and the integrating factor algorithm in [48], we solve equation
(2.24)1 and substitute its general solution into equation (2.24)2 . Consequently under
transformations (2.2) and the conditions 0)( hf is an arbitrary function of ,h
0)( =hg and ,)( 1jhj = the similarity solution of the thin film equation (2.1) in tandem
with group (2.23) is
);(),( uytxh = (2.25)
satisfying the equations
( ) ( ) ,0)(4
1)()()()()( )4( =+ uyuuyuyuyfuyuyf ,11Dt>
(2.26)
( ) ( ) ,0)(4
1
)()()()()(
)4(=++
uyuuyuyuyfuyuyf .11Dt
(2.30)
[ ]{ }20)(2120)4( )()()()(3)( 0 uyguyeDuyuyguy uyg +++
,0)(1
)()(3
1514
1
13
)(3 00 =
+
uyguygeDuyDu
fuyDe .11Dt<
In results (2.29)-(2.30), ,02 D ,01
01
15 =gf
D ,01 f ,00 g ,6D ,7D
,02
711
gDDD = ,
1
112
fgD = ,
1
013
fjD = ,
1
114
fjD = ,
0
016
gjD = ,
02
617
gDDD = ,1g 0j and
1j are arbitrary constants, 0ln 111611
1716
++= DtD
Dt
DtDxu and
( )( ) .0171611 ++ DtDxDt Furthermore, 0)( uy owing to the requirement .0xh
Case (2) 00 =g
We consider the subcases
(i) ,07 D (ii) .07 =D
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Subcase (i) 007 =gD
Via the method of Lagrange [24], we solve equation (2.28)1 and substitute its general
solution into equation (2.28)2 . Hence under transformations (2.2) and the conditions
,0)( 1 = fhf 1)( ghg = and ,)( 10 jhjhj += the similarity solution of the thin film
equation (2.1) in tandem with group (2.27) and the constraints 007 =gD is
;0)(),( 11 += tDuytxh (2.31)
satisfying
.0)()()()()( 17161514)4(
=++++ DuyDuyuyDuyDuy (2.32)
In relations (2.31)-(2.32), 0)( uy owing to the requirement 0xh while ,02 D
,07 D ,07
211 =
DDD ,0
17
217 =
fDDD ,01 f ,6D ,
2 7
0212
DjDD = ,
7
613
DDD =
,1
114
f
gD = ,
1
015
f
jD = ,
17
61716
fD
DjDD
= ,1g 0j and 1j are arbitrary constants and
.0132
12 ++= tDtDxu
Subcase (ii) 007 ==gD
We directly solve equation (2.28)1 and substitute its general solution into equation
(2.28)2 , solving the resulting equation using the integrating factor algorithm [48].
Therefore under transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and
,)( 10 jhjhj += the similarity solution of the thin film equation (2.1) in tandem with
group (2.27) and the constraints 007 ==gD is
( );0),(
602
1112
+
+=
DtjD
DtjxDtxh (2.33)
where ,02 D ,6D ,11D 0j and 1j are arbitrary constants such that 0602 +DtjD and
( ) .01112 + DtjxD
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satisfying
[ ]
+
+++210)(
130
)4( )(3
2)()()()( 14 uy
jfuyeDuyuyfuy
uyD
[ ] ,0)()(
18)(
17)(
150016
=+++ uyfuyfuyD eDuyueDeD ,12Dt>
(2.37)
[ ]
+
+++210)(
130
)4( )(3
2)()()()( 14 uy
jfuyeDuyuyfuy
uyD
[ ] ,0)( )(18)(17)(15 0016 =+ uyfuyfuyD eDuyueDeD .12Dt<
In relations (2.36)-(2.37), 0)( uy owing to the requirement .0xh Furthermore,
,02
D ,04
3
1011
=jfD ( ) ,03
21014
= jfD ,00116
= fjD
( ),0
4 110
1017
=
fjf
jfD
( ),0
4
3
110
18
=fjf
D ,04 10
1021
=
jf
jfD ,01 f ,01 j
,6D ,7D ( ),
4
3
102
712
jfD
DD
= ,
1
113
f
gD = ,
1
015
f
jD = ,
3
10
2119
jf
jjD
=
( ),
3
102
620
jfD
DD
=
( )
( ),
3
102
27622
jfD
jDDD
= ,0f ,1g 0j and 2j are arbitrary constants
with ( )( ) .04 1010 jfjf In addition, 02019 ++ DtDx and
( ) .02221221
+= DtjxDtu D
Case (2) 010 = jf
Via the method of Lagrange [24], we solve equation (2.35)1 and substitute its general
solution into equation (2.35)2 . Hence under transformations (2.2) and the conditions
,0)(1
1 =
hj
efhf
hj
eghg1
1)( =
and ,)( 201
jejhj
hj+=
the similarity solution of the
thin film equation (2.1) in tandem with group (2.34) and the constraint 010 = jf is
;0ln1
)(),( 111
= Dtj
uytxh (2.38)
satisfying
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[ ]21121)4( )()()()()( uyjuyDuyuyjuy +++
[ ] ,0)( )(15)(1413 11 =+++ uyjuyj eDuyeDD ,11Dt> (2.39)
[ ]
2
1121
)4(
)()()()()( uyjuyDuyuyjuy +++
[ ] ,0)( )(15)(1413 11 =+ uyjuyj eDuyeDD .11Dt<
In relations (2.38)-(2.39), 0)( uy owing to the requirement .0xh Furthermore,
,02 D ,01
11
15 =jf
D ,01 f ,01 j ,6D ,7D ,12
711
jD
DD = ,
1
112
f
gD =
,1
013
f
jD = ,
112
27614
fjD
jDDD
= ,
12
27616
jD
jDDD
= ,1g 0j and 2j are arbitrary constants,
011
Dt and ( ) .0ln 1116112 +=
DtDDtjxu
Case (3) 04 10 = jf
We consider the subcases (i) ,07 D (ii) .07 =D
Subcase (i) ,04 10 = jf 07 D
By the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.35)1and substitute its general solution into equation (2.35)2. Therefore under
transformations (2.2) and the conditions ,0)( 14
1 = hj
efhf hj
eghg 12
1)( = and
,)( 201 jejhj
hj+= the similarity solution of the thin film equation (2.1) in connection
with group (2.34) and the constraints 04 10 = jf and 07 D is
;0)(),( 11 += tDuytxh (2.40)
satisfying
[ ]21)(2
121
)4( )(2)()()(4)( 1 uyjuyeDuyuyjuy uyj +++
[ ] .0)( )(415)(414)(313 111 =+++ uyjuyjuyj eDuyueDeD (2.41)
In relations (2.40)-(2.41), ,02 D ,07 D ,07
211 =
D
DD ,0
17
1214 =
fD
jDD
,017
215 =
fD
DD ,0
7
1216 =
D
jDD ,01 f ,01 j ,6D ,
1
112
f
gD = ,
1
013
f
jD =
,12
27617
jD
jDDD
= ,
12
618
jD
DD = ,1g 0j and 2j are arbitrary constants, 0182 + Dtjx
and ( ) .017216 += Dtjxeu
tD Furthermore, 0)( uy owing to the requirement
.0xh
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Subcase (ii) 04 710 == Djf
We directly solve equation (2.35)1, substituting its general solution into equation (2.35)2.
Hence under transformations (2.2) and the conditions ,0)( 141 = hjefhf hjeghg 121)( =
and ,)( 201 jejhj hj += the similarity solution of thin film equation (2.1) in association
with group (2.34) and the constraints 04 710 == Djf is
[ ] ;0)(ln1),( 1121
+= tzDtjxj
txh (2.42)
such that
[ ] [ ] [ ] ,0)(2)()()( 513
1
2
0 =++ tzftzgtzjtz ,0112 >+ Dtjx
(2.43)
[ ] [ ] [ ] ,0)(2)()()( 513
1
2
0 =+ tzftzgtzjtz .0112 tz and .0112 + Dtjx
GROUP (V)
Subject to the conditions ( ) ,0)( 0321 += g
fhfhf ( ) 021)( g
fhghg += and
,ln)( 120 jfhjhj ++= the thin film equation (2.1) admits Lie classical group (V),
namely
( ) ,),,( 6001 DtjxgDhtx ++= ,),,( 701 DtgDhtx += ( ) ;0),,( 21 += fhDhtx
(2.44)
where ,01 D ,01 f ,6D ,7D ,2f ,0g ,1g 0j and 1j are arbitrary constants while
.02 + fh
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Similarity Solutions
Group (2.44), the invariant surface condition (2.3) and the thin film equation (2.1) give
( ) ( ) ,021201100 +=++++ fhhDtghDtjxg tx
(2.45)
( ) ( ) ( )
+++
+++
2
2
021
2
03
2100
3xxx
g
xxxxxxxx
gh
fh
ghfhghh
fh
ghfhf
( ) ;0ln 120 =++++ tx hhjfhj
where ,01 D ,01 f ,6D ,7D ,1
611
D
DD = ,
1
712
D
DD = ,2f ,0g ,1g 0j and 1j are
arbitrary constants while .02 + fh As 0=xh forces 0=th in equation (2.45)2 ,
rendering equation (2.45)1inconsistent, we require .0xh
We consider the cases
(1) ,00 g (2) .00 =g
Case (1) 00 g
Via the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.45)1 , substituting its general solution into equation (2.45)2 . Hence under
transformations (2.2) and the conditions ( ) ,0)( 0321 += g
fhfhf ( ) 021)( g
fhghg +=
and ,ln)( 120 jfhjhj ++= the similarity solution of the thin film equation (2.1) in
association with group (2.44) and the constraint 00 g is
;0)(),( 2/1
13
0= fDtuytxh
g (2.46)
satisfying
[ ]
[ ]
++
+
)(
)()(
)()(
)()(3)(
2
02
14
0
)4(
0 uy
uyguy
uy
D
uy
uyuyguy
g
[ ] [ ] ,0)(1)()(ln
)(
1
0
103
10
=
+++ uyg
uyujuyjuyf g
,13Dt>
(2.47)
[ ]
[ ]
++
+)(
)()(
)()(
)()(3)(
2
02
140
)4(
0 uy
uyguy
uy
D
uy
uyuyguy
g
[ ] [ ] ,0)(1)()(ln
)(
1
0
103
10
=
++ uyg
uyujuyjuyf g
.13Dt<
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In relations (2.46)-(2.47), 0)( uy owing to the requirement .0xh Furthermore,
,01 D ,01 f ,00 g ,6D ,7D ,01
713
gD
DD = ,
1
114
f
gD = ,
0
015
g
jD = ,
01
616
gD
DD =
,2
f ,1
g 0
j and1
j are arbitrary constants, 0ln1315
13
1615
++= DtD
Dt
DtDxu and
( )( ) .0161513 ++ DtDxDt
Case (2) 00 =g
We consider the subcases (i) ,07 D (ii) .07 =D
Subcase (i) 007 =gD
By the method of Lagrange [24], we solve equation (2.45)1 , substituting its general
solution into equation (2.45)2. Therefore under transformations (2.2) and the conditions
,0)( 1 = fhf 1)( ghg = and ,ln)( 120 jfhjhj ++= the similarity solution of the thin
film equation (2.1) in tandem with group (2.44) and the constraints 007 =gD is
;0)(),( 2/ 12 = feuytxh Dt (2.48)
satisfying
[ ] .0)()()(ln)()( 16151413)4(
=++++ uyDuyDuyDuyDuy (2.49)
In relations (2.48)-(2.49), 0)( uy owing to the requirement .0xh Furthermore,
,01 D ,07 D ,01
712 =
D
DD ,0
17
116 =
fD
DD ,01 f ,6D ,
1
113
f
gD = ,
1
014
f
jD =
,17
61715
fD
DjDD
= ,
2 7
0117
D
jDD = ,
7
618
D
DD = ,2f ,1g 0j and 1j are arbitrary
constants and .0182
17 ++= tDtDxu
Subcase (ii) 007 ==gD
We directly solve equation (2.45)1, substituting its general solution into equation (2.45)2.
Hence under transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and
,ln)( 120 jfhjhj ++= the similarity solution of the thin film equation (2.1) in
connection with group (2.44) and the constraints 007 ==gD is
;0)(),( 2601
1
= +
fetytxh DtjD
xD
(2.50)
satisfying
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( )
.0)()(ln)(3
601
14
601
1310
601
1=
++
+++
++ ty
DtjD
D
DtjD
Djtyj
DtjD
Dty (2.51)
In relations (2.50)-(2.51), ,01 D ,03
1114 = DfD ,01 f ,6D ,1113 gDD = ,2f ,1g
0j and 1j are arbitrary constants such that 0601 +DtjD while 0)( ty owing to the
requirement .0xh
GROUP (VI)
Under the conditions ( ) ,0)( 021 += f
fhfhf ( ) 32
21
10
)(jf
fhghg+
+= and
( ) ,)( 2201 jfhjhj
j++= the thin film equation (2.1) yields Lie classical group (VI),
namely
( )[ ] ,33
),,( 621101 Dtjjxjf
Dhtx += ( ) ,4
3),,( 710
1 DtjfD
htx +=
(2.52)
( ) ;0),,( 21 += fhDhtx
where ,01 D ,01 f ,01 j ,6D ,7D ,0f ,2f ,1g 0j and 2j are arbitrary constants
while .02 + fh
Similarity Solutions
Group (2.52), the invariant surface condition (2.3) and the thin film equation (2.1) imply
( ) ( ) ,021514131211 +=++++ fhhDtDhDtDxD tx (2.53)
( ) ( )( )
( ) txxx
jf
xxxxxxxx
fhh
fh
jfhfhghh
fh
fhfhf +
+
+++
+++
+2
2
103
2
21
2
021
3
2100
( )[ ] ;0220 1 =+++ xj hjfhj
where ,01 D ,01 f ,01 j ,6D ,7D ,3
1011
jfD
= ,2112 jjD = ,
1
613
D
DD =
,3
4 1014
jfD
= ,
1
715
D
DD = ,0f ,2f ,1g 0j and 2j are arbitrary constants while
.02 + fh Since 0=xh forces 0=th in equation (2.53)2 , rendering equation (2.53)1
inconsistent, we require .0
xh
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We consider the cases
(1) ( )( ) ,04 1010 jfjf (2) ,010 = jf (3) .04 10 = jf
Case (1) ( )( ) 04 1010 jfjf
Via the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.53)1 , substituting its general solution into equation (2.53)2 . Hence under
transformations (2.2) and the conditions ( ) ,0)( 021 += f
fhfhf ( ) 32
21
10
)(jf
fhghg+
+=
and ( ) ,)( 2201 jfhjhj
j++= the similarity solution of the thin film equation (2.1) in
association with group (2.52) and the constraints ( )( ) 04 1010 jfjf is
;0)(),( 2/1
1614
= fDtuytxh D
(2.54)
such that
[ ] [ ]
+
++
+)(
)(
3
2)()(
)(
)()()(
2
10170
)4( 18
uy
uyjfuyuyD
uy
uyuyfuy
D
[ ] [ ] [ ] ,0)()()()( 0020 1222119 =+++ ffD
uyDuyuyuDuyD ,16Dt>
(2.55)
[ ] [ ]
++++
)()(
32)()(
)()()()(
2
10170
)4( 18
uyuyjfuyuyD
uyuyuyfuy
D
[ ] [ ] [ ] ,0)()()()( 0020 1222119 =+ ffD
uyDuyuyuDuyD .16Dt<
In relations (2.54)-(2.55), 0)( uy owing to the requirement .0xh Furthermore,
,01 D ,03
4 1014
=
jfD ( ) ,0
3
21018 = jfD ,00120 = fjD
( ),0
4 110
1021
=
fjfjfD
( ),0
43
110
22
=fjf
D ,04 10
1025
=
jfjfD ,01 f
,01 j ,6D ,7D ( ),
4
3
101
716
jfD
DD
= ,
1
117
f
gD = ,
1
019
f
jD = ,
3
10
2123
jf
jjD
=
( ),
3
101
624
jfD
DD
=
( )
( ),
3
101
27626
jfD
jDDD
= ,0f ,2f ,1g 0j and 2j are arbitrary constants
with ( )( ) ,04 1010 jfjf 02423 ++ DtDx and ( ) .02621625
+= DtjxDtu D
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Case (2) 010 = jf
By the method of Lagrange [24], we solve equation (2.53)1 , substituting its general
solution into equation (2.53)2 . Hence under transformations (2.2) and the conditions
( ) ,0)(121 +=
j
fhfhf ( )121)(
j
fhghg += and ( ) ,)( 2201 jfhjhj
j
++= the similarity
solution of the thin film equation (2.1) in tandem with group (2.52) and the constraint
010 = jf is
;0)(),( 2/1
16
1=
fDtuytxh j
(2.56)
satisfying
[ ]
++
+)(
)()(
)(
)()()(
2
1171
)4(
uy
uyjuyD
uy
uyuyjuy [ ] )()( 11819 uyuyDD
j++
[ ] ,0)( 1120 =+ j
uyD ,16Dt>
(2.57)
[ ]
++
+)(
)()(
)(
)()()(
2
1171
)4(
uy
uyjuyD
uy
uyuyjuy [ ] )()( 11819 uyuyDD
j+
[ ] ,0)( 1120 = j
uyD .16Dt<
In relations (2.56)-(2.57), 0)( uy owing to the requirement .0xh Furthermore,
,01
D ,0
1
1120
=jfD ,01
f ,01
j ,6D ,7D ,11
7
16 jD
D
D =
,1
1
17 f
g
D =
,111
27618
fjD
jDDD
= ,
1
019
f
jD = ,
11
27621
jD
jDDD
= ,2f ,1g 0j and 2j are arbitrary
constants, 016 Dt and ( ) .0ln 1621162 += DtDDtjxu
Case (3) 04 10 = jf
We consider the subcases (i) ,07 D (ii) .07 =D
Subcase (i) ,04 10
= jf 07
D
Via the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.53)1, substituting its general solution into equation (2.53)2. Therefore under
transformations (2.2) and the conditions ( ) ,0)( 1421 += j
fhfhf ( ) 1221)( j
fhghg +=
and ( ) ,)( 2201 jfhjhj
j++= the similarity solution of the thin film equation (2.1) in
connection with group (2.52) and the constraints 04 10 = jf and 07 D is
;0)(),(2
/ 15 = feuytxh Dt
(2.58)
such that
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[ ] [ ]
++
+
)(
)(2)()(
)(
)()(4)(
2
1
2
161
)4( 1
uy
uyjuyuyD
uy
uyuyjuy
j
[ ] [ ] )()()( 11 4183
17 uyuyuDuyD jj
++ [ ] .0)( 14119 =+
juyD (2.59)
In relations (2.58)-(2.59), 0)( uy owing to the requirement .0xh Furthermore,
,01 D ,07 D ,01
715 =
D
DD ,0
17
1118 =
fD
jDD ,0
17
119 =
fD
DD ,0
7
1122 =
D
jDD
,01 f ,01 j ,6D ,1
116
f
gD = ,
1
017
f
jD = ,
11
620
jD
DD = ,
11
27621
jD
jDDD
= ,2f ,1g
0j and 2j are arbitrary constants, 0202 + Dtjx and ( ) .022212 += tDeDtjxu
Subcase (ii) 04 710 == Djf
We directly solve equation (2.53)1, substituting its general solution into equation (2.53)2.
Hence under transformations (2.2) and the conditions ( ) ,0)( 1421 += j
fhfhf
( ) 1221)( j
fhghg += and ( ) ,)( 2201 jfhjhj
j++= the similarity solution of the thin film
equation (2.1) in tandem with group (2.52) and the constraints 04 710 == Djf is
;0)(),( 2/1
162
1+= fDtjxtytxh
j (2.60)
satisfying
[ ] [ ] [ ] ,0)()()()(
)(111 4
19
2
1817 =+++ jjj
tyDtyDtyDty
ty ,0162 >+ Dtjx
(2.61)
[ ] [ ] [ ] ,0)()()()(
)(111 4
19
2
1817 =++ jjj
tyDtyDtyDty
ty .0162
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GROUP (VII)
Under the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the thin film equation (2.1)
generates Lie classical group (VII), namely
( ) ,34
),,( 613 Dtjx
Dhtx ++= ,),,( 73 DtDhtx += );,(),,( 1 txbhDhtx += (2.62)
such that
;011 =++ txxxxx bbjbf (2.63)
where ,01 f ,1D ,3D ,6D 7D and 1j are arbitrary constants.
Equation (2.63) admits the travelling wave solution of velocity ,1j namely
( )=
=3
0
1 ;),(n
n
n tjxdtxb (2.64)
where ,0d ,1d ,2d 3d and 1j are arbitrary constants.
Similarity Solutions
We construct similarity solutions of the thin film equation (2.1) for the cases
(a) ,03 D (b) ,031 DD ,)(),( 812 DtjxDtxb +=
(c) ,03 =D (d) ,031 =DD ,)(),( 812 DtjxDtxb +=
(e) ,013 =DD ,)(),( 812 DtjxDtxb += (f) ,031 ==DD .)(),( 812 DtjxDtxb +=
Similarity Solutions for Case (a)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) imply
( ) ( ) ),,(34
173613 txbhDhDtDhDtjx
Dtx +=++
++ ;011 =++ txxxxx hhjhf (2.65)
where ,03 D ,01 f ,1D ,6D 7D and 1j are arbitrary constants while ),( txb satisfies
equation (2.63), 073 +DtD and ( ) .034
613
++ DtjxD
As 0=xh forces 0=th in
equation (2.65)2, giving =),( txh constant, we require 0xh for system (2.65) to yield
nonconstant similarity solutions.
As case (e) includes the subcase ,0),(1 =+ txbhD we consider only .0),(1 + txbhD
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Similarity Solutions for Case (b)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) give
( ) ( ),)(3
4 812173613 DtjxDhDhDtDhDtjx
D
tx ++=++
++ ;0
11 =++
txxxxx hhjhf
(2.68)
where ,01 D ,03 D ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants while
073 +DtD and ( ) .034
613
++ DtjxD
As 0=xh gives 0=th in equation (2.68)2 ,
giving =),( txh constant, we require 0xh for system (2.68) to admit nonconstant
solutions.
We consider the subcases
(1) ( ) 8121 DtjxDhD ++ ,0 (2) ( ) 8121 DtjxDhD ++ .0=
Subcase (1) ( ) 8121 DtjxDhD ++ 0
Via the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.68)1 , substituting its general solution into equation (2.68)2 . Hence under
transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the
similarity solution of the thin film equation (2.1) in tandem with group (2.62) and the
constraints ,031 DD ( ) 08121 ++ DtjxDhD and 812 )(),( DtjxDtxb += is
( ) ,0)(),( 13121119110
++++= DDtjxDDtuytxh D
,4 13 DD
(2.69)
( ) ,0ln)(),( 1614121154/1
142 +++++= DDtDtjxDDtuytxh ;04 13 = DD
satisfying
,0)()(4
1)( 1101
)4(
11 =+ uyDuyuuyf ,0)(4
1)(
4
1)( 1522
)4(
21 =++ uDuyuyuuyf
,9Dt >
(2.70)
,0)()(4
1)( 1101
)4(
11 =+ uyDuyuuyf ,0)(4
1)(
4
1)( 1522
)4(
21 =+ uDuyuyuuyf
.9Dt <
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In relations (2.69)-(2.70), ,01 D ,03 D ,03
110 =
D
DD ,01 f ,1j ,2D ,6D ,7D
,8D ,3
79
D
DD = ,
4
4
13
211
DD
DD
= ,
1
17612
D
jDDD
= ,
1
813
D
DD = ,
4 1
714
D
DD =
,4 1
215
D
DD = ( ) ,2
1
81176216
D
DDjDDDD = ( ) ,4
3
17617
D
jDDD = ( )131
3218
4DDD
DDD
=
and( )
( )131
176219
4
4
DDD
jDDDD
= are arbitrary constants. Furthermore, ( ) 03
461
3++ Dtjx
D
and ( ) .01714/1
9 ++=
DtjxDtu
As 0=xh gives 0=th in equation (2.68)2 , rendering equation (2.68)1 inconsistent for
this subcase, we require .0xh Accordingly, 0)( 1114/1
9
10++
DuyDt D
and
.0ln)( 14152 ++ DtDuy In addition, ( ) 0)( 191189110
+++ DtjxDDtuy D
and
( )( ) .04ln)( 14121154/1
142 +++++ DtDtjxDDtuy
Subcase (2) ( ) 08121 =++ DtjxDhD
As the constraint ( ) 8121 DtjxDhD = identically satisfies equation (2.68)2but forces
02 =D in equation (2.68)1, system (2.68) yields only the constant solution. Hence under
transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the sole
similarity solution of the thin film equation (2.1) in tandem with group (2.62) and the
constraints ( ) 0812131 =++ DtjxDhDDD and 812 )(),( DtjxDtxb += is the
constant solution.
Similarity Solutions for Case (c)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) imply
),,(176 txbhDhDhD tx +=+ ;011 =++ txxxxx hhjhf (2.71)
where ,01 f ,1D ,6D 7D and 1j are arbitrary constants while ),( txb satisfies
equation (2.63). As 0=xh forces 0=th in equation (2.71)2, giving =),( txh constant,
we require 0xh for system (2.71) to admit nonconstant solutions.
The subcases arising are
(1) [ ] ,0),(17
+ txbhDD (2) [ ] ,0),(716
=+ DtxbhDD (3) ,0),(17
=+ txbhDD
(4) ,0),(176 =+= txbhDDD (5) .0),(176 =+== txbhDDD
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[ ] [ ] .0),(),()()( 88 912111019 =+++++++
x
c
D
t
xD
xxxxxx detbDbetxbDbDbDbftyDty
(2.75)
In results (2.74)-(2.75), [ ] 0),()()( + txKtyty and [ ] ,0),(),()( 88 ++ xD
etxbtxKtyD
noting that as 0=xh gives 0=th in equation (2.71)2 , rendering equation (2.71)1
inconsistent for this subcase, we require .0xh Furthermore, ( ) ,,),(8
=
x
c
DdtbetxK
),( txb satisfies equation (2.63) and ,06 D ,01 f ,c ,1D ,6
1
8D
DD =
,4
6
4
11
3
611
9D
DfDDjD
+= ,
6
11
10D
fDD = ,
2
6
2
11
11D
DfD =
3
6
3
11
3
61
12D
DfDjD
+= and 1j are
arbitrary constants.
Similarity Solutions for Case (d)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) give
( ) ,812176 DtjxDhDhDhD tx ++=+ ;011 =++ txxxxx hhjhf (2.76)
where ,01 D ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants. As 0=xh
forces 0=th in equation (2.76)2 , giving =),( txh constant, we require 0xh for
system (2.76) to generate nonconstant solutions.
We consider the subcases
(1) ,07 D ( ) ,08121 ++ DtjxDhD (2) ,076 =DD ( ) ,08121 ++ DtjxDhD
(3) ( ) .08121 =++ DtjxDhD
Subcase (1) ,07 D ( ) 08121 ++ DtjxDhD
By the method of Lagrange [24], the integrating factor algorithm [48] and the
Mathematica program [54], we obtain the general solution of system (2.76). Hence under
transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the
similarity solution of the thin film equation (2.1) in tandem with group (2.62) and the
constraints ,0371 =DDD ( ) 8121 DtjxDhD ++ 0 and ( ) 812),( DtjxDtxb += is
( ) ( ) ;0),( 1214
1
11109 ++=
=
DtjxDedetxh
n
tDxc
n
tD n (2.77)
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where ,01 D ,07 D ,07
1
9 =D
DD ,01 f ,2D ,6D ,8D ,
7
6
10D
DD = ,
1
2
11D
DD =
( ),
2
1
176281
12D
jDDDDDD
+=
( ),
1
1762
13D
jDDDD
= ,1j nc and nd are arbitrary
constants for all { }.4,3,2,1n
In addition, the travelling waves of velocity ,10D namely( )
,04
1
10 =
n
tDxc
nned are such
that ( ) 04
1
10 =
n
tDxc
nnnedc as the contradiction 01 =D otherwise occurs. Furthermore,
( ).013
4
1
1109 +
=
DedeD
n
tDxc
n
tD n We require ( ) 011
4
1
109 +=
Dedce
n
tDxc
nn
tD n as 0xh is
necessary for equation (2.76)1to be consistent for this subcase.
For the scenario ,176 jDD = ,0
4/1
17
11
=
fD
Dc ,0
4/1
17
12
=
fD
Dc
0
4/1
17
13
=
fD
Dic and ,0
4/1
17
14
=
fD
Dic where .1=i
Subcase (2) ,076 =DD ( ) 08121 ++ DtjxDhD
We solve system (2.76) via the method of Lagrange [24] and the integrating factor
algorithm [48]. Hence under transformations (2.2) and the conditions ,0)( 1 = fhf
0)( =hg and ,)( 1jhj = the similarity solution of the thin film equation (2.1) in tandem
with group (2.62) and the constraints ,07361 == DDDD ( ) 08121 ++ DtjxDhD
and ( ) 812),( DtjxDtxb += is
( ) ( ) ;0),( 1311291110 ++=
DtjxDeDtxh
tDxD (2.78)
where ,01 D ,06 D ,09 D ,06
110 = D
DD ,01 f ,2D ,8D ,3
6
3
61
3
1111
D
DjDfD
+
=
,1
212
D
DD =
2
1
628113
D
DDDDD
+= and 1j are arbitrary constants. Furthermore,
( ) 0121091110 +
DeDD
tDxD as we require 0xh for equation (2.76)1to be consistent for
this subcase.
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Subcase (3) ( ) 08121 =++ DtjxDhD
The constraint ( ) 08121 =++ DtjxDhD identically satisfies equation (2.76)2 but
causes equation (2.76)1to give the scenarios
(i) ,02 =D (ii) .176 jDD =
Scenario (i) 0281 ==+ DDhD
Under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj =
the similarity solution of the thin film equation (2.1) in connection with group (2.62) and
the constraints 081321 =+== DhDDDD and 8),( Dtxb = is the constant solution.
Scenario (ii) ,176 jDD = ( ) 08121 =++ DtjxDhD
Under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj =
the similarity solution of the thin film equation (2.1) in association with group (2.62) and
the constraints ,031 =DD ,176 jDD = ( ) 08121 =++ DtjxDhD and
( ) 812),( DtjxDtxb += is the travelling wave of velocity ,1j namely
( ) ;),( 1019 DtjxDtxh += (2.79)
where ,01 D ,2D ,8D ,1
29
DDD =
1
810
DDD = and 1j are arbitrary constants. We
require 09 D for solution (2.79) to be nonconstant.
From the constraint 176 jDD = on this case, it follows that 067 =DD forces ,01 =j
reducing the travelling wave (2.79) to a steady state solution.
Similarity Solutions for Case (e)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) imply
( ) ( ) ( ) ,34
81273613 DtjxDhDtDhDtjx
Dtx +=++
++ ;011 =++ txxxxx hhjhf (2.80)
where ,03 D ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants while
073 +DtD and ( ) .034
613
++ DtjxD
As 0=xh gives 0=th in equation (2.80)2 ,
forcing =),( txh constant, we require 0xh for system (2.80) to admit nonconstant
solutions.
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By the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.80)1 , substituting its solution into equation (2.80)2 . Therefore under
transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the
similarity solution of the thin film equation (2.1) in tandem with group (2.62) and theconstraints 013 =DD and ( ) 812),( DtjxDtxb += is
( ) ;ln)(),( 91211110 DtDDtjxDuytxh ++++= (2.81)
satisfying
,0)(4
1)( 12
)4(
1 =+ Duyuuyf ,9Dt >
(2.82)
,0)(4
1
)( 12)4(
1 =+ Duyuuyf .9Dt <
In relations (2.81)-(2.82), ,03 D ,01 f ,2D ,6D ,7D ,8D ,3
79
D
DD = ,
4
3
210
D
DD =
( ),
4
3
17611
D
jDDD
=
( )2
3
83617212
4
D
DDDjDDD
+= and 1j are arbitrary constants,
( ) 034
613
++ DtjxD
and ( ) .01114/1
9 ++=
DtjxDtu For solution (2.81) to be
nonconstant, we require .0)( 104/1
9 ++
DuyDt
Similarity Solutions for Case (f)
Group (2.62), the invariant surface condition (2.3) and the thin film equation (2.1) give
( ) ,81276 DtjxDhDhD tx +=+ ;011 =++ txxxxx hhjhf (2.83)
where ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants. As 0=xh forces 0=th
in equation (2.83)2 , forcing =),( txh constant, we require 0xh for system (2.83) to
generate nonconstant solutions.
As no similarity solutions arise for the thin film equation (2.1) when ,076 ==DD we
consider only the subcases
(1) ,07 D (2) .076 =DD
Subcase (1) 07 D
Via the method of Lagrange [24] and the Mathematica program [54], we obtain the
general solution of system (2.83). Hence under transformations (2.2) and the conditions,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the similarity solution of thin film equation (2.1)
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in connection with group (2.62) and the constraints 0317 == DDD and
( ) 812),( DtjxDtxb += is
( ) ( ) ( ) ( ) ,),( 2141312
2
91191021
91 tDtDxDtDxDtDxDeddtxh tDxc
++++++=
,176 jDD
(2.84)
( ) ( )[ ] ,),( 131125
0
1 tDtjxDtjxdtxhn
n
n ++==
.176 jDD =
In solutions (2.84), ,0
3/1
17
6171
=
fD
DjDc ,07 D ,01 f ,2D ,6D ,8D
,7
69
D
D
D = ,176
810
jDD
D
D = ( ) ,2 176
211
jDD
D
D = ,7
212
D
D
D = ,7
813
D
D
D =
( ),
22
7
176214
D
jDDDD
+= ,0d ,1d ,2d ,3d ,
24 17
84
fD
Dd =
17
25
120 fD
Dd = and 1j are
arbitrary constants.
Furthermore, ( ) ( ) 022 9121101191211 +++ tDxc
edcDtDDDxD and
( ) 0125
1
1
1 +=
tDtjxnd
n
n
n as we require 0xh for solutions (2.84) to be nonconstant.
In addition, ( ) ( ) 0291
2110911 ++ tDxcedcDtDxD and ( ) .05
1
11 =
n
nn tjxnd
Subcase (2) 076 =DD
We directly solve equation (2.83)1, substituting its general solution into equation (2.83)2.
Therefore under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and
,)( 1jhj = the similarity solution of thin film equation (2.1) in tandem with group (2.62)
and the constraints 07316 === DDDD and ( ) 812),( DtjxDtxb += is the
travelling wave of velocity ,1j namely
( ) ( ) ;),( 111102
19 DtjxDtjxDtxh ++= (2.85)
where ,06 D ,2D ,8D ,2 6
29
D
DD = ,
6
810
D
DD = 11D and 1j are arbitrary constants. For
solution (2.85) to be nonconstant requires ( ) .0812 + DtjxD
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GROUP (VIII)
Under conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the thin film equation (2.1)
yields Lie classical group (VIII), namely
,),,( 6Dhtx = ,),,( 7Dhtx = );,(),,( 1 txbhDhtx += (2.86)
such that
;0111 =++ txxxxxxx bbjbgbf (2.87)
where ,01 f ,1D ,6D ,7D 1g and 1j are arbitrary constants.
Equation (2.87) admits the travelling wave solution of velocity ,1j namely
( ) ( ) ( );),( 1515 43121tjxdtjxd
ededtjxddtxb
+++= (2.88)
where ,01
15 =
f
gd ,01 f ,01 g ,1d ,2d ,3d 4d and 1j are arbitrary constants.
As equation (2.63) is a special case of equation (2.87) with ,01 =g solution (2.64) of
equation (2.63) is also a solution of equation (2.87) under the restriction .01 =g
Similarity Solutions
We obtain similarity solutions of the thin film equation (2.1) for the cases
(a) ),( txb is an arbitrary solution of equation (2.87),
(b) ,01 D ,)(),( 812 DtjxDtxb += (c) ,01 =D .)(),( 812 DtjxDtxb +=
Similarity Solutions for Case (a)
Group (2.86), the invariant surface condition (2.3) and the thin film equation (2.1) give
),,(176 txbhDhDhD tx +=+ ;0111 =++ txxxxxxx hhjhghf (2.89)
where ,01 f ,1D ,6D ,7D 1g and 1j are arbitrary constants while ),( txb is an
arbitrary solution of equation (2.87). As 0=xh gives 0=th in equation (2.89)2, forcing
=),( txh constant, we require 0xh for system (2.89) to generate nonconstant
solutions.
The subcases occurring are
(1) [ ] ,0),(17 + txbhDD (2) [ ] ,0),( 716 =+ DtxbhDD (3) ,0),(17 =+ txbhDD
(4) ,0),(176 =+= txbhDDD (5) .0),(176 =+== txbhDDD
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As case (c) includes subcases (3)-(5), we consider only subcases (1) and (2).
Subcase (1) [ ] 0),(17 + txbhDD
By the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.89)1 , substituting its general solution into equation (2.89)2 . Hence under
transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the
similarity solution of the thin film equation (2.1) in association with group (2.86) and the
constraints [ ] 0),(17 + txbhDD (with ),( txb an arbitrary solution of equation (2.87)) is
[ ] ;0),()(1
),( 8
7
+= utKuyeD
txh tD
(2.90)
satisfying
.0),(),()()()()( 8891)4(1 8 =++++ utKDetxbuyDuyDuyguyf tD (2.91)
In relations (2.90)-(2.91), ,07 D ,01 f ,c ,1D ,6D ,7
18
D
DD = ,
7
6179
D
DjDD
=
,7
610
D
DD = 1g and 1j are arbitrary constants and .10tDxu = Furthermore,
( ) ,,),( 10108 +=
t
c
DdDtDxbeutK [ ] 0),(),()(88 ++ txbutKuyeD
tDand ),( txb
is an arbitrary solution of equation (2.87).
In addition, 0)( uy is a travelling wave of velocity 10D such that 0)( uy as
equation (2.91) otherwise leads to the contradiction [ ] .0),(),()(88 =++ txbutKuyeD tD
Furthermore, 0)(
+
x
Kuy as we require 0xh for equation (2.89)1to be consistent
for this subcase.
This subcase includes subcase (1) of case (c) in relation to group (2.62) as results (2.72)-
(2.73) and equation (2.63) are a special case of results (2.90)-(2.91) and equation (2.87)
respectively under the restriction .01 =g
Subcase (2) [ ] 0),( 716 =+ DtxbhDD
Via the method of Lagrange [24] and the integrating factor algorithm [48], we solve
equation (2.89)1, substituting its general solution into equation (2.89)2. Therefore under
transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the
similarity solution of the thin film equation (2.1) in tandem with group (2.86) and theconstraints [ ] 0),( 716 =+ DtxbhDD (with ),( txb satisfying equation (2.87)) is
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[ ] ;0),()(1
),( 8
6
+= txKtyeD
txh xD (2.92)
such that
[ ] [ ] .0),(),()()(88
912111019 =+++++++
x
c
D
t
xD
xxxxxx detbDbetxbDbDbDbftyDty
(2.93)
In results (2.92)-(2.93), [ ] 0),()()( + txKtyty and [ ] ,0),(),()( 88 ++ xD
etxbtxKtyD
noting that we require 0xh for equation (2.89)1 to be consistent for this subcase.
Furthermore, ( ) ,,),( 8
=
x
c
DdtbetxK ),( txb satisfies equation (2.87) and ,06 D
,01 f ,c ,1g ,1j ,1D ,6
18
D
DD =
( ),
4
6
3
611
2
611
4
119
D
DDjDDgDfD
+= ,
6
1110
D
fDD =
2
6
2
61
2
1111
D
DgDfD
= and
3
6
3
61
2
611
3
1112
D
DjDDgDfD
+= are arbitrary constants.
This subcase includes subcase (2) of case (c) in relation to group (2.62) as results (2.74)-
(2.75) and equation (2.63) are a special case of results (2.92)-(2.93) and equation (2.87)
respectively under the restriction .01 =g
Similarity Solutions for Case (b)
Group (2.86), the invariant surface condition (2.3) and the thin film equation (2.1) imply
( ) ,812176 DtjxDhDhDhD tx ++=+ ;0111 =++ txxxxxxx hhjhghf (2.94)
where ,01 D ,01 f ,2D ,6D ,7D ,8D 1g and 1j are arbitrary constants. As 0=xh
forces 0=th in equation (2.94)2 , giving =),( txh constant, we require 0xh for
system (2.94) to admit nonconstant solutions.
We consider the subcases
(1) ,07 D ( ) ,08121 ++ DtjxDhD (2) ,076 =DD ( ) ,08121 ++ DtjxDhD
(3) ( ) .08121 =++ DtjxDhD
Subcase (1) ,07 D ( ) 08121 ++ DtjxDhD
By the method of Lagrange [24], the integrating factor algorithm [48] and the
Mathematica program [54], we solve system (2.94). Hence under transformations (2.2)
and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the similarity solution of the
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thin film equation (2.1) in conjunction with group (2.86) and the constraints ,071 DD
( ) 08121 ++ DtjxDhD and ( ) 812),( DtjxDtxb += is
( ) ( ) ;0),( 1214
1
11109 ++=
=
DtjxDedetxh
n
tDxc
n
tD n (2.95)
where ,01 D ,07 D ,07
19 =
D
DD ,nc ,nd ,1j ,2D ,6D ,8D ,
7
610
D
DD =
,1
211
D
DD =
( )2
1
17628112
D
jDDDDDD
+= and
( )
1
176213
D
jDDDD
= are arbitrary
constants for all { }.4,3,2,1n
Furthermore, ( ) 04
1
10 =
n
tDxc
nnnedc as the contradiction 01 =D otherwise arises. In
addition,( )
013
4
1
1109 +
=
DedeD
n
tDxc
n
tD n and as 0xh is necessary for equation (2.94)1
to be consistent for this subcase, ( ) .011
4
1
109 +=
Dedce
n
tDxc
nn
tD n
This subcase includes subcase (1) of case (d) for group (2.62) as solution (2.77) is a
special case of solution (2.95) with .01 =g
Subcase (2) ,076 =DD ( ) 08121 ++ DtjxDhD
Via the method of Lagrange [24] and the integrating factor algorithm [48], we obtain the
general solution of system (2.94). Therefore under transformations (2.2) and the
conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the similarity solution of the thin
film equation (2.1) in tandem with group (2.86) and the constraints ,0761 =DDD
( ) 08121 ++ DtjxDhD and ( ) 812),( DtjxDtxb += is the sum of two travelling
waves with respective velocities 1j and ,11D namely
( ) ( ) ;0),( 1311291110 ++=
DtjxDeDtxh
tDxD (2.96)
where ,01 D ,06 D ,09 D ,06
110 =
D
DD ,01 f ,2D ,8D
,3
6
3
61
2
611
3
1111
D
DjDDgDfD
+= ,
1
212
D
DD = ,
2
1
628113
D
DDDDD
+= 1g and 1j are
arbitrary constants.
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Furthermore, ( ) 0121091110 +
DeDD tDxD
as ( ) 08121 ++ DtjxDhD and as we require
0xh for equation (2.94)1 to be consistent for this subcase. For the case
,02
1011 = Dfg solution (2.96) reduces to a single travelling wave of velocity .1j
This subcase incorporates subcase (2) of case (d) for group (2.62) as solution (2.78) is a
special case of solution (2.96) with .01 =g
Subcase (3) ( ) 08121 =++ DtjxDhD
The constraint ( ) 08121 =++ DtjxDhD identically satisfies equation (2.94)2 but
causes equation (2.94)1to give the scenarios
(i) ,02 =D (ii) .176 jDD =
Scenario (i) 0281 ==+ DDhD
Under transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj =
the constant solution is the sole similarity solution of the thin film equation (2.1) in
tandem with group (2.86) and the constraints 08121 =+= DhDDD and .),( 8Dtxb =
Scenario (ii) ( ) ,08121 =++ DtjxDhD 176 jDD
=
Under transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj =
the similarity solution of the thin film equation (2.1) in connection with group (2.86) and
constraints ,01761 = jDDD ( ) 08121 =++ DtjxDhD and ( ) 812),( DtjxDtxb +=
is the travelling wave of velocity ,1j namely
( ) ;),( 1019 DtjxDtxh += (2.97)
where ,01 D ,2D ,8D ,1
29
D
DD =
1
810
D
DD = and 1j are arbitrary constants. For
solution (2.97) to be nonconstant requires .09 D
Subcase (3) of case (b) for group (2.86) generates results identical to those of subcase (3)
for case (d) in relation to group (2.62).
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Similarity Solutions for Case (c)
Group (2.86), the invariant surface condition (2.3) and the thin film equation (2.1) give
( ) ,81276 DtjxDhDhD tx +=+ ;0111 =++ txxxxxxx hhjhghf (2.98)
where ,01 f ,2D ,6D ,7D ,8D 1g and 1j are arbitrary constants. As 0=xh gives
0=th in equation (2.98)2 , forcing =),( txh constant, we require 0xh for system
(2.98) to yield nonconstant solutions.
As no similarity solutions occur for the thin film equation (2.1) when ,076 ==DD we
consider only the subcases
(1) ,07 D (2) .076 =DD
Subcase (1) 07 D
By the method of Lagrange [24] and the Mathematica program [54], we solve system
(2.98) for the case .01 g Hence under transformations (2.2) and the conditions
,0)( 1 = fhf 0)( 1 =ghg and ,)( 1jhj = the similarity solution of the thin film
equation (2.1) in tandem with group (2.86) and the constraints 017 =DD and
( ) 812),( DtjxDtxb += is
( ) ( ) ( ) ( ) ,),( 21413122
911910
4
2
19 tDtDxDtDxDtDxDeddtxh
n
tDxc
nn ++++++=
=
,176 jDD
(2.99)
( ) ( ) ( ) ( )[ ] ,),( 131126
5
14
1
1
1115 tDtjxDedtjxdtxh
n
tjxD
n
n
n
n
n
+++= =
=
.176 jDD =
In solutions (2.99), ,07
D ,01
1
15
=
f
gD ,0
1
f ,01
g ,2
c ,3
c ,4
c ,n
d ,1
j ,2
D
,6D ,8D ,7
69
D
DD =
( )
( ),
2
176
172176810
jDD
gDDjDDDD
=
( ),
2 176
211
jDD
DD
= ,
7
212
D
DD =
7
813
D
DD = and
( )2
7
176214
2D
jDDDD
+= are arbitrary constants for all { }6,5,4,3,2,1n .
For solutions (2.99) to be nonconstant, we require .0xh Therefore,
( ) ( ) ( ) ( ) ( ) 011 126
5
1
15
4
2
2
1115 ++
=
=
tDeDdtjxdn
n
tjxDn
n
n
n
n
n
and
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( ) ( ) .022 1011912114
2
9 +++=
DtDDDxDedc
n
tDxc
nnn Nonconstancy of solutions (2.99)1
and (2.99)2 further requires( ) ( ) 02 10911
4
2
9 ++=
DtDxDedc
n
tDxc
nnn and
( ) ( ) ( ) ( ) ( ) 011 6
5
1
15
4
2
2
1115 +
=
=
n
tjxDn
n
n
n
n
n
eDdtjxdn respectively.
Solutions (2.84) are the similarity solutions of the thin film equation (2.1) for this
subcase when 01 =g (in tandem with group (2.86) under transformations (2.2) and the
conditions ,0)( 1 = fhf 0)( =hg and 1)( jhj = ).
Subcase (2) 076 =DD
We directly solve equation (2.98)1, substituting its general solution into equation (2.98)2
and solving the resulting equation. Hence under transformations (2.2) and the conditions
,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the similarity solution of the thin film equation
(2.1) in association with group (2.86) and the constraints 0716 == DDD and
( ) 812),( DtjxDtxb += is
( ) ( ) ;),( 12111102
19 DtDtjxDtjxDtxh +++= (2.100)
where ,06 D ,2D ,8D ,2 6
29
D
DD = ,
6
810
D
DD = ,
6
1211
D
gDD = ,12D 1g and 1j are
arbitrary constants. For solution (2.100) to be nonconstant requires ( ) .0812 + DtjxD
This subcase includes subcase (2) of case (f) for group (2.62) as solution (2.85) is a
special case of solution (2.100) with .01 =g
The infinitesimal generators 821 ,...,, VVV denote the Lie algebras for the respective Lie
groups (I), (II),, (VIII); (see Gandarias [27]). These generators are as follows.
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A List of Infinitesimal Generators for Groups (I)-(VIII)
The generators 821 ,...,, VVV for the respective groups (I), (II),, (VIII) are
,761 tDxDV
+
=
( )[ ] ( ) ,43 746142t
DtDx
DtjxDV
++
++=
( )[ ] ( ) ,270260023h
Dt
DtgDx
DtjxgDV
+
++
++=
( ) ,433
27102
62110
24h
Dt
DtjfD
xDtjjx
jfDV
+
++
+
=
( )[ ] ( ) ( ) ,2170160015h
fhDt
DtgDx
DtjxgDV
++
++
++=
( )[ ] ( ) ( ) ,43
33
217101
621101
6h
fhDt
DtjfD
xDtjjxjf
DV
++
++
+=
( ) ( ) ( )[ ] ,,34
173613
7h
txbhDt
DtDx
DtjxD
V
++
++
++=
( )[ ] ;,1768h
txbhD
t
D
x
DV
++
+
=
where details of 821 ,...,, VVV relate to the respective groups (I), (II),, (VIII).
Next, we present four tables of results. Table 1 features the functions ),(hf )(hg and
)(hj (distinguishing the Lie classical symmetries of the thin film equation (2.1)) with
their associated infinitesimal generators .iV Table 2 is a dimensional classification of the
mathematical structure of groups (I)-(VIII) and the corresponding iV. Table 3 displays
the similarity solutions ),( txh with their similarity variables (where applicable) for the
thin film equation (2.1) in conjunction with groups (I)-(VIII). Table 4 shows the defining
ordinary differential equations (ODEs) for the functions within the functional forms of
),( txh relating to groups (I)-(VIII) in table 3.
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2.3 TABLES OF RESULTS
Table 1.Each row lists functions ),(hf )(hg and )(hj (distinguishing the Lie classical
symmetries of thin film equation (2.1)) with the associated infinitesimal generator .i
V
Group )(hf )(hg )(hj i
V
I arbitrary 0 arbitrary arbitrary1V
II arbitrary 0 01j 2V
III 003
1 hg
ef hg
eg 01 10 jhj + 3V
IV 001 hf
ef hjf
eg 32
1
10+
20
1 jej hj
+ 4V
V ( ) 00321 + g
fhf ( ) 021g
fhg + 120 ln jfhj ++ 5V
VI ( ) 00
21 + f
fhf ( ) 322110
jf
fhg+
+ ( ) 2201 jfhj
j
++ 6V
VII 01 f 0 1j 7V
VIII 01 f 1g 1j 8V
Table 2.A dimensional classification of the mathematical structure of groups (I)-(VIII)
(Lie classical symmetries of thin film equation (2.1)) with their associated infinitesimal
generators .iV
Group ),,( htx ),,( htx ),,( htx iV
I6D 7D 0 1V
II ( ) 03 614 ++ DtjxD 04 74 +DtD 0 2V
III ( ) 6002 DtjxgD ++ 702 DtgD + 02 D 3V
IV621
102
3Dtjjx
jfD +
( ) 710
2 43
DtjfD
+ 02 D 4V
V ( ) 6001 DtjxgD ++ 701 DtgD + ( ) 021 + fhD 5V
VI( )[ ] 62110
1 33
DtjjxjfD
+ ( ) 7101 4
3Dtjf
D+
( ) 021 + fhD 6V
VII( ) 61
3 34
DtjxD
++ 73 DtD + ( )txbhD ,1 + 7V
VIII6D 7D ( )txbhD ,1 + 8V
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Table 3.All rows show the similarity solutions ),( txh and any corresponding similarity
variables iu for the thin film equation (2.1) in connection with groups (I)-(VIII). The
cases 2, 3(1), 7(c1) and 8(b3i) refer to group (II), group (III) case (1), group (VII) case
(c) subcase (1) and group (VIII) case (b) subcase (3) scenario (i) respectively. Other
similarly-named cases in this table use the same denotation pattern.
Case ),( txh iu
1(1) )(uy under the constraint 07 D tDx 11
1(2) constant under the constraints 076 =DD
2 )(uy ( ) 01214/1
11 +
DtjxDt
3(1)0ln
1)( 11
0
+ Dt
g
uy
under the constraint 00 g
0ln 111611
1716
++DtD
Dt
DtDx
3(2i) 0)( 11 + tDuy
under the constraints 007 =gD
0132
12 ++ tDtDx
3(2ii) (