symmetry-enhancing for a thin film equation tanya l.m…

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

i

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 new

Lie 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.

ii

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

31st March 2008

iii

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.

iv

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 Student’s Prayer”

Author unknown

v

TABLE OF CONTENTS

Page

CHAPTER 1: INTRODUCTION 1

CHAPTER 2: LIE CLASSICAL SYMMETRIES FOR THE

THIN FILM EQUATION 10

2.1 Introduction 10

2.2 The Classical Procedure 11

2.3 Tables Of Results 48

2.4 Concluding Remarks 55

CHAPTER 3: NON-CLASSICAL SYMMETRIES FOR THE


3.1 Introduction 56

3.2 The Non-Classical Procedure 57



CHAPTER 4: CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE THIN FILM EQUATION 76

4.1 Introduction 76

4.2 The Method Of Classical Symmetry-

Enhancing Constraints 77



CHAPTER 5: NON-CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE THIN FILM EQUATION 101

5.1 Introduction 101

5.2 The Method Of Non-Classical Symmetry-

Enhancing Constraints 102



vi

CHAPTER 6: CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE THIN FILM EQUATION

INVOLVING ARBITRARY FUNCTIONS 163


6.2 Classical Symmetry-Enhancing Constraints 164



CHAPTER 7: LOCATING POTENTIAL SYMMETRIES FOR THE



7.2 The Method Of Obtaining Potential Symmetries 199


CHAPTER 8: CONCLUSION 201

BIBLIOGRAPHY 204

1

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]. Burde’s

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 Burde’s approach was justified as it enabled Burde to recover new, exact,

explicit, physically significant similarity solutions for the two-dimensional steady-state

2

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 hhhjhhghhf

x (1.1)

where h denotes the height of a thin viscous droplet (or film) as a function of time t and

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].

3

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 4<N for the thin film equation subjected to

Navier’s slip condition or even weaker slip conditions [32]. Via formal asymptotic

expansions and homogenisation theory, Bayada and Chambat examined the asymptotic

behaviour of the Stokes equation (where the roughness spacing and gap height approach

zero) in order to focus on the thin-film hydrodynamic lubrication of rough surfaces [1].

We adopt the restriction 0)( ≠hf in this thesis since the thin film equation (1.1)

generalises the fourth order nonlinear diffusion equation, a special case of equation (1.1)

with .0)()( == hjhg

The case of the thin film equation (1.1) with nhhf =)( and 0)()( == hjhg occurs in

[7, 34, 37, 38, 39, 43, 53] where 0≥h denotes “the thickness of a (surface tension

driven) fluid film” or droplet height [26, 43] and 0>n 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.

4

Bernis, Peletier and Williams considered the critical value 2

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 ;0≥h [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 dynamics

of 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

5

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<n

corresponds to fast diffusion, ),0( ∞∈n denotes slow diffusion with finite speed of

propagation and )0,1(−∈n represents fast diffusion with infinite speed of propagation

while the range 1−≤n has not been considered to such an extent [7].

From a modelling perspective where 0≥h is a requirement and the physically relevant

dimensions are either 1 or 2, some authors studied the full range ];3,0(∈n [5, 10, 12].

Bernis et al. constructed self-similar source-type solutions of obstacle-type free boundary

problems associated with this case of thin film equation (1.1) for the range ;4−≤n [7].

6

Bernis et al. studied the case of one spatial dimension [8]. For the case of two spatial

dimensions and with ,3=n source-type solutions with a jump discontinuity are studied

in [19, 49, 50], forming a special case of the source solution ),( txhk for all 0>n and

for all ),,0( ∞∈k proven to exist and be discontinuous at the free boundary [26]. If one

assumes no slip on the substrate, the case 3=n arises.

Bernoff and Witelski studied the special case of the thin film equation (1.1) with

,)( nhhf = 0)()( == hjhg and ),3,0(∈n using linear stability analysis to demonstrate

the linear stability of the resulting source-type similarity solutions [9]. They derived an

exact polynomial similarity solution for this case with .1=n Polynomial similarity

solutions of the thin film equation (1.1) also occur in chapter 5 of this thesis.

Khayat and Kim observed that in the case of transient two-dimensional thin film flow,

the intensity of the initial gradient in velocity (and film thickness) diminishes with time

[41]. Kim and Khayat examined the two-dimensional non-Newtonian flow of a thin fluid

film emerging from a channel and moving on a solid and stationary substrate [42]. They

stated the flow to be induced by the pressure gradient within the channel where fully

developed Poiseuille conditions are assumed to prevail. They further mentioned that

while the steady-state film thickness for viscous flow tends to increase with distance

from the channel exit, the thickness of a highly elastic film diminishes rapidly. They

observed the substrate geometry to influence mean flow only in the presence of gravity

[42].

Bernis and Ferreira sought radial, self-similar source solutions for the special case of the

thin film equation (1.1) with nhhf =)( and ;0)()( == hjhg [6]. Boatto, Kadanoff and

Olla remarked that the difficulty of studying this case of equation (1.1) lies in its singular

behaviour for 0=h and that an approach to the problem has been to study similarity

solutions for this case of equation (1.1) [18]. Boatto et al. focused on travelling-wave

solutions [18]. We recover such solutions to this and other cases of the thin film equation

(1.1) and include these solutions in this thesis.

Kondic applied the similarity method to explore how the film thinning process evolves

in time [44]. Via linear stability analysis and the similarity method, Kondic studied the

equation comparable to the thin film equation (1.1) with ,)()( 3hhghf == 23)( hhj =

and the size of the normal component of gravity equalling one [44]. Snapshots of fluid

7

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 [ )3,2∈n 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 zero

contact 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 )(ε Lie group of general infinitesimal transformations in ,x t and ,h

namely

( ) ( )( ) ( )( ) ( ).,,

,,,

,,,

2

1

2

1

2

1

εεζ

εεη

εεξ

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 film

equation (1.1).

8

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 hhghhf

x (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

xxxxxx (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.

9

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].

10

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 hhhjhhg

xhhf

x (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 )(ε Lie group of general infinitesimal transformations in

,x t and ,h namely

( ) ( )( ) ( )( ) ( );,,

,,,

,,,

2

1

2

1

2

1

εεζ

εεη

εεξ

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(2.2)

preserving the thin film equation (2.1).

Hence if ),,( txh φ= then from ),,( 111 txh φ= evaluating the expansion of ε∂

∂ 1h at 0=ε

gives the invariant surface condition

).,,(),,(),,( htxt

hhtx

x

hhtx ζηξ =

∂

∂+

∂

∂ (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].

11

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

derivative 4

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 ε in the infinitesimal version of equation (2.1). Terms of order 2ε are

neglected in these calculations since they involve relations between the group generators

,ξ η and ζ already considered in the coefficient of ,ε 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 ),,,( htxξ ),,( htxη and ),,( htxζ satisfy the determining

equations

,0=hξ ,0== xh ηη ,0=hhζ ,0)(

)(=�

�

��

� ′ζ

hf

hf

dh

d ( ) ,0)( =−′

xxxhhf ξζ

,0)()()( =+−+ xxxxxxxt hfhghj ζζζζ ,0)(

)()(4 =

′−′−

hf

hftx ζηξ

[ ] xxxxxxtx hfhghgdh

d

hf

hj

dh

dhfhj ζξζζξξ )()()(2

)(

)()()(3 ′++−�

�

��

�+−

( ) ,04)( =−+ xxxxxxxhhf ξζ (2.4)

,064)(

)(=−+

′xxxhx

hf

hfξζζ ,0

)(

)(

)(

)(246 =�

�

��

�−−−

hf

hg

dh

d

hf

hgxxxxxxh ζξξζ

( ) ( ) .0)(

)(2

)(

)(3

)(

)(=�

�

��

� ′−+

′−−

′

hf

hg

dh

d

hf

hg

hf

hfxhxxxxxh ζξζξζ

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.

12

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 ξξ = ),(),,( thtx ηη = );,(),(),,( txbhtxahtx +=ζ (2.5)

where ),( txa and ),( txb are arbitrary functions of x and .t

By results (2.5)1 and (2.5)3 , equation (2.4)5 gives ( ) ,0)( =−′xxxahf ξ generating cases

(1) ),,(),( txtxa xxx ξ= (2) .0)( =′ hf

We present the derivation of results for case (1) only.

Case (1) ),(),( txtxa xxx ξ=

It follows that

);(),(),( ttxtxa x δξ += (2.6)

where )(tδ is an arbitrary function of .t

Results (2.5)3 and (2.6) cause equation (2.4)9 to give [ ] ,)()(2)( xxx bhfhfhfh ′−=−′ ξ

integrating which with respect to x implies

[ ] );,(),()()(2)( htctxbhfhfhfh x =′+−′ ξ (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)7 gives ,)(

)()(),()()(2

hf

thfhhtcttx

δφηξ

′+==′− so

),(2

)()(),( tx

tttx α

ηφξ +

′+= );()()()(),( thfhthfhtc δφ ′−= (2.8)

where ),(tα )(tη and )(tφ are arbitrary functions of .t

13

Results (2.8) cause relations (2.5)3 , (2.6) and (2.7) to give

),,(2

)(2)()(),,( txbh

ttthtx +

+′+=

δηφζ

(2.9)

[ ] ).(2

)(2)()()()(2)(),()( hfh

ttttthftxbhf ′

+′+−′+=′

δηφηφ

As equation (2.9)2 gives ,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 α

ηφξ +

′+= ),(

2

)(2)()(),(),,( tbh

ttththtx +

+′+==

δηφζζ

(2.10)

[ ] );(2

)(2)()()()(2)()()( hfh

ttttthftbhf ′

+′+−′+=′

δηφηφ

where )(tb is an arbitrary function of .t

Substituting result (2.10)2 into equation (2.4)6 gives

),()(2)( 1 ttdt φδη −−=′ ;)( 2dtb = (2.11)

where 1d and 2d are arbitrary constants.

Results (2.10) and (2.11) give

[ ] ),()(),( 1 txtetx αδξ +−= ),()(2)( 1 ttdt φδη −−=′ ,)(),( 21 dhehht +== ζζ

(2.12)

( ) [ ] );()(2)()( 121 hfdtthfdhe +−=′+ δφ

where .2

11

de =

As equation (2.12)4 has the form ),()( tmhk = giving ,0)()( =′=′ tmhk it follows that

,)(2)( 31 ddtt =+− δφ ( ) );()( 321 hfdhfdhe =′+ (2.13)

where 3d is an arbitrary constant.

14

Results (2.12) and (2.13) yield

),(2

)(),( 3 tx

tdtx α

φξ +

−= ),(2)( 3 tdt φη −=′ ,)( 21 dheh +=ζ

(2.14)

( ) ).()( 321 hfdhfdhe =′+

By results (2.14)1 , (2.14)3 and (2.14)4 , equation (2.4)8 gives

( ) ).(2

)()(

2

)(3)( 3

21 txt

hjtd

hjdhe αφφ

′=′

+−

+′+ (2.15)

Setting to zero the coefficient of x in equation (2.15) yields

;)( 4dt =φ (2.16)

where 4d is an arbitrary constant.

In view of result (2.16), equation (2.15) gives

;)( 65 dtdt +=α (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 ++=ξ ,)( 73 DtDt +=η ,)( 21 DhDh +=ζ

( ) ),()( 821 hfDhfDhD =′+ ( ) ),()( 921 hgDhgDhD =′+ (2.18)

( ) ;)()( 51021 DhjDhjDhD =+′+

where iD is an arbitrary constant for all { }10,...,2,1∈i with ,4 348 DDD −=

349 2 DDD −= and .4310 DDD −=

In view of equations (2.18)4 and (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 ≠

15

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 =ξ ,),,( 7Dhtx =η ;0),,( =htxζ (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)()()( =++−∂


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 0≠xh 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)1 and 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)

16

satisfying

( ) ( ) ( ) ( )[ ]2)4( )()()()()()()()()( uyuyguyuyguyuyuyfuyuyf ′′−′′−′′′′′+

( )[ ] .0)()( 11 =′−+ uyDuyj (2.22)

In relations (2.21)-(2.22), ,07 ≠D 6D and 7

6

11D

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 ≠++= DtjxDhtxξ ,04),,( 74 ≠+= DtDhtxη ;0),,( =htxζ (2.23)

where ,04 ≠D ,6D 7D and 1j are arbitrary constants.


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 0≠xh for system (2.24) to admit nonconstant solutions.

17

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 <

In results (2.25)-(2.26), ,04 ≠D ,6D ,7D ,4 4

7

11D

DD −= ,

4

176

12D

jDDD

−=

4

6

13D

DD =

and 1j are arbitrary constants, ,03 131 ≠++ Dtjx ( ) 0121

4/1

11 ≠+−−=−

DtjxDtu and

( ) 0)( ≠uyf is an arbitrary function of ).(uy We require 0)( ≠′ uy for solution (2.25) to

be nonconstant.

GROUP (III)

Subject to the conditions ,0)( 03

1 ≠= hgefhf

hgeghg 0

1)( = and ,)( 10 jhjhj += the thin

film equation (2.1) admits Lie classical group (III), namely

( ) ,),,( 6002 DtjxgDhtx ++=ξ ,),,( 702 DtgDhtx +=η ;0),,( 2 ≠= Dhtxζ

(2.27)

where ,02 ≠D ,01 ≠f ,6D ,7D ,0g ,1g 0j and 1j are arbitrary constants.



( )[ ] ( ) ,027026002 ≠=++++ DhDtgDhDtjxgD tx

(2.28)

( ) ( )[ ] ( ) ;03 10

2

010

3

100 =++++−+ txxxx

hg

xxxxxxxx

hghhjhjhgheghhghef

where ,02 ≠D ,01 ≠f ,6D ,7D ,0g ,1g 0j and 1j are arbitrary constants.

18

As 0=xh forces 0=th in equation (2.28)2 , rendering equation (2.28)1 inconsistent, we

require .0≠xh

We consider only the cases

(1) ,00 ≠g (2) .00 =g

Case (1) 00 ≠g

By the method of Lagrange [24] and the integrating factor algorithm [48], we solve

equation (2.28)1 and substitute its general solution into equation (2.28)2 . Therefore under

transformations (2.2) and the conditions ,0)( 03

1 ≠= hgefhf

hgeghg 0

1)( = and

,)( 10 jhjhj += the similarity solution of the thin film equation (2.1) in association with

group (2.27) and the constraint 00 ≠g is

;0ln1

)(),( 11

0

≠−+= Dtg

uytxh (2.29)

satisfying the equations

[ ]{ }2

0

)(2

120

)4( )()()()(3)( 0 uyguyeDuyuyguyuyg ′+′′+′′′′+ −

,0)(1

)()(3

1514

1

13

)(3 00 =+′��

��

�+−+ −− uyguyg

eDuyDuf

uyDe ,11Dt >

(2.30)

[ ]{ }2

0

)(2

120


,0)(1

)()(3

1514

1

13

)(3 00 =−′��

��

�+−− −− uyguyg

eDuyDuf

uyDe .11Dt <

In results (2.29)-(2.30), ,02 ≠D ,01

01

15 ≠=gf

D ,01 ≠f ,00 ≠g ,6D ,7D

,02

7

11gD

DD −= ,

1

112

f

gD −= ,

1

0

13f

jD = ,

1

114

f

jD = ,

0

0

16g

jD = ,

02

6

17gD

DD = ,1g 0j and

1j are arbitrary constants, 0ln 1116

11

1716 ≠−−−

++= DtD

Dt

DtDxu and

( )( ) .0171611 ≠++− DtDxDt Furthermore, 0)( ≠′ uy owing to the requirement .0≠xh

Case (2) 00 =g

We consider the subcases

(i) ,07 ≠D (ii) .07 =D

19

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 0≠xh while ,02 ≠D

,07 ≠D ,07

211 ≠=

D

DD ,0

17

217 ≠=

fD

DD ,01 ≠f ,6D ,

2 7

02

12D

jDD −= ,

7

6

13D

DD −=

,1

114

f

gD −= ,

1

0

15f

jD = ,

17

617

16fD

DjDD

−= ,1g 0j and 1j are arbitrary constants and

.013

2

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

20

GROUP (IV)

Under the conditions ,0)( 0

1 ≠= hfefhf

hjf

eghg 3

2

1

10

)(

+

= and ,)( 201 jejhjhj += the

thin film equation (2.1) yields Lie classical group (IV), namely

,3

),,( 62110

2 Dtjjxjf

Dhtx +��

�

�−

−=ξ ( ) ,4

3),,( 710

2 DtjfD

htx +−=η

(2.34)

;0),,( 2 ≠= Dhtxζ

where ,02 ≠D ,01 ≠f ,01 ≠j ,6D ,7D ,0f ,1g 0j and 2j are arbitrary constants.



( ) ,0433

27102

621

10

2 ≠=��

��

�+−+�

�

��

�+��

�

�−

−DhDtjf

DhDtjjx

jfD tx

(2.35)

( ) ( ) ( ) ;03

220

2103

2

1011

10

0 =+++��

��

� ++−+

+

tx

hj

xxx

hjf

xxxxxxxx

hfhhjejh

jfheghhfhef

where ,02 ≠D ,01 ≠f ,01 ≠j ,6D ,7D ,0f ,1g 0j and 2j are arbitrary constants. As

0=xh gives 0=th in equation (2.35)2 , rendering equation (2.35)1 inconsistent, we

require .0≠xh

We consider the cases

(1) ( )( ) ,04 1010 ≠−− jfjf (2) ,010 ≠= jf (3) .04 10 ≠= jf

Case (1) ( )( ) 04 1010 ≠−− jfjf




1 ≠= hfefhf

hjf

eghg 3

2

1

10

)(

+

= and

,)( 201 jejhjhj += the similarity solution of the thin film equation (2.1) in connection

with group (2.34) and the constraints ( )( ) 04 1010 ≠−− jfjf is

;0ln)(),( 1211 ≠−+= DtDuytxh (2.36)

21

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

)(

150016 =−′−+ −− uyfuyfuyD

eDuyueDeD .12Dt <

In relations (2.36)-(2.37), 0)( ≠′ uy owing to the requirement .0≠xh Furthermore,

,02 ≠D ,04

3

10

11 ≠−

=jf

D ( ) ,03

21014 ≠−−= jfD ,00116 ≠−= fjD

( ),0

4 110

10

17 ≠−

−−=

fjf

jfD

( ),0

4

3

110

18 ≠−

=fjf

D ,04 10

10

21 ≠−

−−=

jf

jfD ,01 ≠f ,01 ≠j

,6D ,7D ( )

,4

3

102

7

12jfD

DD

−−= ,

1

113

f

gD −= ,

1

0

15f

jD = ,

3

10

2119

jf

jjD

−−=

( ),

3

102

6

20jfD

DD

−=

( )( )

,3

102

276

22jfD

jDDD

−

−= ,0f ,1g 0j and 2j are arbitrary constants

with ( )( ) .04 1010 ≠−− jfjf In addition, 02019 ≠++ DtDx and

( ) .022212

21 ≠+−−= DtjxDtuD

Case (2) 010 ≠= jf

Via the method of Lagrange [24], we solve equation (2.35)1 and substitute its general


,0)( 1

1 ≠= hjefhf hj

eghg 1

1)( = and ,)( 201 jejhjhj += the similarity solution of the

thin film equation (2.1) in tandem with group (2.34) and the constraint 010 ≠= jf is

;0ln1

)(),( 11

1

≠−−= Dtj

uytxh (2.38)

satisfying

22

[ ]{ }2

1121

)4( )()()()()( uyjuyDuyuyjuy ′+′′+′′′′+

[ ] ,0)()(

15

)(

141311 =+′++ −− uyjuyj

eDuyeDD ,11Dt >

(2.39)

[ ]{ }2

1121

)4( )()()()()( uyjuyDuyuyjuy ′+′′+′′′′+

[ ] ,0)()(

15

)(

141311 =−′−+ −− uyjuyj

eDuyeDD .11Dt <


,02 ≠D ,01

11

15 ≠−=jf

D ,01 ≠f ,01 ≠j ,6D ,7D ,12

7

11jD

DD = ,

1

112

f

gD −=

,1

0

13f

jD = ,

112

276

14fjD

jDDD

−= ,

12

276

16jD

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




1 ≠= hjefhf hj

eghg 12

1)( = and


with group (2.34) and the constraints 04 10 ≠= jf and 07 ≠D is

;0)(),( 11 ≠+= tDuytxh (2.40)

satisfying

[ ]{ }2

1

)(2

121

)4( )(2)()()(4)( 1 uyjuyeDuyuyjuyuyj ′+′′+′′′′+ −

[ ] .0)()(4

15

)(4

14

)(3

13111 =+′++ −−− 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

0

13f

jD =

,12

276

17jD

jDDD

−= ,

12

6

18jD

DD = ,1g 0j and 2j are arbitrary constants, 0182 ≠+− Dtjx

and ( ) .017216 ≠+−= Dtjxeu

tD Furthermore, 0)( ≠′ uy owing to the requirement

.0≠xh

23

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)( 14

1 ≠= hjefhf hj

eghg 12

1)( =

and ,)( 201 jejhjhj += the similarity solution of thin film equation (2.1) in association

with group (2.34) and the constraints 04 710 =≠= Djf is

[ ] ;0)(ln1

),( 112

1

≠+−= tzDtjxj

txh (2.42)

such that

[ ] [ ] [ ] ,0)(2)()()( 5

1

3

1

2

0 =+−+′ tzftzgtzjtz ,0112 >+− Dtjx

(2.43)

[ ] [ ] [ ] ,0)(2)()()( 5

1

3

1

2

0 =+−−′ tzftzgtzjtz .0112 <+− Dtjx

In relations (2.42)-(2.43), ,02 ≠D ,01 ≠f ,01 ≠j ,6D ,12

6

11jD

DD = ,1g 0j and 2j are

arbitrary constants, 0)( >tz and .0112 ≠+− Dtjx

GROUP (V)

Subject to the conditions ( ) ,0)( 03

21 ≠+=g

fhfhf ( ) 0

21)(g

fhghg += and

,ln)( 120 jfhjhj ++= the thin film equation (2.1) admits Lie classical group (V),

namely

( ) ,),,( 6001 DtjxgDhtx ++=ξ ,),,( 701 DtgDhtx +=η ( ) ;0),,( 21 ≠+= fhDhtxζ

(2.44)

where ,01 ≠D ,01 ≠f ,6D ,7D ,2f ,0g ,1g 0j and 1j are arbitrary constants while

.02 ≠+ fh

24



( ) ( ) ,021201100 ≠+=++++ fhhDtghDtjxg tx

(2.45)

( ) ( ) ( ) ��

��

�

+++−��

�

�

�

+++

2

2

0

21

2

03

2100

3xxx

g

xxxxxxxx

gh

fh

ghfhghh

fh

ghfhf

( ) ;0ln 120 =++++ tx hhjfhj

where ,01 ≠D ,01 ≠f ,6D ,7D ,1

6

11D

DD = ,

1

7

12D

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)1 inconsistent, we require .0≠xh


(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)( 03

21 ≠+=g

fhfhf ( ) 0

21)(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 ≠−−= fDtuytxhg

(2.46)

satisfying

[ ]

[ ]

��

�� ′

+′′+′′′′

+)(

)()(

)()(

)()(3)(

2

02

140

)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 <

25


,01 ≠D ,01 ≠f ,00 ≠g ,6D ,7D ,01

7

13gD

DD −= ,

1

1

14f

gD −= ,

0

0

15g

jD = ,

01

6

16gD

DD =

,2f ,1g 0j and 1j are arbitrary constants, 0ln 1315

13

1615 ≠−−−

++= DtD

Dt

DtDxu and

( )( ) .0161513 ≠++− DtDxDt

Case (2) 00 =g


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 ≠−= feuytxhDt (2.48)

satisfying

[ ] .0)()()(ln)()( 16151413

)4( =+′++′′+ uyDuyDuyDuyDuy (2.49)


,01 ≠D ,07 ≠D ,01

7

12 ≠=D

DD ,0

17

1

16 ≠=fD

DD ,01 ≠f ,6D ,

1

1

13f

gD −= ,

1

0

14f

jD =

,17

617

15fD

DjDD

−= ,

2 7

01

17D

jDD −= ,

7

6

18D

DD −= ,2f ,1g 0j and 1j are arbitrary

constants and .018

2

17 ≠++= tDtDxu

Subcase (ii) 007 == gD


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

26

( )

.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 .0≠xh

GROUP (VI)

Under the conditions ( ) ,0)( 0

21 ≠+=f

fhfhf ( ) 3

2

21

10

)(jf

fhghg+

+= and

( ) ,)( 2201 jfhjhjj

++= the thin film equation (2.1) yields Lie classical group (VI),

namely

( )[ ] ,33

),,( 621101 Dtjjxjf

Dhtx +−−=ξ ( ) ,4

3),,( 710

1 DtjfD

htx +−=η

(2.52)

( ) ;0),,( 21 ≠+= fhDhtxζ

where ,01 ≠D ,01 ≠f ,01 ≠j ,6D ,7D ,0f ,2f ,1g 0j and 2j are arbitrary constants

while .02 ≠+ fh



( ) ( ) ,021514131211 ≠+=++++ fhhDtDhDtDxD tx

(2.53)

( ) ( )( )

( )txxx

jf

xxxxxxxx

fhh

fh

jfhfhghh

fh

fhfhf +�

�

��

�

+

+++−��

�

�

�

+++

+2

2

103

2

21

2

0

213

2100

( )[ ] ;02201 =+++ x

jhjfhj

where ,01 ≠D ,01 ≠f ,01 ≠j ,6D ,7D ,3

1011

jfD

−= ,2112 jjD −= ,

1

6

13D

DD =

,3

4 1014

jfD

−= ,

1

7

15D

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

27


(1) ( )( ) ,04 1010 ≠−− jfjf (2) ,010 ≠= jf (3) .04 10 ≠= jf

Case (1) ( )( ) 04 1010 ≠−− jfjf




21 ≠+=f

fhfhf ( ) 3

2

21

10

)(jf

fhghg+

+=

and ( ) ,)( 2201 jfhjhjj

++= 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

16

14 ≠−−= fDtuytxhD

(2.54)

such that

[ ] [ ]

��

�� ′+

+′′+′′′′

+)(

)(

3

2)()(

)(

)()()(

2

10170

)4( 18

uy

uyjfuyuyD

uy

uyuyfuy

D

[ ] [ ]{ } [ ] ,0)()()()( 0020 1

222119 =+′++ −− ffDuyDuyuyuDuyD ,16Dt >

(2.55)

[ ] [ ]

��

�� ′+

+′′+′′′′

+)(

)(

3

2)()(

)(

)()()(

2

10170

)4( 18

uy

uyjfuyuyD

uy

uyuyfuy

D

[ ] [ ]{ } [ ] ,0)()()()( 0020 1

222119 =−′−+ −− ffDuyDuyuyuDuyD .16Dt <


,01 ≠D ,03

4 1014 ≠

−=

jfD ( ) ,0

3

21018 ≠−−= jfD ,00120 ≠−= fjD

( ),0

4 110

10

21 ≠−

−−=

fjf

jfD

( ),0

4

3

110

22 ≠−

=fjf

D ,04 10

10

25 ≠−

−−=

jf

jfD ,01 ≠f

,01 ≠j ,6D ,7D ( )

,4

3

101

7

16jfD

DD

−−= ,

1

117

f

gD −= ,

1

0

19f

jD = ,

3

10

2123

jf

jjD

−−=

( ),

3

101

6

24jfD

DD

−=

( )( )

,3

101

276

26jfD

jDDD

−

−= ,0f ,2f ,1g 0j and 2j are arbitrary constants

with ( )( ) ,04 1010 ≠−− jfjf 02423 ≠++ DtDx and ( ) .026216

25 ≠+−−= DtjxDtuD

28

Case (2) 010 ≠= jf

By the method of Lagrange [24], we solve equation (2.53)1 , substituting its general


( ) ,0)( 1

21 ≠+=j

fhfhf ( ) 1

21)(j

fhghg += and ( ) ,)( 2201 jfhjhjj

++= 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 ≠−−=−

fDtuytxhj

(2.56)

satisfying

[ ]

��

�� ′

+′′+′′′′

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ]{ } )()( 1

1819 uyuyDDj ′++

−

[ ] ,0)( 11

20 =+ − juyD ,16Dt >

(2.57)

[ ]

��

�� ′

+′′+′′′′

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ]{ } )()( 1

1819 uyuyDDj ′−+

−

[ ] ,0)( 11

20 =− − juyD .16Dt <


,01 ≠D ,01

11

20 ≠−=jf

D ,01 ≠f ,01 ≠j ,6D ,7D ,11

7

16jD

DD = ,

1

117

f

gD −=

,111

276

18fjD

jDDD

−= ,

1

0

19f

jD = ,

11

276

21jD

jDDD

−= ,2f ,1g 0j and 2j are arbitrary

constants, 016 ≠− Dt and ( ) .0ln 1621162 ≠−+−−= DtDDtjxu

Case (3) 04 10 ≠= jf


Subcase (i) ,04 10 ≠= jf 07 ≠D


equation (2.53)1 , substituting its general solution into equation (2.53)2 . Therefore under


21 ≠+=j

fhfhf ( ) 12

21)(j

fhghg +=

and ( ) ,)( 2201 jfhjhjj

++= 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 ≠−= feuytxhDt

(2.58)

such that

29

[ ] [ ]

��

�� ′

+′′+′′′′

+−

)(

)(2)()(

)(

)()(4)(

2

1

2

161

)4( 1

uy

uyjuyuyD

uy

uyuyjuy

j

[ ] [ ]{ } )()()( 11 4

18

3

17 uyuyuDuyDjj ′++

−− [ ] .0)( 141

19 =+ − juyD (2.59)


,01 ≠D ,07 ≠D ,01

7

15 ≠=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

0

17f

jD = ,

11

6

20jD

DD = ,

11

276

21jD

jDDD

−= ,2f ,1g

0j and 2j are arbitrary constants, 0202 ≠+− Dtjx and ( ) .022

212 ≠+−= tDeDtjxu

Subcase (ii) 04 710 =≠= Djf


Hence under transformations (2.2) and the conditions ( ) ,0)( 14

21 ≠+=j

fhfhf

( ) 12

21)(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 ≠−+−= fDtjxtytxhj

(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 <+− Dtjx

In relations (2.60)-(2.61), 0)( ≠ty owing to the requirement .0≠xh Furthermore,

,01 ≠D ,01 ≠f ,01 ≠j ,6D ,11

6

16jD

DD = ,

1

0

17j

jD =

( ),

12

1

1118

j

jgD

+−= ,2f ,1g ,0j

2j and ( )( )( )

4

1

111119

1121

j

jjjfD

+−−= are arbitrary constants and .0162 ≠+− Dtjx

30

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

),,( 61

3 DtjxD

htx ++=ξ ,),,( 73 DtDhtx +=η );,(),,( 1 txbhDhtx +=ζ (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.


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)


( ) ( ) ),,(34

17361

3 txbhDhDtDhDtjxD

tx +=++��

��

�++ ;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

61

3 ≠++ DtjxD

As 0=xh forces 0=th in

equation (2.65)2 , giving =),( txh constant, we require 0≠xh for system (2.65) to yield

nonconstant similarity solutions.

As case (e) includes the subcase ,0),(1 =+ txbhD we consider only .0),(1 ≠+ txbhD

31



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 03 ≠D and 0),(1 ≠+ txbhD is

[ ] ;0),()(1

),(9

8

3

≠++= utKuyDtD

txhD

(2.66)

such that

,0),(),()()(4

1)( 989

)4(

1

9 =++++′−−

utKDtxbDtuyDuyuuyfD

,8Dt −>

(2.67)

,0),(),()()(4

1)( 989

)4(

1

9 =−+−−′+−

utKDtxbDtuyDuyuuyfD

.8Dt −<

In relations (2.66)-(2.67), ,03 ≠D ,01 ≠f ,c ,1D ,6D ,7D ,3

7

8D

DD = ,

3

19

D

DD =

( )

3

176

10

4

D

jDDD

−= and 1j are arbitrary constants. Furthermore, ),( txb satisfies

equation (2.63), ( ) ,034

613 ≠++ Dtjx

D ( ) 0101

4/1

8 ≠+−+=−

DtjxDtu and

[ ] 0),(),()(9

89 ≠+++ txbutKuyDtDD

while 0)( ≠′ uy as equations (2.67) otherwise

generate the contradiction [ ] .0),(),()(9

89 =+++ txbutKuyDtDD

In addition,

( ) ( )� −++++=−−

t

c

DdDjuDbDDutK .,),( 101

4/1

8

1

88

9 ωωωωωω As 0=xh forces

0=th in equation (2.65)2 , rendering equation (2.65)1 inconsistent for this subcase, we

require .0≠xh Owing to this requirement, [ ] .0),()( ≠+∂

∂utKuy

x

32

Similarity Solutions for Case (b)


( ) ( ) ,)(34

812173613 DtjxDhDhDtDhDtjx

Dtx +−+=++�

�

��

�++ ;011 =++ 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 ≠++ Dtjx

D As 0=xh gives 0=th in equation (2.68)2 ,

giving =),( txh constant, we require 0≠xh for system (2.68) to admit nonconstant

solutions.


(1) ( ) 8121 DtjxDhD +−+ ,0≠ (2) ( ) 8121 DtjxDhD +−+ .0=

Subcase (1) ( ) 8121 DtjxDhD +−+ 0≠




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)(),( 131211191

10 ≠++−++= DDtjxDDtuytxhD

,4 13 DD ≠

(2.69)

( ) ,0ln)(),( 161412115

4/1

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 −<

33

In relations (2.69)-(2.70), ,01 ≠D ,03 ≠D ,03

110 ≠=

D

DD ,01 ≠f ,1j ,2D ,6D ,7D

,8D ,3

7

9D

DD = ,

4

4

13

211

DD

DD

−= ,

1

176

12D

jDDD

−= ,

1

8

13D

DD −= ,

4 1

7

14D

DD =

,4 1

215

D

DD =

( ),

2

1

811762

16D

DDjDDDD

−−=

( ),

4

3

176

17D

jDDD

−=

( )131

32

184DDD

DDD

−=

and ( )( )131

1762

194

4

DDD

jDDDD

−

−= are arbitrary constants. Furthermore, ( ) 03

461

3 ≠++ DtjxD

and ( ) .0171

4/1

9 ≠+−+=−

DtjxDtu

As 0=xh gives 0=th in equation (2.68)2 , rendering equation (2.68)1 inconsistent for

this subcase, we require .0≠xh Accordingly, 0)( 111

4/1

9

10 ≠+′+−

DuyDtD

and

.0ln)( 14152 ≠++′ DtDuy In addition, ( ) 0)( 1911891

10 ≠+−++ DtjxDDtuyD

and

( )( ) .04ln)( 1412115

4/1

142 ≠+++−++ DtDtjxDDtuy

Subcase (2) ( ) 08121 =+−+ DtjxDhD

As the constraint ( ) 8121 DtjxDhD −−−= identically satisfies equation (2.68)2 but 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


constraints ( ) 0812131 =+−+≠ DtjxDhDDD and 812 )(),( DtjxDtxb +−= is the

constant solution.

Similarity Solutions for Case (c)


),,(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 0≠xh 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

34

As case (f) includes subcases (3)-(5), we consider only subcases (1) and (2).

Subcase (1) [ ] 0),(17 ≠+ txbhDD




similarity solution of the thin film equation (2.1) in conjunction with group (2.62) and

the constraints 037 =≠ DD and 0),(1 ≠+ txbhD is

[ ] ;0),()(1

),( 8

7

≠+= utKuyeD

txhtD

(2.72)

such that

.0),(),()()()( 889

)4(

18 =+++′+ −

utKDetxbuyDuyDuyftD

(2.73)

In relations (2.72)-(2.73), ,07 ≠D ,01 ≠f ,c ,1D ,6D ,7

1

8D

DD = ,

7

617

9D

DjDD

−=

7

6

10D

DD = and 1j are arbitrary constants. Furthermore, ,10tDxu −=

( ) ,,),( 10108� +−= −

t

c

DdDtDxbeutK ωωωω

[ ] ,0),(),()(8

8 ≠++ txbutKuyeDtD

),( txb

satisfies equation (2.63) and 0)( ≠uy is a travelling wave of velocity .10D

As 0=xh forces 0=th in equation (2.71)2 , rendering equation (2.71)1 inconsistent for

this subcase, we require .0≠xh Owing to this requirement, [ ] .0),()( ≠+∂

∂utKuy

x

In addition, 0)( ≠′ uy as equation (2.73) otherwise gives rise to the contradiction

[ ] .0),(),()(8

8 =++ txbutKuyeDtD

Subcase (2) ,076 =≠ DD 0),(1 ≠+ txbhD





constraints 0736 ==≠ DDD and 0),(1 ≠+ txbhD is

[ ] ;0),()(1

),( 8

6

≠+= txKtyeD

txhxD

(2.74)

such that

35

[ ] [ ] .0),(),()()( 88

912111019 =+++++++′ �−−

x

c

D

t

xD

xxxxxx detbDbetxbDbDbDbftyDty ωω ω

(2.75)

In results (2.74)-(2.75), [ ] 0),()()( ≠+ txKtyty and [ ] ,0),(),()( 8

8 ≠++ − xDetxbtxKtyD

noting that as 0=xh gives 0=th in equation (2.71)2 , rendering equation (2.71)1

inconsistent for this subcase, we require .0≠xh Furthermore, ( ) ,,),( 8�−=

x

c

DdtbetxK ωωω

),( 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)


( ) ,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 0≠xh for

system (2.76) to generate nonconstant solutions.


(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



constraints ,0371 =≠ DDD ( ) 8121 DtjxDhD +−+ 0≠ and ( ) 812),( DtjxDtxb +−= is

( ) ( ) ;0),( 121

4

1

11109 ≠+−+= �

=

−DtjxDedetxh

n

tDxc

n

tD n (2.77)

36

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 { }.4,3,2,1∈n

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 0≠xh is

necessary for equation (2.76)1 to be consistent for this subcase.

For the scenario ,176 jDD = ,0

4/1

17

1

1 ≠

��

�−−=

fD

Dc ,0

4/1

17

1

2 ≠

��

�−=

fD

Dc

0

4/1

17

1

3 ≠

��

�−−=

fD

Dic and ,0

4/1

17

1

4 ≠

��

�−=

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),( 131129

1110 ≠+−+= −DtjxDeDtxh

tDxD (2.78)

where ,01 ≠D ,06 ≠D ,09 ≠D ,06

1

10 ≠=D

DD ,01 ≠f ,2D ,8D ,

3

6

3

61

3

11

11D

DjDfD

+=

,1

2

12D

DD −=

2

1

6281

13D

DDDDD

+−= and 1j are arbitrary constants. Furthermore,

( )012109

1110 ≠+−DeDD

tDxD as we require 0≠xh for equation (2.76)1 to be consistent for

this subcase.

37


The constraint ( ) 08121 =+−+ DtjxDhD identically satisfies equation (2.76)2 but

causes equation (2.76)1 to 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

2

9D

DD −=

1

8

10D

DD −= 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)


( ) ( ) ( ) ,34

8127361

3 DtjxDhDtDhDtjxD

tx +−=++��

��

�++ ;011 =++ txxxxx hhjhf (2.80)

where ,03 ≠D ,01 ≠f ,2D ,6D ,7D 8D and 1j are arbitrary constants while

073 ≠+ DtD and ( ) .034

61

3 ≠++ DtjxD

As 0=xh gives 0=th in equation (2.80)2 ,

forcing =),( txh constant, we require 0≠xh for system (2.80) to admit nonconstant

solutions.

38


equation (2.80)1 , substituting its solution into equation (2.80)2 . Therefore under



constraints 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

7

9D

DD = ,

4

3

2

10D

DD =

( ),

4

3

176

11D

jDDD

−=

( )2

3

836172

12

4

D

DDDjDDD

+−= and 1j are arbitrary constants,

( ) 034

61

3 ≠++ DtjxD

and ( ) .0111

4/1

9 ≠+−+=−

DtjxDtu For solution (2.81) to be

nonconstant, we require .0)( 10

4/1

9 ≠+′+−

DuyDt

Similarity Solutions for Case (f)


( ) ,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 0≠xh 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)

39

in connection with group (2.62) and the constraints 0317 ==≠ DDD and

( ) 812),( DtjxDtxb +−= is

( ) ( ) ( ) ( ) ,),( 2

141312

2

9119102191 tDtDxDtDxDtDxDeddtxhtDxc +++−+−++= −

,176 jDD ≠

(2.84)

( ) ( )[ ] ,),( 13112

5

0

1 tDtjxDtjxdtxhn

n

n +−+−=�=

.176 jDD =

In solutions (2.84), ,0

3/1

17

617

1 ≠

��

� −−=

fD

DjDc ,07 ≠D ,01 ≠f ,2D ,6D ,8D

,7

6

9D

DD = ,

176

8

10jDD

DD

−=

( ),

2 176

2

11jDD

DD

−= ,

7

2

12D

DD = ,

7

8

13D

DD =

( ),

22

7

1762

14D

jDDDD

+−= ,0d ,1d ,2d ,3d ,

24 17

8

4fD

Dd −=

17

2

5120 fD

Dd −= and 1j are


Furthermore, ( ) ( )022 91

21101191211 ≠++−+ − tDxcedcDtDDDxD and

( ) 012

5

1

1

1 ≠+−�=

−tDtjxnd

n

n

n as we require 0≠xh for solutions (2.84) to be nonconstant.

In addition, ( ) ( )02 91

2110911 ≠++− − tDxcedcDtDxD and ( ) .0

5

1

1

1 ≠−�=

−

n

n

n tjxnd

Subcase (2) 076 =≠ DD


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

( ) ( ) ;),( 11110

2

19 DtjxDtjxDtxh +−+−= (2.85)

where ,06 ≠D ,2D ,8D ,2 6

2

9D

DD = ,

6

8

10D

DD = 11D and 1j are arbitrary constants. For

solution (2.85) to be nonconstant requires ( ) .0812 ≠+− DtjxD

40

GROUP (VIII)

Under conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the thin film equation (2.1)

yields Lie classical group (VIII), namely

,),,( 6Dhtx =ξ ,),,( 7Dhtx =η );,(),,( 1 txbhDhtx +=ζ (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

43121

tjxdtjxdededtjxddtxb

−−− ++−+= (2.88)

where ,01

1

5 ≠=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


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 +−=



),,(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 0≠xh 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

41

As case (c) includes subcases (3)-(5), we consider only subcases (1) and (2).

Subcase (1) [ ] 0),(17 ≠+ txbhDD



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

txhtD

(2.90)

satisfying

.0),(),()()()()( 8891

)4(

18 =+++′+′′− −

utKDetxbuyDuyDuyguyftD

(2.91)

In relations (2.90)-(2.91), ,07 ≠D ,01 ≠f ,c ,1D ,6D ,7

1

8D

DD = ,

7

617

9D

DjDD

−=

,7

6

10D

DD = 1g and 1j are arbitrary constants and .10tDxu −= Furthermore,

( ) ,,),( 10108� +−= −

t

c

DdDtDxbeutK ωωωω

[ ] 0),(),()(8

8 ≠++ txbutKuyeDtD

and ),( 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),(),()(8

8 =++ txbutKuyeDtD

Furthermore, 0)( ≠∂

∂+′

x

Kuy as we require 0≠xh for equation (2.89)1 to 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



transformations (2.2) and the conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the


constraints [ ] 0),( 716 =≠+ DtxbhDD (with ),( txb satisfying equation (2.87)) is

42

[ ] ;0),()(1

),( 8

6

≠+= txKtyeD

txhxD

(2.92)

such that

[ ] [ ] .0),(),()()( 88

912111019 =+++++++′ �−−

x

c

D

t

xD


(2.93)

In results (2.92)-(2.93), [ ] 0),()()( ≠+ txKtyty and [ ] ,0),(),()( 8

8 ≠++ − xDetxbtxKtyD

noting that we require 0≠xh for equation (2.89)1 to be consistent for this subcase.

Furthermore, ( ) ,,),( 8�−=

x

c

DdtbetxK ωωω

),( txb satisfies equation (2.87) and ,06 ≠D

,01 ≠f ,c ,1g ,1j ,1D ,6

1

8D

DD =

( ),

4

6

3

611

2

611

4

11

9D

DDjDDgDfD

+−= ,

6

11

10D

fDD =

2

6

2

61

2

11

11D

DgDfD

−= and

3

6

3

61

2

611

3

11

12D

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



( ) ,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 0≠xh for

system (2.94) to admit nonconstant solutions.


(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

43

thin film equation (2.1) in conjunction with group (2.86) and the constraints ,071 ≠DD

( ) 08121 ≠+−+ DtjxDhD and ( ) 812),( DtjxDtxb +−= is

( ) ( ) ;0),( 121

4

1

11109 ≠+−+= �

=

−DtjxDedetxh

n

tDxc

n

tD n (2.95)

where ,01 ≠D ,07 ≠D ,07

1

9 ≠=D

DD ,nc ,nd ,1j ,2D ,6D ,8D ,

7

6

10D

DD =

,1

2

11D

DD −=

( )2

1

176281

12D

jDDDDDD

−+−= and

( )

1

1762

13D

jDDDD

−−= are arbitrary

constants for all { }.4,3,2,1∈n

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 0≠xh 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),( 131129

1110 ≠+−+= −DtjxDeDtxh

tDxD (2.96)

where ,01 ≠D ,06 ≠D ,09 ≠D ,06

1

10 ≠=D

DD ,01 ≠f ,2D ,8D

,3

6

3

61

2

611

3

11

11D

DjDDgDfD

+−= ,

1

2

12D

DD −= ,

2

1

6281

13D

DDDDD

+−= 1g and 1j are


44

Furthermore, ( )

0121091110 ≠+−

DeDDtDxD

as ( ) 08121 ≠+−+ DtjxDhD and as we require

0≠xh 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



The constraint ( ) 08121 =+−+ DtjxDhD identically satisfies equation (2.94)2 but

causes equation (2.94)1 to 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 =


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

2

9D

DD −=

1

8

10D

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).

45



( ) ,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 0≠xh 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

( ) ( ) ( ) ( ) ,),( 2

141312

2

911910

4

2

19 tDtDxDtDxDtDxDeddtxh

n

tDxc

nn +++−+−++= �

=

−

,176 jDD ≠

(2.99)

( ) ( ) ( ) ( )[ ] ,),( 13112

6

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 ,01 ≠f ,01 ≠g ,2c ,3c ,4c ,nd ,1j ,2D

,6D ,8D ,7

6

9D

DD =

( )( )

,2

176

1721768

10jDD

gDDjDDDD

−

−−=

( ),

2 176

2

11jDD

DD

−= ,

7

2

12D

DD =

7

8

13D

DD = and

( )2

7

1762

142D

jDDDD

+−= are arbitrary constants for all { }6,5,4,3,2,1∈n .

For solutions (2.99) to be nonconstant, we require .0≠xh Therefore,

( ) ( ) ( ) ( ) ( )011 12

6

5

1

15

4

2

2

1115 ≠+−+−− ��

=

−−

=

−tDeDdtjxdn

n

tjxDn

n

n

n

n

n

and

46

( ) ( ) .022 101191211

4

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

( ) ( ) ;),( 1211110

2

19 DtDtjxDtjxDtxh ++−+−= (2.100)

where ,06 ≠D ,2D ,8D ,2 6

2

9D

DD = ,

6

8

10D

DD = ,

6

12

11D

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


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.

47

A List of Infinitesimal Generators for Groups (I)-(VIII)

The generators 821 ,...,, VVV for the respective groups (I), (II),…, (VIII) are

,761t

Dx

DV∂

∂+

∂

∂=

( )[ ] ( ) ,43 746142t

DtDx

DtjxDV∂

∂++

∂

∂++=

( )[ ] ( ) ,270260023h

Dt

DtgDx

DtjxgDV∂

∂+

∂

∂++

∂

∂++=

( ) ,433

2710

2

621

10

24h

Dt

DtjfD

xDtjjx

jfDV

∂

∂+

∂

∂��

��

�+−+

∂

∂��

��

�+

��

�−

−=

( )[ ] ( ) ( ) ,2170160015h

fhDt

DtgDx

DtjxgDV∂

∂++

∂

∂++

∂

∂++=

( )[ ] ( ) ( ) ,43

33

21710

1

62110

1

6h

fhDt

DtjfD

xDtjjxjf

DV

∂

∂++

∂

∂��

��

�+−+

∂

∂

��

��

+−−=

( ) ( ) ( )[ ] ,,34

17361

3

7h

txbhDt

DtDx

DtjxD

V∂

∂++

∂

∂++

∂

∂��

��

�++=

( )[ ] ;,1768h

txbhDt

Dx

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.

48

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 .iV

Group )(hf )(hg )(hj iV

I arbitrary 0≠ arbitrary arbitrary 1V

II arbitrary 0≠ 0 1j 2V

III 003

1 ≠hgef

hgeg 0

1 10 jhj + 3V

IV 00

1 ≠hfef h

jf

eg 3

2

1

10 +

20

1 jejhj + 4V

V ( ) 003

21 ≠+g

fhf ( ) 0

21

gfhg + 120 ln jfhj ++ 5V

VI ( ) 00

21 ≠+f

fhf ( ) 3

2

21

10 jf

fhg+

+ ( ) 2201 jfhjj

++ 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 ),,( htxξ ),,( htxη ),,( htxζ iV

I 6D 7D 0

1V

II ( ) 03 614 ≠++ DtjxD 04 74 ≠+ DtD 0 2V

III ( ) 6002 DtjxgD ++ 702 DtgD + 02 ≠D 3V

IV 621

10

23

Dtjjxjf

D +

��

�−

− ( ) 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

VIII 6D 7D ( )txbhD ,1 + 8V

49

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 ( ) 0121

4/1

11 ≠+−−−

DtjxDt

3(1) 0ln

1)( 11

0

≠−+ Dtg

uy

under the constraint 00 ≠g

0ln 1116

11

1716 ≠−−−

++DtD

Dt

DtDx

3(2i) 0)( 11 ≠+ tDuy

under the constraints 007 =≠ gD

013

2

12 ≠++ tDtDx

3(2ii) ( )0

602

1112 ≠+

+−

DtjD

DtjxD

under the constraints 007 == gD

4(1) 0ln)( 1211 ≠−+ DtDuy under the

constraints ( )( ) 04 1010 ≠−− jfjf

( ) 022212

21 ≠+−− DtjxDtD

4(2) 0ln

1)( 11

1

≠−− Dtj

uy

under the constraint 010 ≠= jf

( ) 0ln 1116112 ≠−+−− DtDDtjx

4(3i) 0)( 11 ≠+ tDuy under the constraints

04 10 ≠= jf and 07 ≠D

( ) 017216 ≠+− Dtjxe

tD

4(3ii) [ ] 0)(ln1

112

1

≠+− tzDtjxj

under the

constraints 04 710 =≠= Djf

t

5(1) 0)( 2

/1

13

0 ≠−− fDtuyg

under the constraint 00 ≠g

0ln 1315

13

1615 ≠−−−

++DtD

Dt

DtDx

5(2i) 0)( 2

/ 12 ≠− feuyDt

under the constraints 007 =≠ gD

018

2

17 ≠++ tDtDx

5(2ii) 0)( 2

601

1

≠−+fety

DtjD

xD

under the constraints 007 == gD

t

50

Table 3. Continued.

Case ),( txh iu

6(1) 0)( 2

/1

16

14 ≠−− fDtuyD

under the constraints ( )( ) 04 1010 ≠−− jfjf

( ) 026216

25 ≠+−− DtjxDtD

6(2) 0)( 2

/1

16

1 ≠−−−

fDtuyj

under the constraint 010 ≠= jf

( ) 0ln 1621162 ≠−+−− DtDDtjx

6(3i) 0)( 2

/ 15 ≠− feuyDt

under the constraints 04 10 ≠= jf and 07 ≠D

( ) 022

212 ≠+− tDeDtjx

6(3ii) 0)( 2

/1

162

1 ≠−+− fDtjxtyj

under the constraints 04 710 =≠= Djf

t

7(a) [ ] 0),()(

1 9

8

3

≠++ utKuyDtD

D

under the constraints 03 ≠D and 0),(1 ≠+ txbhD

( ) 0101

4/1

8 ≠+−+−

DtjxDt

7(b1) ( ) 0)( 131211191

10 ≠++−++ DDtjxDDtuyD

under the constraints ,031 ≠DD ,4 13 DD ≠

( ) 08121 ≠+−+ DtjxDhD and

( ) ,),( 812 DtjxDtxb +−=

( ) 0ln)( 161412115

4/1

142 ≠+++−++ DDtDtjxDDtuy

under the constraints ,04 13 ≠= DD

( ) 08121 ≠+−+ DtjxDhD and

( ) 812),( DtjxDtxb +−=

( ) 0171

4/1

9 ≠+−+−

DtjxDt

7(b2) constant under the constraints

( ) 0812131 =+−+≠ DtjxDhDDD and

( ) 812),( DtjxDtxb +−=

7(c1) [ ] 0),()(

18

7

≠+ utKuyeD

tD under the constraints

037 =≠ DD and 0),(1 ≠+ txbhD

tDx 10−

7(c2) [ ] 0),()(

18

6

≠+ txKtyeD

xD under the constraints

0736 ==≠ DDD and 0),(1 ≠+ txbhD

t

7(d1) ( ) ( ) 0121

4

1

11109 ≠+−+�

=

−DtjxDede

n

tDxc

n

tD n under the

constraints ,0371 =≠ DDD ( ) 08121 ≠+−+ DtjxDhD

and ( ) 812),( DtjxDtxb +−=

51

Table 3. Continued.

Case ),( txh iu

7(d2) ( ) ( ) 01311291110 ≠+−+−

DtjxDeDtDxD

under constraints

,07361 ==≠ DDDD ( ) 08121 ≠+−+ DtjxDhD


7(d3i) constant under the constraints

081321 =+==≠ DhDDDD and 8),( Dtxb =

7(d3ii) ( ) 1019 DtjxD +− under the constraints ,031 =≠ DD

,176 jDD = ( ) 08121 =+−+ DtjxDhD


7(e) ( ) 91211110 ln)( DtDDtjxDuy +++−+ under the

constraints 013 =≠ DD and ( ) 812),( DtjxDtxb +−=

( ) 0111

4/1

9 ≠+−+−

DtjxDt

7(f1) ( ) ( ) ( )2

9119102191 tDxDtDxDeddtDxc −+−++ −

( ) 2

141312 tDtDxD +++ under the constraints

,0317 ==≠ DDD 176 jDD ≠ and

( ) ,),( 812 DtjxDtxb +−=

( ) ( )[ ]tDtjxDtjxdn

n

n 13112

5

0

1 +−+−�=

under the

constraints ,0317 ==≠ DDD 176 jDD = and

( ) 812),( DtjxDtxb +−=

7(f2) ( ) ( ) 11110

2

19 DtjxDtjxD +−+− under the constraints

07316 ===≠ DDDD and ( ) 812),( DtjxDtxb +−=

8(a1) [ ] 0),()(

18

7

≠+ utKuyeD

tD under the constraints

[ ] 0),(17 ≠+ txbhDD with

),( txb an arbitrary solution of equation (2.87)

tDx 10−

8(a2) [ ] 0),()(

18

6

≠+ txKtyeD

xD under the constraints

[ ] 0),( 716 =≠+ DtxbhDD with

),( txb satisfying equation (2.87)

t

8(b1) ( ) ( ) 0121

4

1

11109 ≠+−+�

=

−DtjxDede

n

tDxc

n

tD n under the

constraints ,071 ≠DD ( ) 08121 ≠+−+ DtjxDhD


8(b2) ( ) ( ) 01311291110 ≠+−+−

DtjxDeDtDxD

under constraints

,0761 =≠ DDD ( ) 08121 ≠+−+ DtjxDhD and

( ) 812),( DtjxDtxb +−=

52

Table 3. Continued.

Case ),( txh iu

8(b3i) constant under the constraints 08121 =+=≠ DhDDD and 8),( Dtxb =

8(b3ii) ( ) 1019 DtjxD +− under the constraints ,01761 =−≠ jDDD

( ) 08121 =+−+ DtjxDhD and ( ) 812),( DtjxDtxb +−=

8(c1) ( ) ( ) ( ) ( ) 2

141312

2

911910

4

2

19 tDtDxDtDxDtDxDedd

n

tDxc

nn +++−+−++�

=

−

under the constraints ,0117 =≠ DgD 176 jDD ≠ and

( ) ,),( 812 DtjxDtxb +−=

( ) ( ) ( ) ( )[ ]tDtjxDedtjxdn

tjxD

n

n

n

n

n

13112

6

5

14

1

1

1115 +−++− ��

=

−−

=

− under the

constraints ,0117 =≠ DgD 176 jDD = and ( ) 812),( DtjxDtxb +−=

8(c2) ( ) ( ) 1211110

2

19 DtDtjxDtjxD ++−+− under the constraints

0716 ==≠ DDD and ( ) 812),( DtjxDtxb +−=

Table 4. All rows show the defining ODEs for the functions iy within the functional

forms of ),( txh in connection with groups (I)-(VIII) in table 3. For case 4(3ii), .zyi =

For case 7(b1), ., 21 yyyi = In all other cases, .yyi = Cases 2, 3(1) and 7(c1) refer to

group (II), group (III) case (1) and group (VII) case (c) subcase (1) respectively. Other

similarly-named cases in this table use the same denotation pattern.

Case ( ) 0,,,,)4(

=′′′′′′iiiii yyyyyA

1(1) ( ) ( ) ( ) ( )[ ]2)4( )()()()()()()()()( uyuyguyuyguyuyuyfuyuyf ′′−′′−′′′′′+

( )[ ] 0)()( 11 =′−+ uyDuyj with ( ) 0)( ≠uyf

2 ( ) ( ) ,0)(

4

1)()()()()( )4( =′−′′′′′+ uyuuyuyuyfuyuyf ,11Dt >

( ) ( ) ,0)(4

1)()()()()( )4( =′+′′′′′+ uyuuyuyuyfuyuyf ;11Dt <

with ( ) 0)( ≠uyf

3(1) [ ]{ }2

0

)(2

120


,0)(1

)()(3

1514

1

13

)(3 00 =+′��

��

�+−+ −− uyguyg

eDuyDuf

uyDe ,11Dt >

[ ]{ }2

0

)(2

120


,0)(1

)()(3

1514

1

13

)(3 00 =−′��

��

�+−− −− uyguyg

eDuyDuf

uyDe 11Dt <

53

Table 4. Continued. For case 4(3ii), .zyi = In all other cases except 7(b1), .yyi =

Case ( ) 0,,,,)4(


3(2i) 0)()()()()( 17161514

)4( =+′+′+′′+ DuyDuyuyDuyDuy

4(1) [ ]

��

��

′+

+′′+′′′′+210)(

130

)4( )(3

2)()()()( 14 uy

jfuyeDuyuyfuy

uyD

[ ] ,0)()(

18

)(

17

)(

150016 =+′++ −− uyfuyfuyD

eDuyueDeD

,12Dt >

[ ]��

��

′+

+′′+′′′′+210)(

130

)4( )(3

2)()()()( 14 uy

jfuyeDuyuyfuy

uyD

[ ] ,0)()(

18

)(

17

)(

150016 =−′−+ −− uyfuyfuyD

eDuyueDeD 12Dt <

4(2) [ ]{ }2

1121

)4( )()()()()( uyjuyDuyuyjuy ′+′′+′′′′+

[ ] ,0)()(

15

)(

141311 =+′++ −− uyjuyj

eDuyeDD ,11Dt >

[ ]{ }2

1121

)4( )()()()()( uyjuyDuyuyjuy ′+′′+′′′′+

[ ] ,0)()(

15

)(

141311 =−′−+ −− uyjuyj

eDuyeDD 11Dt <

4(3i) [ ]{ }2

1

)(2

121

)4( )(2)()()(4)( 1 uyjuyeDuyuyjuyuyj ′+′′+′′′′+ −

[ ] 0)()(4

15

)(4

14

)(3

13111 =+′++ −−− uyjuyjuyj

eDuyueDeD

4(3ii) [ ] [ ] [ ] ,0)(2)()()( 5

1

3

1

2

0 =+−+′ tzftzgtzjtz ,0112 >+− Dtjx

[ ] [ ] [ ] ,0)(2)()()( 5

1

3

1

2

0 =+−−′ tzftzgtzjtz 0112 <+− Dtjx

5(1) [ ] [ ]

��

�� ′

+′′+′′′′

+−

)(

)()()(

)(

)()(3)(

2

0

2

140

)4( 0

uy

uyguyuyD

uy

uyuyguy

g

[ ] [ ] ,0)(1

)()(ln)(1

0

10

3

1

0 =��

��

+′−++−

uyg

uyujuyjuyf

g ,13Dt >

[ ] [ ]

��

�� ′

+′′+′′′′

+−

)(

)()()(

)(

)()(3)(

2

0

2

140

)4( 0

uy

uyguyuyD

uy

uyuyguy

g

[ ] [ ] ,0)(1

)()(ln)(1

0

10

3

1

0 =��

��

+′−+−−

uyg

uyujuyjuyf

g 13Dt <

5(2i) [ ] 0)()()(ln)()( 16151413

)4( =+′++′′+ uyDuyDuyDuyDuy

5(2ii)

( )0)()(ln)(

3

601

14

601

1310

601

1 =��

�

��

�

++

+++

++′ ty

DtjD

D

DtjD

Djtyj

DtjD

Dty

6(1) [ ] [ ]

��

�� ′+

+′′+′′′′

+)(

)(

3

2)()(

)(

)()()(

2

10170

)4( 18

uy

uyjfuyuyD

uy

uyuyfuy

D

[ ] [ ]{ } [ ] ,0)()()()( 0020 1

222119 =+′++ −− ffDuyDuyuyuDuyD ,16Dt >

[ ] [ ]

��

�� ′+

+′′+′′′′

+)(

)(

3

2)()(

)(

)()()(

2

10170

)4( 18

uy

uyjfuyuyD

uy

uyuyfuy

D

[ ] [ ]{ } [ ] ,0)()()()( 0020 1

222119 =−′−+ −− ffDuyDuyuyuDuyD 16Dt <

54

Table 4. Continued. For case 7(b1), ., 21 yyyi = In all other cases except 4(3ii), .yyi =

Case ( ) 0,,,,)4(


6(2) [ ]

��

�� ′

+′′+′′′′

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ]{ } )()( 1

1819 uyuyDDj ′++

−

[ ] ,0)( 11

20 =+ − juyD ,16Dt >

[ ]

��

�� ′

+′′+′′′′

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ]{ } )()( 1

1819 uyuyDDj ′−+

−

[ ] ,0)( 11

20 =− − juyD 16Dt <

6(3i) [ ] [ ]

��

�� ′

+′′+′′′′

+−

)(

)(2)()(

)(

)()(4)(

2

1

2

161

)4( 1

uy

uyjuyuyD

uy

uyuyjuy

j

[ ] [ ]{ } )()()( 11 4

18

3

17 uyuyuDuyDjj ′++

−− [ ] 0)( 141

19 =+ − juyD

6(3ii) [ ] [ ] [ ] ,0)()()(

)(

)(111 4

19

2

1817 =+++′ jjj

tyDtyDtyDty

ty ,0162 >+− Dtjx

[ ] [ ] [ ] ,0)()()()(

)(111 4

19

2

1817 =++−′ jjj

tyDtyDtyDty

ty 0162 <+− Dtjx

7(a) ,0),(),()()(

4

1)( 989

)4(

1

9 =++++′−−

utKDtxbDtuyDuyuuyfD

,8Dt −>

,0),(),()()(4

1)( 989

)4(

1

9 =−+−−′+−

utKDtxbDtuyDuyuuyfD

8Dt −<

7(b1) ,0)()(

4

1)( 1101

)4(

11 =+′− uyDuyuuyf ,0)(4

1)(

4

1)( 1522

)4(

21 =++′− uDuyuyuuyf

for all ,9Dt −>

,0)()(4

1)( 1101

)4(

11 =−′+ uyDuyuuyf ,0)(4

1)(

4

1)( 1522

)4(

21 =−−′+ uDuyuyuuyf

for all 9Dt −<

7(c1) 0),(),()()()( 889

)4(

18 =+++′+ −

utKDetxbuyDuyDuyftD

7(c2) [ ] [ ] 0),(),()()( 88

912111019 =+++++++′ �−−

x

c

D

t

xD


7(e) ,0)(

4

1)( 12

)4(

1 =+′− Duyuuyf ,9Dt −>

,0)(4

1)( 12

)4(

1 =−′+ Duyuuyf 9Dt −<

8(a1) 0),(),()()()()( 8891

)4(

18 =+++′+′′− −

utKDetxbuyDuyDuyguyftD

8(a2) [ ] [ ] 0),(),()()( 88

912111019 =+++++++′ �−−

x

c

D

t

xD


55

2.4 CONCLUDING REMARKS

We applied Lie classical analysis to the thin film equation (2.1), obtaining eight Lie

classical symmetry groups, namely groups (I)-(VIII).

In association with groups (I)-(VIII), we derived similarity solutions for the thin film

equation (2.1). We recovered several of these solutions via Mathematica [54]. All

physically relevant solutions of the thin film equation (2.1) are space-dependent as the

thin film equation (2.1) reduces solutions independent of x to constant solutions.

Sums of travelling waves with differing velocities and combinations of travelling waves

and polynomials involving space and time variables are among the similarity solutions

occurring for the thin film equation (2.1) in connection with groups (I)-(VIII) and their

accompanying conditions.

In the following chapter we study the thin film equation (2.1) using the non-classical

symmetry method of Bluman and Cole [16]. This non-classical procedure enables us to

determine if the thin film equation (2.1) admits non-classical symmetries arising beyond

its classical symmetries obtained in this chapter. In chapter 4 of this thesis, we consider

the thin film equation (2.1) from the perspective of the method of classical symmetry-

enhancing constraints introduced and developed by Goard and Broadbridge [29]. This

allows us to ascertain the existence of classical symmetries extending beyond the

confines of those retrievable by Lie classical analysis for the thin film equation (2.1).

56

CHAPTER 3

NON-CLASSICAL SYMMETRIES FOR THE

THIN FILM EQUATION

3.1 INTRODUCTION

By the non-classical procedure, we deduce the non-classical symmetry groups for the

thin film equation (2.1) given by

[ ] ;0)()()( =++−∂


x (3.1)

where .0)( ≠hf The thin film equations (3.1) and (2.1) are identical.


,x t and ,h namely

( ) ( )( ) ( )( ) ( );,,

,,,

,,,

2

1

2

1

2

1

εεζ

εεη

εεξ

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(3.2)

preserving the thin film equation (3.1).

Hence if ),,( txh φ= then from ),,( 111 txh φ= evaluating the expansion of ε∂

∂ 1h at 0=ε

gives the invariant surface condition

).,,(),,(),,( htxt

hhtx

x

hhtx ζηξ =

∂

∂+

∂

∂ (3.3)

Solutions of the invariant surface condition (3.3) are functional forms of similarity

solutions for the thin film equation (3.1).

The following section contains a brief description of the non-classical method. This

technique enables recovery of the non-classical symmetry groups leaving the thin film

equation (3.1) invariant. The non-classical method generalises the Lie classical

procedure and includes the classical groups as special cases. The non-classical approach

is outlined in Bluman and Cole [16] and Hill [36].

57

3.2 THE NON-CLASSICAL PROCEDURE

Commencing the non-classical method, we introduce for transformations (3.2) the terms

),,( htxA and ),,( htxB defined as

,),,(

),,(),,(

htx

htxhtxA

η

ζ= ;

),,(

),,(),,(

htx

htxhtxB

η

ξ= (3.4)

so that the invariant surface condition (3.3) becomes

.x

hBA

t

h

∂

∂−=

∂

∂ (3.5)

To obtain non-trivial transformations, we require .0),,( ≠htxη

The non-classical method uses equation (3.5) to express all derivatives of h with respect

to t in terms of derivatives of h with respect to .x In the invariance requirement derived

for the thin film equation (3.1) via the Lie classical procedure in chapter 2 of this thesis,

these new expressions then replace all derivatives of h with respect to t so that the left-

hand side of the invariance requirement depends on h only via h itself and derivatives

of h with respect to .x

As we express all derivatives of h with respect to t in terms of derivatives of h with

respect to ,x fewer derivatives of h occur in the resulting invariance requirement than is

the case for the classical method. Hence, fewer restrictions apply to the non-classical

group generators ),,( htxA and ),,( htxB than to the classical group generators

),,,( htxξ ),,( htxη and ),,( htxζ when equating to zero the coefficients of all the

derivatives of h and the sum of all remaining terms not involving derivatives of h in the

invariance requirement. This enables the non-classical procedure to generalise the

classical method. Following the replacements mentioned, setting to zero the coefficients

of all derivatives of h and the sum of all remaining terms not involving derivatives of h

in the invariance requirement of thin film equation (3.1) yields the determining equations

58

,0=hhA ,0=hB ,0)(

)(=�

�

��

� ′A

hf

hf

dh

d ( ) ,0)( =−′

xxxh BAhf

,0)(

)(4)()()( 2 =

′−+++− A

hf

hfABAAhjAhgAhf xtxxxxxxx

,0)(

)(

)(

)(246 =�

�

��

�−−−

hf

hg

dh

dAB

hf

hgBA xxxxxxh ,0

)(

)(64 =

′+− xxxxh A

hf

hfBA

( ) ( ) ,0)(

)(2

)(

)(3

)(

)(=�

�

��

� ′−+

′−−

′

hf

hg

dh

dABA

hf

hgBA

hf

hfxhxxxxxh (3.6)

[ ] ( ) ( ))(

)(4)(2)(4)(3

hf

hfABBAhfABhgBBBhj xxxxxxxhxhxxtx

′+−+−+−−

Ahf

hj

dh

dhf �

�

��

�+

)(

)()( .0)(2)( =′−′+ xxxx AhgAhf

All subscripts in system (3.6) denote partial differentiation with ,x t and h as

independent variables. Throughout this chapter, primes represent differentiation with

respect to the argument indicated.

System (3.6) enables retrieval of all the non-classical symmetries and corresponding

conditions on ,0)( ≠hf )(hg and )(hj for the thin film equation (3.1) under

transformations (3.2) via the non-classical procedure.

The following pages feature a description of the nine non-classical groups (I) – (IX)

arising for the thin film equation (3.1). Where applicable, a listing of the special cases

occurring for each of groups (I) – (IX) follows. Finally, we present any new similarity

solutions for the thin film equation (3.1) in association with each of these groups.

GROUP (I)

Under the conditions ,0)( ≠hf )(hg and )(hj are arbitrary functions of ,h the thin

film equation (3.1) admits non-classical group (I), namely

,0),,( =htxA ;),,( 4chtxB = (3.7)

where 4c is an arbitrary constant.

59



,04 =+ xt hch [ ] ;0)()()( =++−∂


x (3.8)

where 4c is an arbitrary constant while ,0)( ≠hf )(hg and )(hj are arbitrary functions

of .h

System (3.8) is a special case of system (2.20) with .17 =D It follows that under

transformations (3.2) and with ,0)( ≠hf )(hg and )(hj arbitrary functions of ,h the

similarity solution of the thin film equation (3.1) in association with group (3.7) is a

special case of results (2.21)-(2.22) with .17 =D

GROUP (II)

Subject to the conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 10 jhjhj += the thin film

equation (3.1) yields non-classical group (II), namely

,0),,( 2 ≠= chtxA ;),,( 402 ctjchtxB += (3.9)

where ,02 ≠c ,01 ≠f ,4c ,1g 0j and 1j are arbitrary constants.



( ) ,02402 ≠=++ chctjch xt ( ) ;01011 =+++− txxxxxxx hhjhjhghf (3.10)

where ,02 ≠c ,01 ≠f ,4c ,1g 0j and 1j are arbitrary constants. As 0=xh forces

0=th in equation (3.10)2 , rendering equation (3.10)1 inconsistent, we require .0≠xh

Equations (3.10) are a special case of equations (2.28) with 00 =g and .17 =D Hence

under transformations (3.2) and the conditions ,0)( 1 ≠= fhf 1)( ghg = and

,)( 10 jhjhj += the similarity solution of the thin film equation (3.1) in tandem with

group (3.9) is a special case of results (2.31)-(2.32) with .17 =D

60

By the Mathematica program [54], we solve the case 00 =j of the relations obtained

here; (which corresponds to results (2.31)-(2.32) with 00 =j and 17 =D ). These

solutions are

( ) ( ) ,0),( 1110

9

7

54 ≠+−+=�

=

−− dtjxcedtxh

n

tcxc

nn ,05 ≠c

( ) ( ) ( ) ,0),( 162

2

412

4

3

1 411 ≠+++−+=�=

−−dtdxdtcxcedtxh

n

tcxc

n

n

,051 =≠ cg (3.11)

( ) ,0),( 2

5

1

1

4 ≠+−=�=

−tctcxdtxh

n

n

n .051 == cg

In solutions (3.11), ,02 ≠c ,014

210 ≠

−=

jc

cc ,0

1

111 ≠=

f

gc ,0

2 1

212 ≠=

g

cc ,01 ≠f

,024 1

25 ≠−=

f

cd ,4c ,415 cjc −= ,7c ,8c ,9c ,nd 1g and 1j are arbitrary constants for

all { }.6,4,3,2,1∈n

Furthermore, ( )

,010

9

7

54 ≠+�

=

−− cedc

n

tcxc

nnn ( ) ( ) 01

5

1

2

4 ≠−−�=

−

n

n

n tcxdn and

( ) ( ) ( ) 02)1( 2412

4

3

1

11411 ≠+−+−�

=

−−dtcxcedc

n

tcxc

n

nn

owing to the requirement .0≠xh

GROUP (III)

Under the conditions ,0)( 1 ≠= fhf 1)( ghg = and ,ln)( 210 jjhjhj ++= the thin

film equation (3.1) admits non-classical group (III), namely

( ) ,0),,( 11 ≠+= jhchtxA ;),,( 401 ctjchtxB += (3.12)

where ,01 ≠c ,01 ≠f ,4c ,1g ,0j 1j and 2j are arbitrary constants while .01 ≠+ jh


Group (3.12), the invariant surface condition (3.5) and the thin film equation (3.1) yield

( ) ( ) ,011401 ≠+=++ jhchctjch xt ( ) ;0ln 21011 =++++− txxxxxxx hhjjhjhghf

(3.13)

where ,01 ≠c ,01 ≠f ,4c ,1g ,0j 1j and 2j are arbitrary constants while .01 ≠+ jh

As 0=xh forces 0=th in equation (3.13)2 , rendering equation (3.13)1 inconsistent,

0≠xh is a requirement.

61

System (3.13) is a special case of system (2.45) with 00 =g and .17 =D Therefore

under transformations (3.2) and the conditions ,0)( 1 ≠= fhf 1)( ghg = and

,ln)( 210 jjhjhj ++= the similarity solution of thin film equation (3.1) in connection

with group (3.12) is a special case of relations (2.48)-(2.49) with .17 =D

Via the Mathematica program [54], we solve the case 00 =j of the relations obtained

here; (which corresponds to results (2.48)-(2.49) with 00 =j and 17 =D ). The solution

arising is

( )

;0),( 1

10

7

641 ≠−= �

=

−− jedetxh

n

tcxc

n

tc n (3.14)

where ,01 ≠c ,4c ,7c ,8c ,9c ,10c ,1d ,2d ,3d 4d and 1j are arbitrary constants.

Furthermore, ( )

010

7

64 ≠�

=

−−

n

tcxc

nnnedc owing to the requirement .0≠xh

GROUP (IV)


1 ≠= hfefhf

hjf

eghg 3

2

1

10

)(

+

= and ,)( 201 jejhjhj += the

thin film equation (3.1) yields the non-classical group (IV), namely

,0

3

4

1),,(

210

≠

+−

=

ctjf

htxA ;

3

43),,(

210

52110

ctjf

ctjjxjf

htxB

+−

+−−

= (3.15)

where ,01 ≠f ,01 ≠j ,2c ,5c ,0f ,1g 0j and 2j are arbitrary constants while

.03

42

10 ≠+−

ctjf

62



,0

3

4

1

3

43

210

210

52110

≠

+−

=

+−

+−−

+

ctjf

h

ctjf

ctjjxjf

h xt

(3.16)

( ) ( ) ( ) ;03

220

2103

2

1011

10

0 =+++��

��

� ++−+

+

tx

hj

xxx

hjf

xxxxxxxx

hfhhjejh

jfheghhfhef

where ,01 ≠f ,01 ≠j ,2c ,5c ,0f ,1g 0j and 2j are arbitrary constants while

.03

42

10 ≠+−

ctjf

As 0=xh gives 0=th in equation (3.16)2 , causing equation (3.16)1

to be inconsistent, 0≠xh is a requirement.

System (3.16) is a special case of system (2.35) with 12 =D and .03

47

10 ≠+−

Dtjf

We examine the cases

(1) ,04 10 ≠= jf ,02 ≠c (2) ,010 ≠= jf (3) ( )( ) .04 1010 ≠−− jfjf

Case (1) ,04 10 ≠= jf 02 ≠c

Under the constraints 04 10 ≠= jf and ,02 ≠c system (3.16) is a special case of system

(2.35) with ,12 =D 07 ≠D and .04 10 ≠= jf Hence under transformations (3.2) and

the conditions ,0)( 14

1 ≠= hjefhf hj

eghg 12

1)( = and ,)( 201 jejhjhj += the similarity

solution of the thin film equation (3.1) in tandem with group (3.15) and constraints

04 10 ≠= jf and 02 ≠c is a special case of results (2.40)-(2.41) with .12 =D

Case (2) 010 ≠= jf

Subject to the constraint ,010 ≠= jf system (3.16) is a special case of system (2.35)

with ,12 =D 010 ≠= jf and .071 ≠+− Dtj Thus under transformations (3.2) and the

conditions ,0)( 1

1 ≠= hjefhf hj

eghg 1

1)( = and ,)( 201 jejhjhj += the similarity

solution of thin film equation (3.1) in association with group (3.15) and the constraint

010 ≠= jf is a special case of relations (2.38)-(2.39) with .12 =D

63

Case (3) ( )( ) 04 1010 ≠−− jfjf

Under the constraints ( )( ) ,04 1010 ≠−− jfjf system (3.16) is a special case of system

(2.35) with ,12 =D ( )( ) 04 1010 ≠−− jfjf and .03

47

10 ≠+−

Dtjf

Therefore under


1 ≠= hfefhf

hjf

eghg 3

2

1

10

)(

+

= and


with group (3.15) and constraints ( )( ) 04 1010 ≠−− jfjf is a special case of relations

(2.36)-(2.37) with .12 =D

GROUP (V)


1 ≠= hgefhf

hgeghg 0

1)( = and ,)( 20 jhjhj += the thin

film equation (3.1) admits non-classical group (V), namely

,01

),,(20

≠+

=ctg

htxA ;),,(20

500

ctg

ctjxghtxB

+

++= (3.17)

where ,01 ≠f ,2c ,5c ,0g ,1g 0j and 2j are arbitrary constants with .020 ≠+ ctg



,01

2020

500 ≠+

=+

+++

ctgh

ctg

ctjxgh xt

(3.18)

( ) ( )[ ] ( ) ;03 20

2

010

3

100 =++++−+ txxxx

hg

xxxxxxxx

hghhjhjhgheghhghef

where ,01 ≠f ,2c ,5c ,0g ,1g 0j and 2j are arbitrary constants with .020 ≠+ ctg As

0=xh forces 0=th in equation (3.18)2 , rendering equation (3.18)1 inconsistent, we

require .0≠xh

System (3.18) is a special case of system (2.28) with 12 =D and .070 ≠+ Dtg

We study the cases

(1) ,00 ≠g (2) ≠2c .00 =g

64

Case (1) 00 ≠g

Under the constraint ,00 ≠g system (3.18) is a special case of system (2.28) with

,12 =D 00 ≠g and .070 ≠+ Dtg Hence under transformations (3.2) and the conditions

,0)( 03

1 ≠= hgefhf

hgeghg 0

1)( = and ,)( 20 jhjhj += the similarity solution of the thin

film equation (3.1) in association with group (3.17) and the constraint 00 ≠g is a special

case of relations (2.29)-(2.30) with .12 =D

Case (2) ≠2c 00 =g

Subject to the constraints ≠2c ,00 =g system (3.18) is equivalent to system (3.10).

Hence this case is identical to that of group (II) in this chapter.

GROUP (VI)


21 ≠+=f

fhfhf ( ) 3

2

21

10

)(jf

fhghg+

+= and

( ) ,)( 2201 jfhjhjj

++= the thin film equation (3.1) admits the non-classical group (VI),

namely

,0

3

4),,(

2

10

2 ≠

+−

+=

ctjf

fhhtxA ;

3

43),,(

2

10

521

10

ctjf

ctjjxjf

htxB

+−

+−−

= (3.19)

where ,01 ≠f ,01 ≠j ,2c ,5c ,0f ,2f ,1g 0j and 2j are arbitrary constants such that

03

42

10 ≠+−

ctjf

while .02 ≠+ fh

65



,0

3

4

3

43

2

10

2

2

10

521

10

≠

+−

+=

+−

+−−

+

ctjf

fhh

ctjf

ctjjxjf

h xt

(3.20)

( ) ( )( )

( ) ��

��

�

+

+++−��

��

+++

+

2

2

10

21

2

0

213

23

120

0

xxxxxxxxxxx

fh

fh

jfhfhghh

fh

fhfhf

jf

th+ ( )[ ] ;02201 =+++ x

jhjfhj

where ,01 ≠f ,01 ≠j ,2c ,5c ,0f ,2f ,1g 0j and 2j are arbitrary constants such that

03

42

10 ≠+−

ctjf

while .02 ≠+ fh As 0=xh gives 0=th in equation (3.20)2 ,

rendering equation (3.20)1 inconsistent, 0≠xh is a requirement.

System (3.20) is a special case of system (2.53) with ( ) .043

7101 ≠+− Dtjf

D

We examine the cases

(1) ,02 ≠c ,04 10 ≠= jf (2) ,010 ≠= jf (3) ( )( ) .04 1010 ≠−− jfjf

Case (1) ,02 ≠c 04 10 ≠= jf

Under constraints 02 ≠c and ,04 10 ≠= jf system (3.20) is a special case of system

(2.53) with 07 ≠D and .04 10 ≠= jf This case accordingly corresponds to case (3)

subcase (i) for group (VI) in chapter 2. Hence under transformations (3.2) and conditions

( ) ,0)( 14

21 ≠+=j

fhfhf ( ) 12

21)(j


++= the


constraints 02 ≠c and 04 10 ≠= jf is equivalent to results (2.58)-(2.59).

66

Case (2) 010 ≠= jf

Under the constraint ,010 ≠= jf system (3.20) is a special case of system (2.53) with

010 ≠= jf and .0711 ≠+− DtjD This case thus corresponds to case (2) of group (VI)

in chapter 2. Therefore under transformations (3.2) and the conditions

( ) ,0)( 1

21 ≠+=j

fhfhf ( ) 1

21)(j


++= the similarity

solution of thin film equation (3.1) in association with group (3.19) and the constraint

010 ≠= jf is equivalent to relations (2.56)-(2.57).

Case (3) ( )( ) 04 1010 ≠−− jfjf

Under the constraints ( )( ) ,04 1010 ≠−− jfjf system (3.20) is a special case of system

(2.53) with ( )( ) 04 1010 ≠−− jfjf and ( ) .043

7101 ≠+− Dtjf

D Hence this case is

equivalent to case (1) of group (VI) in chapter 2. Hence under transformations (3.2) and

conditions ( ) ,0)( 0

21 ≠+=f

fhfhf ( ) 3

2

21

10

)(jf

fhghg+

+= and ( ) ,)( 2201 jfhjhjj

++=


the constraints ( )( ) 04 1010 ≠−− jfjf is equivalent to results (2.54)-(2.55).

GROUP (VII)


21 ≠+=g

fhfhf ( ) 0

21)(g

fhghg += and

,ln)( 220 jfhjhj ++= the thin film equation (3.1) yields non-classical group (VII),

namely

,0),,(20

2 ≠+

+=

ctg

fhhtxA ;),,(

20

500

ctg

ctjxghtxB

+

++= (3.21)

where ,01 ≠f ,2c ,5c ,2f ,0g ,1g 0j and 2j are arbitrary constants such that

020 ≠+ ctg while .02 ≠+ fh

67



,020

2

20

500 ≠+

+=

+

+++

ctg

fhh

ctg

ctjxgh xt

(3.22)

( ) ( ) ( )txxx

g

xxxxxxxx

ghh

fh

ghfhghh

fh

ghfhf +�

�

��

�

+++−��

��

+++

2

2

0

21

2

03

2100

3

( ) ;0ln 220 =+++ xhjfhj

where ,01 ≠f ,2c ,5c ,2f ,0g ,1g 0j and 2j are arbitrary constants with 020 ≠+ ctg

and .02 ≠+ fh As 0=xh forces 0=th in equation (3.22)2 , rendering equation (3.22)1

inconsistent, we require .0≠xh

System (3.22) is a special case of system (2.45) with .0701 ≠+ DtgD

We consider the cases (1) ,00 ≠g (2) ≠2c .00 =g

Case (1) 00 ≠g

Under the constraint ,00 ≠g system (3.22) is a special case of system (2.45) with

00 ≠g and .0701 ≠+ DtgD This case therefore corresponds to case (1) of group (V) in

chapter 2. Hence under transformations (3.2) and the conditions

( ) ,0)( 03

21 ≠+=g

fhfhf ( ) 0

21)(g

fhghg += and ,ln)( 220 jfhjhj ++= the similarity

solution of the thin film equation (3.1) in relation to group (3.21) and the constraint

00 ≠g is equivalent to results (2.46)-(2.47).

Case (2) ≠2c 00 =g

Under constraints ≠2c ,00 =g system (3.22) is equivalent to system (3.13).

Accordingly, this case is identical to that of group (III) in this chapter.

68

GROUP (VIII)

Under conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the thin film equation (3.1)

admits the non-classical group (VIII), namely

),,(),,( 1 txbhchtxA += ;),,( 2chtxB = (3.23)

such that

;0111 =++− txxxxxxx bbjbgbf (3.24)

where ,01 ≠f ,1c ,2c 1g and 1j are arbitrary constants.

As equations (3.24) and (2.87) are identical, equation (3.24) also admits the travelling

wave (2.88) of velocity .1j

Special cases of group (3.23) emerge and are as follows.

Case (1) The case of group (3.23) with ( ) ( )

8437565),( cecectxbtcxctcxc ++= −−−

and

01 =c occurs under conditions identical to those on group (3.23) where ,05 ≠c ,01 ≠f

,3c ,4c ,15

3

5116 gccfjc −+= ,15

3

5117 gccfjc +−= ,8c 1g and 1j are arbitrary

constants such that ( ) ( )

.07565

43 ≠− −−− tcxctcxcecec

Case (2) The case of group (3.23) with 0≠xb arises under conditions identical to

those on group (3.23).



),,(12 txbhchch xt +=+ ;0111 =++− txxxxxxx hhjhghf (3.25)

where ,01 ≠f ,1c ,2c 1g and 1j are arbitrary constants while ),( txb satisfies equation

(3.24). As 0=xh gives 0=th in equation (3.25)2 , forcing =),( txh constant, we

require 0≠xh for system (3.25) to generate nonconstant solutions.

We derive similarity solutions of the thin film equation (3.1) for the cases


(b) ,01 ≠c ,)(),( 413 ctjxctxb +−=

(c) ,01 =c .)(),( 413 ctjxctxb +−=

69


As equations (3.24) and (2.87) are identical defining equations for ),,( txb system (3.25)

is a special case of system (2.89) with .17 =D Hence under transformations (3.2) and

the conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the similarity solutions of thin

film equation (3.1) in relation to group (3.23) for case (a) are the solutions arising for

subcases (1) and (3) of group (VIII) case (a) with 17 =D in chapter 2.


Under constraints 01 ≠c and ,)(),( 413 ctjxctxb +−= system (3.25) is a special case of

system (2.94) with .17 =D Hence under transformations (3.2) and the conditions

,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the similarity solutions of thin film equation

(3.1) in tandem with group (3.23) for case (b) are the constant solution and the special

cases of solutions (2.95) and (2.97) with .17 =D


Under constraints 01 =c and ,)(),( 413 ctjxctxb +−= system (3.25) is a special case of

system (2.98) with .17 =D Therefore under transformations (3.2) and the conditions

,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = the similarity solutions of thin film equation

(3.1) in connection with group (3.23) for case (c) are the special cases of solutions (2.99)

and (2.84) with .17 =D

GROUP (IX)

Under conditions ,0)( 1 ≠= fhf 0)( =hg and ,)( 1jhj = the thin film equation (3.1)

admits the non-classical group (IX), namely

),,(4

),,(2

1 txbhct

chtxA +

+= ;0

4

3),,(

2

31 ≠+

++=

ct

ctjxhtxB (3.26)

such that

;0),(4

4

2

11 =+

+++ txbct

bbjbf txxxxx (3.27)

where ,01 ≠f ,1c ,2c 3c and 1j are arbitrary constants while 03 31 ≠++ ctjx and

.04 2 ≠+ ct

70

Equation (3.27) (whose only constant solution is the trivial one) admits the solution

;4

),(2

7465

ct

cectxb

tcxc

+

+=

+

(3.28)

where ,05 ≠c ,01 ≠f ,2c ,4c ( ),3

51156 cfjcc +−= 7c and 1j are arbitrary constants

while .04 2 ≠+ ct

Special cases of group (3.26) arise and are as follows.

Case (1) The case of group (3.26) with 0),( 1 == ctxb occurs under the conditions

0)( ≠hf is an arbitrary function, 0)( =hg and 1)( jhj = where 1j is an arbitrary

constant.

Case (2) The case of group (3.26) with ( )

04

2),(

20

≠+

−=ctg

txb arises under the

conditions ,0)( 1 ≠= fhf hg

eghg 0

1)( = and 22

3

0

0

)( jejhjhg

+= where ,01 ≠f ,00 ≠g

,2c ,1g 0j and 2j are arbitrary constants and .04 2 ≠+ ct

Case (3) The case of group (3.26) with ( )20

2

4

2),(

ctg

gtxb

+−= emerges under

conditions ,0)( 1 ≠= fhf ( ) 0

21)(g

ghghg += and ( ) 22

3

200)( jghjhj

g++= where

,01 ≠f ,00 ≠g ,2c ,1g ,2g 0j and 2j are arbitrary constants while

( )( ) .04 22 ≠++ ghct

Case (4) The case of group (3.26) with ( )

2

615

4),(

ct

ctjxctxb

+

+−= occurs under

conditions identical to those on group (3.26) where ,2c ,5c 6c and 1j are arbitrary

constants while .04 2 ≠+ ct

Case (5) The case of group (3.26) with

( )[ ] ( )[ ]

2

1065

4),(

37117

37117

ct

cecectxb

tcfjxctcfjxc

+

++=

−−−+−

and 01 =c arises under conditions

identical to those on group (3.26) where ,07 ≠c ,01 ≠f ,2c ,5c ,6c 10c and 1j are

arbitrary constants such that ( )[ ] ( )[ ]

03

71173

7117

65 ≠− −−−+− tcfjxctcfjxcecec and .04 2 ≠+ ct

Case (6) The case of group (3.26) with 0≠xb occurs under conditions identical to

those on group (3.26).

71



),,(44

3

2

1

2

31 txbhct

ch

ct

ctjxh xt +

+=

+

+++ ;011 =++ txxxxx hhjhf (3.29)

where ,01 ≠f ,1c ,2c 3c and 1j are arbitrary constants, ,03 31 ≠++ ctjx 04 2 ≠+ ct

and ),( txb satisfies equation (3.27). As 0=xh forces 0=th in equation (3.29)2 , giving

=),( txh constant, we require 0≠xh for system (3.29) to admit nonconstant solutions.

We obtain similarity solutions of the thin film equation (3.1) for the cases


(b) ,01 ≠c ( )

,4

),(2

615

ct

ctjxctxb

+

+−=

(c) ,01 =c ( )

.4

),(2

615

ct

ctjxctxb

+

+−=


Let ),( txα replace ),( txb in equation (2.63) and the ensuing discussion throughout

group (VII) in chapter 2 of this thesis. Upon letting ( ) ),(4),( 2 txbcttx +=α where this

last-mentioned ),( txb appears in equations (3.26)-(3.27) and (3.29), equations (3.27) and

(2.63) are equivalent. We then find system (3.29) to be a special case of system (2.65)

with 43 =D and ( ) ).,(4),( 2 txbcttx +=α

Hence under transformations (3.2) and conditions ,0)( 1 ≠= fhf 0)( =hg and

,)( 1jhj = the similarity solutions of thin film equation (3.1) in tandem with group (3.26)

for case (a) are the solutions occurring for case (a) of group (VII) with 43 =D and

( ) ),(4),( 2 txbcttx +=α in chapter 2 of this thesis.


Under constraints 01 ≠c and ( )

,4

),(2

615

ct

ctjxctxb

+

+−= system (3.29) is a special case of

system (2.68) with .43 =D Hence under transformations (3.2) and the conditions

,0)( 1 ≠= fhf 0)( =hg and ,)( 1jhj = the similarity solutions of the thin film equation

(3.1) in relation to group (3.26) for case (b) are the constant solution and the special case

of solutions (2.69)-(2.70) with .43 =D

72


Under constraints 01 =c and ( )

,4

),(2

615

ct

ctjxctxb

+

+−= system (3.29) is a special case of

system (2.80) with .43 =D Therefore under transformations (3.2) and the conditions

,0)( 1 ≠= fhf 0)( =hg and ,)( 1jhj = the similarity solution of thin film equation (3.1)

in connection with group (3.26) for case (c) is the special case of solutions (2.81)-(2.82)

with .43 =D

The infinitesimal generators ,1V ,...,2V 9V represent the algebras for the respective non-

classical symmetry groups (I), (II),…, (IX); (see Gandarias [28]). These generators are as

follows.

A List of Infinitesimal Generators for Groups (I)-(IX)

The generators ,1V ,...,2V 9V for the respective non-classical symmetry groups (I),

(II),…, (IX) are

,41tx

cV∂

∂+

∂

∂=

( ) ,24022h

ctx

ctjcV∂

∂+

∂

∂+

∂

∂+=

( ) ( ) ,114013h

jhctx

ctjcV∂

∂++

∂

∂+

∂

∂+=

,

3

4

1

3

43

2

10

2

10

521

10

4h

ctjftx

ctjf

ctjjxjf

V∂

∂

+−

+∂

∂+

∂

∂

+−

+−−

=

,1

2020

500

5hctgtxctg

ctjxgV

∂

∂

++

∂

∂+

∂

∂

+

++=

,

3

4

3

43

2

10

2

2

10

521

10

6h

ctjf

fh

txct

jf

ctjjxjf

V∂

∂

+−

++

∂

∂+

∂

∂

+−

+−−

=

,20

2

20

500

7hctg

fh

txctg

ctjxgV

∂

∂

+

++

∂

∂+

∂

∂

+

++=

[ ] ,),(128h

txbhctx

cV∂

∂++

∂

∂+

∂

∂=

.),(44

3

2

1

2

31

9h

txbhct

c

txct

ctjxV

∂

∂��

��

�+

++

∂

∂+

∂

∂

+

++=

73

We now present three tables of results. Table 1 lists the functions ),(hf )(hg and )(hj

(distinguishing the non-classical symmetries of the thin film equation (3.1)) with their

associated infinitesimal generators iV for all }.9,...,2,1{∈i Table 2 is a dimensional

classification of the mathematical structure of groups (I)-(IX) and the corresponding iV .

Table 3 displays the similarity solutions ),( txh not featured in chapter 2 for the thin film

equation (3.1) in tandem with groups (I)-(IX).


Table 1. Each row lists the functions ),(hf )(hg and )(hj (distinguishing the non-

classical symmetries of the thin film equation (3.1)) with the associated infinitesimal

generator .iV


I arbitrary 0≠ arbitrary arbitrary 1V

II 01 ≠f 1g 10 jhj + 2V

III 01 ≠f 1g 210 ln jjhj ++ 3V

IV 00

1 ≠hfef h

jf

eg 3

2

1

10 +

20

1 jejhj + 4V

V 003

1 ≠hgef

hgeg 0

1 20 jhj + 5V

VI ( ) 00

21 ≠+f

fhf ( ) 3

2

21

10 jf

fhg+

+ ( ) 2201 jfhjj

++ 6V

VII ( ) 003

21 ≠+g

fhf ( ) 0

21

gfhg + 220 ln jfhj ++ 7V

VIII 01 ≠f 1g 1j 8V

IX 01 ≠f 0 1j 9V

74

Table 2. A dimensional classification of the mathematical structure of groups (I)-(IX)

(the non-classical symmetries of the thin film equation (3.1)) with their associated

infinitesimal generators .iV

Group ),,( htxA ),,( htxB iV

I 0 4c 1V

II 02 ≠c 402 ctjc + 2V

III ( ) 011 ≠+ jhc 401 ctjc + 3V

IV 0

3

4

1

2

10

≠

+−

ctjf

2

10

521

10

3

43

ctjf

ctjjxjf

+−

+−−

4V

V 0

1

20

≠+ ctg

20

500

ctg

ctjxg

+

++ 5V

VI 0

3

42

10

2 ≠

+−

+

ctjf

fh

2

10

521

10

3

43

ctjf

ctjjxjf

+−

+−−

6V

VII 0

20

2 ≠+

+

ctg

fh

20

500

ctg

ctjxg

+

++ 7V

VIII ( )txbhc ,1 + 2c 8V

IX ( )txbh

ct

c,

4 2

1 ++

04

3

2

31 ≠+

++

ct

ctjx 9V

Table 3. All rows show the similarity solutions ),( txh not featured in chapter 2 for the

thin film equation (3.1) in tandem with groups (I)-(IX).

Group ),( txh

II ( ) ( ) 01110

9

7

54 ≠+−+�

=

−− dtjxced

n

tcxc

nn under the constraints ,005 =≠ jc

( ) ( ) ( ) 0162

2

412

4

3

1 411 ≠+++−+�=

−−dtdxdtcxced

n

tcxc

n

n

under the constraints

,0051 ==≠ jcg

( ) 02

5

1

1

4 ≠+−�=

−tctcxd

n

n

n under the constraints 0015 === jgc

III ( )01

10

7

641 ≠−�

=

−− jede

n

tcxc

n

tc n under the constraint 00 =j

75


In this chapter, we obtained nine non-classical symmetries, namely groups (I)-(IX), of

the thin film equation (3.1) (identical to the thin film equation (2.1)) via the non-classical

symmetry method of Bluman and Cole [16].

A comparison of these nine non-classical symmetry groups with the eight classical Lie

symmetry groups constructed for the equivalent thin film equation (2.1) in chapter 2 of

this thesis showed that the thin film equation (3.1) does not admit any non-classical

symmetries arising beyond its classical symmetries. Applying the non-classical

symmetry method of Bluman and Cole [16] to the thin film equation (3.1) generates no

similarity solutions which are not retrievable via Lie classical analysis of the equivalent

thin film equation (2.1), performed in chapter 2 of this thesis.

In chapter 5 of this thesis, we examine the thin film equation (3.1) via an approach

combining the non-classical symmetry method of Bluman and Cole [16] and the

technique of symmetry-enhancing constraints presented and developed by Goard and

Broadbridge [29]. This combined approach enables us to determine the existence of non-

classical symmetries extending beyond the confines of those obtainable for the thin film

equation (3.1) solely by the non-classical symmetry method of Bluman and Cole [16].

76

CHAPTER 4

CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE

THIN FILM EQUATION

4.1 INTRODUCTION

We apply the method of classical symmetry-enhancing constraints to obtain symmetry

groups for the thin film equation (2.1) given by

[ ] [ ] 0)()()( =++∂

∂−

∂

∂txxxxx hhhjhhg

xhhf

x; (4.1)

where the restriction 0)( ≠hf holds.

The technique of classical symmetry-enhancing constraints, outlined in the next section,

is identical to the method of symmetry-enhancing constraints, introduced and developed

by Goard and Broadbridge [29]. The method of classical symmetry-enhancing

constraints is so termed in order to differentiate it from the method of symmetry-

enhancing constraints [29] which is augmented by the non-classical procedure and

features in chapter 5 of this thesis.

From the perspective of this method, we studied various partitions of the thin film

equation (4.1). Two of these produced new Lie symmetry groups and generate the

respective systems

( ) 0)()(2

=′−+ xxt hhghhjh , [ ] 0)()( =−∂

∂xxxxx hhghhf

x; (4.2)

and

0)()( =+− txxxxxx hhhghhf , ( ) 0)()()(2

=+′−′xxxxxx hhjhhghhhf . (4.3)

On system (4.2), we impose the restrictions

(i) 0)( ≠hf , (ii) 0)( =hg and 0)( =′ hf do not occur jointly,

(iii) 0)( =′ hg and 0)( =hj do not occur together;

to prevent th and xxxxh from vanishing. We place system (4.3) under the condition

0)( ≠′ hf and the restrictions (i) and (iii) to prevent xxxh and xxxxh disappearing.

77

Next, we consider the one-parameter )(ε Lie group of general infinitesimal

transformations in ,x t and ,h namely

( ) ( )( ) ( )( ) ( );,,

,,,

,,,

2

1

2

1

2

1

εεζ

εεη

εεξ

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(4.4)

leaving systems (4.2) and (4.3) invariant.

Hence if ),( txh φ= , then from ),( 111 txh φ= , evaluating the expansion of ε∂

∂ 1h at 0=ε

yields the invariant surface condition

),,(),,(),,( htxt

hhtx

x

hhtx ζηξ =

∂

∂+

∂

∂. (4.5)

Solutions of equation (4.5) are functional forms of similarity solutions for systems (4.2)

and (4.3).

The following section contains a brief description of the technique of classical

symmetry-enhancing constraints. This involves using the Lie classical method to

determine the symmetry groups leaving systems (4.2) and (4.3) invariant.

4.2 THE METHOD OF CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS

Goard and Broadbridge define symmetry-enhancing constraints as “equations whose

addition to a target equation results in the enlarged system having at least one additional

symmetry not possessed by the original target equation on its own” [29]. These authors

describe the concept of the method of symmetry-enhancing constraints in terms of

random equation-splitting, whereby “single partial differential equations (PDEs) are

partitioned into a larger system of PDEs, chosen so that this system has a larger Lie

symmetry group than has the original PDE” [29]. Owing to the derivation of new Lie

symmetry groups from systems (4.2) and (4.3), symmetry-enhancing constraints are

considered to be added to the thin film equation (4.1).

78

The method of symmetry-enhancing constraints involves partitioning the equation of

interest into a larger system of PDEs before applying the classical procedure to this

system to retrieve its Lie classical groups [29]. Should this system yield Lie groups other

than those admitted by the original single equation (and which are not special cases of

groups admitted by the original equation), we may consider a symmetry-enhancing

constraint to be added to the original equation. Accordingly, we apply this treatment to

systems (4.2) and (4.3), briefly describing it with respect to system (4.2). The process is

identical for system (4.3).

The classical method involves equating to zero the infinitesimal version of system (4.2)

without using the invariant surface condition (4.5). In obtaining the infinitesimal version

of system (4.2), the highest order derivative in each equation within system (4.2) is

eliminated by expressing it with respect to all the remaining terms of the equation in

question. In equation (4.2)1 , we choose th for elimination, all derivatives in (4.2)1 being

of equal order. Prolongation of the action of group transformations (4.4) on system (4.2)

generates the invariance requirements, obtained by equating to zero the coefficient of ε

in the infinitesimal version of each of equations (4.2)1 and (4.2)2 . Terms of order 2ε are

neglected in these calculations since they involve relations between the group generators

ξ , η and ζ already considered in the coefficients of ε , the left-hand sides of the

invariance requirements.

System (4.2) remains invariant under group transformations (4.4) provided the group

generators ),,( htxξ , ),,( htxη and ),,( htxζ satisfy the determining equations

0=hξ , 0== xh ηη , 0=hhζ , 0)( =+ xt hj ζζ ,

[ ] 0)()(2)( =′′−−′−′ hgthg hx ζζηξ , [ ] 0)(2)()()( =′−−−′+′xtx hgthjhj ζξξηζ ,

0)(

)(=− xxxxxx

hf

hgζζ , ( ) 03)( =−′

xxxxxhhf ξζ , ( ) 0)( =−′xxxhhf ξζ , (4.6)

( ) 02)(

)(

)(

)(4 =−+

′+− xhxxxxxxxxxxxxh

hf

hg

hf

hfζξζξζ , 0

)(

)(=�

�

��

� ′ζ

hf

hf

dh

d,

0)(

)(2

)(

)(46 =−�

�

��

�−− xxxxxxh

hf

hg

hf

hg

dh

dξζξζ , 0

)(

)(64 =

′+− xxxxh

hf

hfζξζ .

79

Similarly, system (4.3) remains invariant under group transformations (4.4) provided the

group generators ),,( htxξ , ),,( htxη and ),,( htxζ satisfy the determining equations

0=hξ , 0== xh ηη , 0=hhζ , 0=− xxxh ξζ ,

( ) 03)(

)(

)(

)(

)(

)(=−

′−+

′

′−�

�

��

�

′ xhxxxxhf

hj

hf

hg

hf

hj

dh

dξζζζζ ,

0)(

)(

)(

)(23 =�

�

��

�

′

′−

′

′−−

hf

hg

dh

d

hf

hgxxxxxxh ζξξζ , 0

)(

)()(4 =

′−′−

hf

hftx ζηξ ,

0)(

)(2

)(

)(46 =−�

�

��

�−− xxxxxxh

hf

hg

hf

hg

dh

dξζξζ , (4.7)

( ) 064)(

)()()()( =−

′−+− xxxhxxxxxxt

hf

hjhfhfhg ξζζζζ ,

( ) ( ) ( ) 02)(64)(

)()(4)( =−−−−

′

′+− xxxhtxxxhxxxxxxxh hg

hf

hghfhf ξζξξζξζ .


terms not involving derivatives of h within the invariance requirements for each of

systems (4.2) and (4.3) produces systems (4.6) and (4.7) respectively. All subscripts in

systems (4.6) and (4.7) symbolise partial differentiation with ,x t and h as independent

variables. Throughout this chapter, primes represent differentiation with respect to the

argument indicated.

Systems (4.6) and (4.7) enable the retrieval of all Lie classical symmetries and

corresponding conditions on 0)( ≠hf , )(hg and )(hj for the respective systems (4.2)

and (4.3) under transformations (4.4). System (4.2) admits six new Lie classical groups

extending beyond the confines of groups obtainable via the classical method for the thin

film equation (4.1). We recover one such new group for system (4.3). These new groups

enhance the symmetries of the thin film equation (4.1). One may therefore consider

symmetry-enhancing constraints to be added to the thin film equation (4.1).

We describe each of these groups in the following pages. Where applicable, we list the

special cases arising for each such group. Then we present the similarity solutions of

systems (4.2) and (4.3) associated with each of these groups. Solutions of systems (4.2)

and (4.3) are also solutions to the thin film equation (4.1).

80

GROUP (I)

Subject to the conditions 0)( 1 ≠= fhf , 0)( 1 ≠= ghg and 0)( 1 ≠= jhj , system (4.2)

admits classical group (I) given by

,)(),,( 31 ctjhtx += αξ ),(),,( thtx αη =

(4.8)

( )( ) ( )

;),,( 654121

11

11

1

1

cecectjxchchtxtjx

f

gtjx

f

g

+++−+=−−−

ζ

where ,01 ≠f ,01 ≠g 01 ≠j and ic are arbitrary constants for all i ∈ }6,5,4,3,2,1{

while )(tα is an arbitrary function of .t

A special case of group (4.8) emerges and is as follows.

Case (1) The case of group (4.8) with 0=ic for all i ∈ }6,5,4,2,1{ and

7)( ct =β occurs under conditions 0)( 1 ≠= fhf , 0)( 1 ≠= ghg and 0)( ≠hj is an

arbitrary function of h where 01 ≠f , 01 ≠g and 7c are arbitrary constants.


Group (4.8), system (4.2) and the invariant surface condition (4.5) give the equations

,01 =+ xt hjh ,01

1 =− xxxxxx hf

gh

(4.9)

xhc3 ( )( ) ( )

;654121

11

11

1

1

cecectjxchctjx

f

gtjx

f

g

+++−+=−−−

where ,01 ≠f ,01 ≠g 01 ≠j and ic are arbitrary constants for all i ∈ }.6,5,4,3,2,1{ As

0=xh forces 0=th in equation (4.9)1 , giving =),( txh constant, we require 0≠xh for

system (4.9) to yield nonconstant similarity solutions.

Via the method in [24], we find the general solution of equation (4.9)1 to be the

travelling wave of velocity ,01 ≠j namely

);(),( uytxh = (4.10)

where 01 ≠j is an arbitrary constant and )(uy is an arbitrary function of .1tjxu −= We

require 0)( ≠′ uy for solution (4.10) to be nonconstant.

81

Substituting solution (4.10) into equation (4.9)2 and solving the resulting equation via the

method in [48] yields the general solution

;)( 43211

1

1

1

dudededuyu

f

gu

f

g

+++=−

(4.11)

where tjxu 1−= while ,01 ≠f ,01 ≠g ,01 ≠j ,1d ,2d 3d and 4d are arbitrary

constants. For solution (4.11) to be nonconstant requires

.0321

1

1 1

1

1

1

≠+��

�

�

�−

−

dededf

g uf

gu

f

g

For the purpose of obtaining nonconstant similarity solutions from system (4.9), we

impose the restriction .0≠xh Accordingly, equation (4.9)3 generates the cases

(1) 0=ic for all i ∈ },6,5,4,3,2,1{

(2) ,03 ≠c ( )( ) ( )

.0654121

11

11

1

1

≠+++−+−−−

cecectjxchctjx

f

gtjx

f

g

Case (1) 0=ic for all i ∈ }6,5,4,3,2,1{

Under the condition 0=ic for all i ∈ },6,5,4,3,2,1{ equation (4.9)3 vanishes.

Results (4.10) and (4.11) imply that under transformations (4.4) and conditions

,0)( 1 ≠= fhf 0)( 1 ≠= ghg and ,0)( 1 ≠= jhj the similarity solution of system (4.2)

and the thin film equation (4.1) in association with group (4.8) subject to the constraint

0=ic for all i ∈ }6,5,4,3,2,1{ is the travelling wave of velocity ,01 ≠j namely

( ) ( )

( ) ;),( 41321

11

11

1

1

dtjxdededtxhtjx

f

gtjx

f

g

+−++=−−−

(4.12)

where ,01 ≠f ,01 ≠g ,01 ≠j ,1d ,2d 3d and 4d are arbitrary constants. We require

( ) ( )

0321

1

11

1

11

1

1

≠+��

�

�

��

�

�−

−−−

dededf

g tjxf

gtjx

f

g

for solution (4.12) to be nonconstant.

Figure (4a) displays thin film height profiles for solution (4.12) of system (4.2) at the

times =t 0, 1, 2 and 3 respectively.

82

Figure (4a). Thin film height )(h profile versus position )(x at the times =t 0, 1, 2, 3

for 1421111 ====== dddjgf and .13 −=d

Figure (4a) clearly indicates that as time increases, the fluid flows from where its

concentration is greatest to where it is least.

Case (2) ,03 ≠c ( )( ) ( )

0654121

11

11

1

1

≠+++−+−−−

cecectjxchctjx

f

gtjx

f

g

Substituting results (4.10) and (4.11) into equation (4.9)3 yields the subcases

(a) ,04

1

131 =≠= c

f

gcc ,

21

13

5

2

f

gc

cd −= ,

1

13

23

f

gc

cd −= ,

13

1

1621

4gc

f

gccf

d

��

�

�

�+

−=

(b) ,05

1

131 =≠−= c

f

gcc ,

21

13

41

f

gc

cd = ,

1

13

23

f

gc

cd = ,

13

2

1

161

4gc

cf

gcf

d

��

�

�

�−

=

(c) ≠3c ,021 == cc ,

1

13

41

f

gc

cd = ,

1

13

5

2

f

gc

cd −= ,

3

6

3c

cd =

(d) ,01

131 ≠±

f

gcc ,031 ≠cc ,

1

131

41

f

gcc

cd

−

−= ,

1

131

5

2

f

gcc

cd

+

−= ,1

23

c

cd −=

.2

1

3261

4c

ccccd

+−=

83

For subcases (a), (b), (c) and (d), it follows from results (4.10) and (4.11) that under

transformations (4.4) and conditions ,0)( 1 ≠= fhf 0)( 1 ≠= ghg and ,0)( 1 ≠= jhj

the similarity solution of system (4.2) and the thin film equation (4.1) in connection with

group (4.8) and subject to the constraints on each subcase is the travelling wave (4.12)

where ,03 ≠c ,01 ≠f ,01 ≠g ,01 ≠j ,1d ,2d 3d and 4d are arbitrary constants such

that ( ) ( )

0321

1

11

1

11

1

1

≠+��

�

�

��

�

�−

−−−

dededf

g tjxf

gtjx

f

g

for solution (4.12) to be nonconstant.

The combinations of ic expressing ,1d ,2d 3d and 4d in subcases (a), (b), (c) and (d)

form arbitrary constants.

GROUP (II)

Under conditions 0)( 2

0 ≠= hfhf , 0)( =hg and 0)( 1 ≠= jhj , system (4.2) admits

classical group (II) given by

( ) ( ) ,)(2

),,( 3112

2

11 ctjtjxctjx

chtx ++−+−= αξ ),(),,( thtx αη =

(4.13)

( )[ ]hctjxchtx 411),,( +−=ζ ;

where 00 ≠f , 01 ≠j and ic are arbitrary constants for all i ∈ }4,3,2,1{ while )(tα is

an arbitrary function of t . Furthermore, .0≠h

Two special cases of group (4.13) occur and are as follows.

Case (1) The case of group (4.13) with 041 == cc arises under the conditions

0)( ≠hf is an arbitrary function of ,h 0)( =hg and 0)( 1 ≠= jhj where 01 ≠j is an

arbitrary constant and .0)( ≠′ hf

Case (2) The case of group (4.13) with 01 =c appears under conditions

,0)( 5

0 ≠= chfhf 0)( =hg and 0)( 1 ≠= jhj where ,05 ≠c 00 ≠f and 01 ≠j are

arbitrary constants while 0≠h .

84


Group (4.13), system (4.2) and the invariant surface condition (4.5) imply

01 =+ xt hjh , 02

=+ xxxxxxxx hhh

h ,

(4.14)

( ) ( ) ( )[ ] hctjxchctjxctjxc

x 411312

2

1

1

2+−=�

�

��

�+−+− ;

where 01 ≠j and ic are arbitrary constants for all i ∈ }4,3,2,1{ and 0≠h . Since

0=xh forces 0=th in equation (4.14)1 , giving =),( txh constant, we require 0≠xh

for system (4.14) to generate nonconstant similarity solutions.

By the method in [24], we deduce the general solution of equation (4.14)1 to be the

travelling wave solution of velocity 01 ≠j , namely

0)(),( ≠= uytxh ; (4.15)

where 01 ≠j is an arbitrary constant and 0)( ≠uy is an arbitrary function of .1tjxu −=

We require 0)( ≠′ uy for solution (4.15) to be nonconstant.

Substituting result (4.15) into equations (4.14)2 and (4.14)3 yields the cases

(1) 0)( ==′′′icuy for all i ∈ },4,3,2,1{ (2) 0)( =≠′′′

icuy for all i ∈ },4,3,2,1{

(3) .02

32

21 ≠++ cucuc

Case (1) 0)( ==′′′icuy for all i ∈ }4,3,2,1{

Equations (4.14)2 and (4.14)3 vanish under the constraint on this case.

Result (4.15) reflects that under transformations (4.4) and conditions 0)( 2

0 ≠= hfhf ,

0)( =hg and 0)( 1 ≠= jhj , the similarity solution of system (4.2) and the thin film

equation (4.1) in tandem with group (4.13) subject to the constraint 0=ic for all

i ∈ }4,3,2,1{ is the travelling wave of velocity 01 ≠j , namely

( ) ( ) 0),( 716

2

15 ≠+−+−= ctjxctjxctxh ; (4.16)

where 01 ≠j , 5c , 6c and 7c are arbitrary constants. We require ( ) 02 615 ≠+− ctjxc for

solution (4.16) to be nonconstant.

85

Case (2) 0)( =≠′′′icuy for all i ∈ }4,3,2,1{

Under this constraint , equation (4.14)3 vanishes.

Result (4.15) and equation (4.14)2 imply that under transformations (4.4) and conditions

0)( 2

0 ≠= hfhf , 0)( =hg and 0)( 1 ≠= jhj , the similarity solution of system (4.2) and

the thin film equation (4.1) associated with group (4.13) under the constraint 0=ic for

all i ∈ }4,3,2,1{ is the travelling wave (4.15) satisfying the equation

0)()( 5

2 ≠=′′′ cuyuy ; (4.17)

where 05 ≠c and 01 ≠j are arbitrary constants while tjxu 1−= and .0)( ≠′′′ uy

Case (3) 02

32

21 ≠++ cucuc


(a) ,24 cc = (b) ,02 124 =≠= ccc (c) ( ) .022

24

1

4

31 =−+≠ ccc

ccc

Subcase (a) ,02

32

21 ≠++ cucuc

24 cc =

Result (4.15) and equations (4.14)2 and (4.14)3 imply that under transformations (4.4)

and conditions 0)( 2

0 ≠= hfhf , 0)( =hg and 0)( 1 ≠= jhj , the similarity solution of

system (4.2) and the thin film equation (4.1) in conjunction with group (4.13) subject to

the constraints on this subcase is the travelling wave of velocity 01 ≠j , namely

( ) ( ) 02

),( 312

2

1

1

5 ≠��

��

�+−+−= ctjxctjx

cctxh ; (4.18)

where ,05 ≠c 01 ≠j and ic are arbitrary constants for all i ∈ }3,2,1{ while

( ) ( ) .02

312

2

11 ≠+−+− ctjxctjx

c We require ( ) 0211 ≠+− ctjxc for solution (4.18) to

be nonconstant.

86

Subcase (b) ,032 ≠+ cuc 02 124 =≠= ccc

Relations (4.14)2 , (4.14)3 and (4.15) reflect that under transformations (4.4) and

conditions ,0)( 2

0 ≠= hfhf 0)( =hg and ,0)( 1 ≠= jhj the similarity solution of

system (4.2) and the thin film equation (4.1) in connection with group (4.13) subject to

the restriction 02 124 =≠= ccc is the travelling wave of velocity ,01 ≠j namely

( )[ ] ;0),( 2

3125 ≠+−= ctjxcctxh (4.19)

where ,02 ≠c ,05 ≠c 01 ≠j and 3c are arbitrary constants with ( ) .0312 ≠+− ctjxc

Subcase (c) ,02

32

21 ≠++ cucuc

( ) 022

24

1

4

31 =−+≠ ccc

ccc

Equations (4.14)2 , (4.14)3 and (4.15) indicate that under transformations (4.4) and

conditions ,0)( 2

0 ≠= hfhf 0)( =hg and ,0)( 1 ≠= jhj the similarity solution of

system (4.2) and the thin film equation (4.1) in tandem with group (4.13) and the

constraints on this subcase is the travelling wave of velocity ,01 ≠j namely

( ) ;0),(2

615 ≠+−= ctjxctxh (4.20)

where ,01 ≠c ,05 ≠c ,01 ≠j ,2c 4c and 1

42

6

2

c

ccc

−= are arbitrary constants with

.061 ≠+− ctjx

GROUP (III)

Subject to conditions ( ) ,0)(2

20 ≠+= chfhf 0)( =hg and ,0)( 1 ≠= jhj system (4.2)

admits classical group (III) given by

( ) ( ) ,0)(2

),,( 4113

2

11 ≠++−+−= ctjtjxctjx

chtx αξ ),(),,( thtx αη =

(4.21)

( )( ) ;0),,( 5121 ≠+−+= ctjxchchtxζ

where ,01 ≠c ,02 ≠c ,00 ≠f ,01 ≠j ,3c 4c and 5c are arbitrary constants such that

( )( ) 0512 ≠+−+ ctjxch and ( ) ( ) 0)(2

4113

2

11 ≠++−+− ctjtjxctjx

cα while )(tα is an

arbitrary function of .t

87


Group (4.21), system (4.2) and the invariant surface condition (4.5) yield the equations

,01 =+ xt hjh ,02

2

=+

+ xxxxxxxx hhch

h

(4.22)

( ) ( ) ( )( ) ;02

5121413

2

1

1 ≠+−+=��

��

�+−+− ctjxchchctjxctjx

cx

where ,01 ≠c ,02 ≠c ,01 ≠j ,3c 4c and 5c are arbitrary constants with

( )( ) 0512 ≠+−+ ctjxch and ( ) ( ) .02

413

2

11 ≠+−+− ctjxctjx

c For equation (4.22)3 to

be consistent requires ,0≠xh forcing 0≠th in equation (4.22)1 .

By the method in [24], we find the general solution of equation (4.22)1 to be the

travelling wave solution of velocity ,01 ≠j namely

;0)(),( ≠= uytxh (4.23)

where 01 ≠j is an arbitrary constant and 0)( ≠uy is an arbitrary function of tjxu 1−=

such that 0)( ≠′ uy since .0≠xh

Equations (4.22)2 and (4.22)3 yield the cases

(1) ,513 ccc = (2) .2

51

354 ��

�

�−=

ccccc

Case (1) 513 ccc =

Result (4.23) with equations (4.22)2 and (4.22)3 imply that under transformations (4.4)

and the conditions ( ) ,0)(2

20 ≠+= chfhf 0)( =hg and ,0)( 1 ≠= jhj the similarity

solution of system (4.2) and the thin film equation (4.1) in association with group (4.21)

subject to the constraint 513 ccc = is the travelling wave of velocity ,01 ≠j namely

( ) ( )[ ] ;0),( 817

2

16 ≠+−+−= ctjxctjxctxh (4.24)

where ,01 ≠c ,02 ≠c ,06 ≠c ,01 ≠j ,4c ,5c 57 2cc = and 2

1

64

8 2 cc

ccc −= are

arbitrary constants with 051 ≠+− ctjx and ( ) ( ) .02

4151

2

11 ≠+−+− ctjxcctjx

c

88

Case (2) ��

�

�−=

2

51

354

ccccc

Result (4.23) and equations (4.22)2 and (4.22)3 reveal that under transformations (4.4)


20 ≠+= chfhf 0)( =hg and ,0)( 1 ≠= jhj the similarity

solution for system (4.2) and the thin film equation (4.1) in connection with group (4.21)

under the constraint ��

�

�−=

2

51

354

ccccc is the travelling wave of velocity ,01 ≠j namely

( ) ;0),( 2

2

716 ≠−+−= cctjxctxh (4.25)

where ,01 ≠c ,02 ≠c ,06 ≠c ,01 ≠j ,3c 5c and 5

1

3

7

2c

c

cc −= are arbitrary constants

while ( )( ) .05171 ≠+−+− ctjxctjx

GROUP (IV)

Under conditions 0)( ≠hf is an arbitrary function of ,h 1)( ghg = and ,0)( 1 ≠= jhj

system (4.2) admits classical group (IV) given by

,)(),,( 11 ctjhtx += αξ ),(),,( thtx αη = ;0),,( =htxζ (4.26)

where ,01 ≠j 1c and 1g are arbitrary constants while )(tα is an arbitrary function of .t

Furthermore 0)( 1 ==′ ghf does not occur.

A special case of group (4.26) arises and is as follows.

Case (1) The case of group (4.26) with 32)( ctct +=α appears under conditions

,0)( ≠hf )(hg and )(hj are arbitrary functions of h where 2c and 3c are arbitrary

constants. Neither 0)()( ==′ hghf nor 0)()( ==′ hjhg occurs.

89



,01 =+ xt hjh [ ] ,0)( 1 =−∂

∂xxxxx hghhf

x ;01 =xhc (4.27)

where ,01 ≠j 1c and 1g are arbitrary constants with 0)( ≠hf a function of h such that

0)( 1 ==′ ghf does not occur. Since 0=xh forces 0=th in equation (4.27)1 , yielding

only constant solutions for system (4.27), we require 0≠xh for system (4.27) to

generate nonconstant solutions. This requirement forces 01 =c in equation (4.27)3 .

By the method in [24], we solve equation (4.27)1 and substitute its general solution into

equation (4.27)2 . Consequently, it is clear that under transformations (4.4) and the

conditions 0)( ≠hf is an arbitrary function of ,h 1)( ghg = and ,0)( 1 ≠= jhj the

similarity solution of system (4.2) and the thin film equation (4.1) in connection with

group (4.26) and the constraint 01 =c is the travelling wave of velocity ,01 ≠j namely

;)(),( uytxh = (4.28)

such that

( ) ( ) ;0)()()()()()( 1

)4( =′′−′′′′′+ uyguyuyuyfuyuyf (4.29)

where 01 ≠j and 1g are arbitrary constants while tjxu 1−= and ( ) 0)( ≠uyf is a

function of )(uy such that ( ) 0)( 1 ==′ guyf does not occur. We require 0)( ≠′ uy for

solution (4.28) to be nonconstant. As ,01 ≠j solution (4.28) is not reducible to a steady

state solution of system (4.2) and the thin film equation (4.1).

GROUP (V)


1 ≠= hfefhf 1)( ghg = and ,0)( 1 ≠= jhj system (4.2)

admits classical group (V) given by

( ) ,)(2

),,( 211

01 ctjtjxfc

htx ++−= αξ ),(),,( thtx αη = ;0),,( 1 ≠= chtxζ (4.30)

where )(tα is an arbitrary function of t while ,01 ≠c ,01 ≠f ,01 ≠j ,2c 0f and 1g are

arbitrary constants with 001 == fg not occurring.

A special case of group (4.30) emerges and is as follows.

90

Case (1) The case of group (4.30) with 43)( ctct +=α arises under the conditions





,01 =+ xt hjh ( ) ,01010 =−+ xxxxxxxxxx

hfhghhfhef ( ) ;0

2121

01 ≠=��

��

�+− chctjx

fcx

(4.31)

where ,01 ≠c ,01 ≠f ,01 ≠j ,2c 0f and 1g are arbitrary constants with 001 == fg

not occurring and ( ) .02

21

01 ≠+− ctjxfc

Consistency in equation (4.31)3 requires

,0≠xh forcing 0≠th in equation (4.31)1 .

By the method in [24], we find the general solution of equation (4.31)1 to be the

travelling wave of velocity 01 ≠j , namely

;0)(),( ≠= uytxh (4.32)


such that 0)( ≠′ uy since .0≠xh

Equation (4.31)3 yields the cases (1) ,00 ≠f (2) .002 =≠ fc

Case (1) 00 ≠f

Substituting result (4.32) into equation (4.31)3 , directly solving the resulting equation

and substituting its general solution with result (4.32) into equation (4.31)2 forces the

constraint .02 04

11 ≠= fcefg Hence under transformations (4.4) and the conditions

,0)( 0

1 ≠= hfefhf 0)( 1 ≠= ghg and ,0)( 1 ≠= jhj the similarity solution of system

(4.2) and the thin film equation (4.1) in tandem with group (4.30) and the constraints

00 ≠f and 02 04

11 ≠= fcefg is the travelling wave of velocity 01 ≠j , namely

;0ln2

),( 431

0

≠++−= cctjxf

txh (4.33)

where ,01 ≠c ,00 ≠f ,01 ≠f ,01 ≠j ,2c 01

2

3

2

fc

cc = and 4c are arbitrary constants with

.031 ≠+− ctjx

91

Case (2) 002 =≠ fc

We substitute result (4.32) into equation (4.31)3 , directly solve the resulting equation

and substitute its general solution with result (4.32) into equation (4.31)2 , causing

equation (4.31)2 to vanish. Hence under transformations (4.4) and the conditions


and the thin film equation (4.1) in association with group (4.30) and the constraint

002 =≠ fc is the travelling wave of velocity ,01 ≠j namely

( ) ;0),( 413 ≠+−= ctjxctxh (4.34)

where ,01 ≠c ,02 ≠c ,02

1

3 ≠=c

cc 01 ≠j and 4c are arbitrary constants.

GROUP (VI)

Under the conditions ( ) 0)( 0

21 ≠+=f

fhfhf , 1)( ghg = and 0)( 1 ≠= jhj , system (4.2)

admits classical group (VI) given by

( ) ,)(2

),,( 211

01 ctjtjxfc

htx ++−= αξ ),(),,( thtx αη = ( ) ;0),,( 21 ≠+= fhchtxζ

(4.35)

where )(tα is an arbitrary function of t while ,01 ≠c ,01 ≠f ,01 ≠j ,2c ,0f 2f and

1g are arbitrary constants with 001 == fg not occurring and .02 ≠+ fh


Case (1) The case of group (4.35) with 43)( ctct +=α appears under conditions



92


Group (4.35), system (4.2) and the invariant surface condition (4.5) generate equations

,01 =+ xt hjh ( ) ,1

2

0

210

xxxxxxxxxx

fhghh

fh

fhfhf =��

�

�

�

+++

(4.36)

( ) ( ) ;02

2121

01 ≠+=��

��

�+− fhchctjx

fcx

where ( ) 02

21

01 ≠+− ctjxfc

and 02 ≠+ fh while ,01 ≠c ,01 ≠f ,01 ≠j ,2c ,0f 2f

and 1g are arbitrary constants with 001 == fg not occurring. Consistency in equation

(4.36)3 requires ,0≠xh forcing 0≠th in equation (4.36)1 .

Via the method in [24], we deduce the general solution of equation (4.36)1 to be the

travelling wave of velocity ,01 ≠j namely

;0)(),( ≠= uytxh (4.37)


with 0)( ≠′ uy since .0≠xh

Equation (4.36)3 admits the cases (1) ,00 ≠f (2) .002 =≠ fc

Case (1) 00 ≠f


and substituting its general solution with result (4.37) into equation (4.36)2 yields the

subcases (a) ,20 =f (b) ( )( ).122

002

0

31

1

0

−−= fff

cfg

f

Subcase (a) 20 =f

Under transformations (4.4) with the conditions ( ) ,0)(2

21 ≠+= fhfhf 1)( ghg = and

,0)( 1 ≠= jhj the similarity solution of system (4.2) and the thin film equation (4.1)

associated with group (4.35) and the constraint 20 =f is the travelling wave of velocity

,01 ≠j namely

( ) ;0),( 2413 ≠−+−= fctjxctxh (4.38)

where ,01 ≠c ,03 ≠c ,01 ≠j ,2c 1

2

4c

cc = and 2f are arbitrary constants with

.041 ≠+− ctjx

93

Subcase (b) ,00 ≠f ( )( )122

002

0

31

1

0

−−= fff

cfg

f

Subject to transformations (4.4) with conditions ( ) ,0)( 0

21 ≠+=f

fhfhf 1)( ghg = and

,0)( 1 ≠= jhj the similarity solution of system (4.2) and the thin film equation (4.1) in

tandem with group (4.35) and the constraints 00 ≠f and ( )( )122

002

0

31

1

0

−−= fff

cfg

f

is

the travelling wave of velocity ,01 ≠j namely

( ) ;0),( 2

/2

4130 ≠−+−= fctjxctxh

f (4.39)

where ,01 ≠c ,03 ≠c ,00 ≠f ,01 ≠f ,01 ≠j ,2c 01

2

4

2

fc

cc = and 2f are arbitrary

constants with .041 ≠+− ctjx

Case (2) 002 =≠ fc


and substituting its general solution with result (4.37) into equation (4.36)2 yields the

constraint .02

411 ≠= cfg Therefore under transformations (4.4) and the conditions


and the thin film equation (4.1) in conjunction with group (4.35) and the constraints

002 =≠ fc and 02

411 ≠= cfg is the travelling wave of velocity ,01 ≠j namely

( )

;0),( 2314 ≠−= −

fectxhtjxc

(4.40)

where ,01 ≠c ,02 ≠c ,03 ≠c ,02

1

4 ≠=c

cc ,01 ≠f 01 ≠j and 2f are arbitrary

constants.

94

GROUP (VII)

Subject to the conditions ( ) ,0)(4

21 ≠+= fhfhf ( ) 0)(4

2

2

21 ≠+= fhcfhg and

,0)( =hj system (4.3) admits classical group (VII), namely

,),,( 43122 cecechtxxcxc ++= −ξ ,),,( 65 ctchtx +=η

(4.41)

( ) ;04

),,( 2

5

322122 ≠+�

�

�

�−−= −

fhc

eccecchtxxcxcζ

where ,02 ≠c ,01 ≠f ,1c ,3c ,4c ,5c 6c and 2f are arbitrary constants with

( ) 04

5

31222 ≠−− − c

ececcxcxc and .02 ≠+ fh



( ),0

1

4

22

2 =+

+−−

txxxxxx hf

fhhch ( ) ,0

2

2 =− xxxxx hchh

(4.42)

( ) ( ) ( ) ;04

2

5

3221654312222 ≠+�

�

�

�−−=++++ −−

fhc

eccecchctchcececxcxc

tx

xcxc

where ,02 ≠c ,01 ≠f ,1c ,3c ,4c ,5c 6c and 2f are arbitrary constants with

( ) 04

5

31222 ≠−− − c

ececcxcxc and .02 ≠+ fh Since 0=xh forces 0=th in equation

(4.42)1 , generating the contradiction ( ) 04

2

5

322122 =+�

�

�

�−− −

fhc

ecceccxcxc

in equation

(4.42)3 , we require 0≠xh for equation (4.42)3 to be consistent.

The requirement 0≠xh forces equation (4.42)2 to give

;02

2 ≠= xxxx hch (4.43)

where 02 ≠c is an arbitrary constant and .0≠xh

95

As relation (4.43) forces 0=th in equation (4.42)1 , equations (4.42) and (4.43) imply

,0)(),( ≠= xytxh ,0)()(2

2 ≠′=′′′ xycxy

( );04

)(

)(

431

5312

222

22

≠++

−−=

+

′−

−

cecec

cececc

fxy

xyxcxc

xcxc

(4.44)

where ,0)( 2 ≠+ fxy ( ) 04

5312

22 ≠−− − cececc

xcxc and 043122 ≠++ −

cececxcxc

with

,02 ≠c ,1c ,3c 4c and 2f being arbitrary constants. Furthermore 0)( ≠′ xy owing to the

requirement .0≠xh

Result (4.44)1 indicates steady state solutions to be the only similarity solutions of

system (4.3) and the thin film equation (4.1) under the conditions on ,0)( ≠hf )(hg and

)(hj for group (4.41).

Via the method in [48], we find the general solution of equation (4.44)2 to be the steady

state solution

;0)( 98722 ≠++= −

cececxyxcxc

(4.45)

where ,02 ≠c ,7c 8c and 9c are arbitrary constants with .022

87 ≠− − xcxcecec

Substituting result (4.45) into equation (4.44)3 and manipulating the resulting equation

yields the cases

(1) ,0751 ≠ccc ( )

,64

162

21

2

5

2

42

3cc

cccc

−=

( )[ ],

64

282

2

2

1

54242

2

57

8cc

cccccccc

++=

( )

,4

42

21

5427

9 fcc

ccccc −

+=

(2) ,0853 ≠ccc ( )

,64

16

3

2

2

2

5

2

42

1cc

cccc

−=

( )[ ],

64

282

3

2

2

54242

2

5

87cc

cccccccc

−+=

( )

,4

42

32

5428

9 fcc

ccccc −

−=

(3) ,0731854 ===≠ cccccc ,04 4

5

2 ≠=c

cc ,29 fc −=

(4) ,0831754 ===≠ cccccc ,04 4

5

2 ≠−=c

cc ,29 fc −=

(5) ,0571 =≠ ccc ,1

73

8c

ccc = ,2

1

74

9 fc

ccc −=

(6) ,0583 =≠ ccc ,3

81

7c

ccc = .2

3

84

9 fc

ccc −=

96

For cases (1) - (6), relations (4.44)1 and (4.45) imply that under transformations (4.4) and

conditions ( ) ,0)(4

21 ≠+= fhfhf ( ) 0)(4

2

2

21 ≠+= fhcfhg and ,0)( =hj the

similarity solution of system (4.3) and the thin film equation (4.1) in association with

group (4.41) and the constraints on each case is the steady state solution

;0),( 98722 ≠++= −

cecectxhxcxc

(4.46)

where ,02 ≠c ,7c 8c and 9c are arbitrary constants with 022

87 ≠− − xcxcecec since

0≠xh is a requirement. In cases (1) – (6), the combinations of ic expressing ,1c ,3c

,7c 8c and 9c form arbitrary constants.

Figure (4b) displays the thin film height profile of solution (4.46) for case (1) at the

position ∈x [ ]1,0 .

Figure (4b). Thin film height ( )h profile versus position ( )x for .19872 ==== cccc

We present four tables of results at the end of this chapter. Table 1 outlines the functions

),(hf )(hg and )(hj (distinguishing enhanced symmetries of thin film equation (4.1))

with their associated infinitesimal generators .iV Table 2 displays a dimensional

classification of the mathematical structure of groups (I)-(VII) and the corresponding .iV

Table 3 features the similarity solutions ),( txh with their similarity variables ),( txu

(where applicable) for systems (4.2) and (4.3) in tandem with groups (I)-(VII). Table 4

shows the defining ordinary differential equations (ODEs) for the functions )(uy within

the functional forms of ),( txh relating to groups (II) and (IV) in table 3.

97

Each infinitesimal generator iV for all i ∈ }7,...,2,1{ represents the Lie algebra for the

respective Lie groups (I)-(VII); (see Gandarias [27]). These generators are as follows.

A List of the Infinitesimal Generators for Groups (I)-(VII)

The generators iV for all i ∈ }7,...,2,1{ for the respective groups (I)-(VII) are

[ ] ( )( ) ( )

,)()( 654121311

11

11

1

1

hcecectjxchc

tt

xctjV

tjxf

gtjx

f

g

∂

∂

��

�

�

��

�

�+++−++

∂

∂+

∂

∂+=

−−−

αα

( ) ( ) ( )[ ] ,)()(2

4113112

2

1

1

2h

hctjxct

tx

ctjtjxctjxc

V∂

∂+−+

∂

∂+

∂

∂��

��

�++−+−= αα

( ) ( ) ( )( ) ,)()(2

51214113

2

1

1

3h

ctjxchct

tx

ctjtjxctjxc

V∂

∂+−++

∂

∂+

∂

∂��

��

�++−+−= αα

[ ] ,)()( 114t

tx

ctjV∂

∂+

∂

∂+= αα

( ) ,)()(2

1211

01

5h

ct

tx

ctjtjxfc

V∂

∂+

∂

∂+

∂

∂��

��

�++−= αα

( ) ( ) ,)()(2

21211

01

6h

fhct

tx

ctjtjxfc

V∂

∂++

∂

∂+

∂

∂��

��

�++−= αα

( ) ( ) ( ) ;4

2

5

32216543172222

hfh

ceccecc

tctc

xcececV

xcxcxcxc

∂

∂+�

�

�

�−−+

∂

∂++

∂

∂++= −−

where details of each iV for all i ∈ }7,...,2,1{ relate to the respective groups (I)-(VII).


Table 1. Each row lists the functions ),(hf )(hg and )(hj (distinguishing the enhanced

symmetries of thin film equation (4.1)) with the associated infinitesimal generators .iV


I 01 ≠f 01 ≠g 01 ≠j 1V

II 02

0 ≠hf 0 01 ≠j 2V

III ( ) 02

20 ≠+ chf 0 01 ≠j 3V

IV arbitrary 0≠ 1g 01 ≠j 4V

V 00

1 ≠hfef 1g 01 ≠j 5V

VI ( ) 00

21 ≠+f

fhf 1g 01 ≠j 6V

VII ( ) 04

21 ≠+ fhf ( ) 04

2

2

21 ≠+ fhcf 0 7V

98

Table 2. A dimensional classification of the mathematical structure of groups (I)-(VII)

(the enhanced symmetries of the thin film equation (4.1)) with their associated


),,( htxξ ),,( htxη ),,( htxζ iV

31 )( ctj +α )(tα

( )( )tjx

f

g

ectjxchc1

1

1

4121

−

+−+

( )

65

11

1

cectjx

f

g

++−−

1V

( ) ( ) 3112

2

11 )(

2ctjtjxctjx

c++−+− α

)(tα ( )[ ] hctjxc 411 +− 2V

( ) ( ) 0)(2

4113

2

11 ≠++−+− ctjtjxctjx

cα

)(tα ( )( ) 05121 ≠+−+ ctjxchc 3V

11 )( ctj +α )(tα 0 4V

( ) 21101 )(

2ctjtjx

fc++− α

)(tα 01 ≠c 5V

( ) 21101 )(

2ctjtjx

fc++− α

)(tα ( ) 021 ≠+ fhc 6V

43122 cececxcxc ++ −

65 ctc + ( ) 0

42

5

322122 ≠+�

�

�

�−− −

fhc

ecceccxcxc

7V

The entries for ),,,( htxξ ),,( htxη and ),,( htxζ in each of rows 1 – 7 in table 2

respectively relate to Lie classical groups (I) – (VII).

99

Table 3. Rows 1-14 list the similarity solutions ),( txh and any corresponding similarity

variables ),( txu for system (4.2) associated with groups (I)-(VI). Row 15 features the

similarity solution ),( txh for system (4.3) in connection with group (VII). Cases 4, 2(1),

1(1,2) and 6(1a) relate to group (IV), group (II) case (1), group (I) cases (1) and (2) and

group (VI) case (1) subcase (a) respectively. The same denotation pattern applies to

other similarly-named cases in this table.

Case ),( txh ),( txu

1(1,2) ( ) ( )

( ) 41321

11

11

1

1

dtjxdededtjx

f

gtjx

f

g

+−++−−−

2(1) ( ) ( ) 0716

2

15 ≠+−+− ctjxctjxc under the constraints

04321 ==== cccc

2(2) ( ) 0),( ≠txuy under the constraints 0)( 4321 ====≠′′′ ccccuy tjx 1−

2(3a) ( ) ( ) 0

2312

2

1

1

5 ≠��

��

�+−+− ctjxctjx

cc under the constraint 24 cc =

2(3b) ( )[ ] 02

3125 ≠+− ctjxcc under the constraints 02 124 =≠= ccc

2(3c) ( ) 02

615 ≠+− ctjxc under the constraints

( ) 022

24

1

4

31 =−+≠ ccc

ccc

3(1) ( ) ( )[ ] 0817

2

16 ≠+−+− ctjxctjxc under the constraint 513 ccc =

3(2) ( ) 02

2

716 ≠−+− cctjxc under the constraint ��

�

�−=

2

51

354

ccccc

4 ( )),( txuy under the constraint 01 =c tjx 1−

5(1) 0ln

2431

0

≠++− cctjxf


00 ≠f and 02 04

11 ≠= fcefg

5(2) ( ) 0413 ≠+− ctjxc under the constraints ≠12 gc 00 =f

6(1a) ( ) 02413 ≠−+− fctjxc under the constraint 20 =f

6(1b) ( ) 02

/2

4130 ≠−+− fctjxc

f under the constraints

00 ≠f and ( )( )122

002

0

31

1

0

−−= fff

cfg

f

6(2) ( )023

14 ≠−−fec

tjxc under the constraints 002 =≠ fc and 0

2

411 ≠= cfg

7 098722 ≠++ −

cececxcxc

100

Table 4. The rows below list the defining ODEs for the functions ( ),),( txuy the

functional forms of ),( txh connected with group (II) case (2) and group (IV) in table 3.

Cases 2(2) and 4 refer to group (II) case (2) and group (IV) respectively.

Case ( ) 0,,,, )4( =′′′′′′ yyyyyA

2(2) 0)()( 5

2 ≠=′′′ cuyuy

4 ( ) ( ) 0)()()()()()( 1

)4( =′′−′′′′′+ uyguyuyuyfuyuyf

with ( ) 0)( 1 ==′ guyf not occurring and ( ) 0)( ≠uyf


Classical symmetry analysis of partition (4.2) of the thin film equation (4.1) led to the

addition of six symmetry-enhancing constraints. These resulted in travelling waves being

the only similarity solutions available for system (4.2) and the thin film equation (4.1)

under the conditions on ,0)( ≠hf )(hg and )(hj relating to groups (I)-(VI). This

situation arose since 0)()( =′=′ hjhg in the conditions on groups (I)-(VI), causing

equation (4.2)1 to generate a travelling wave solution of velocity .01 ≠j

By Mathematica [54], we attempted to solve equations (4.17) and (4.29), the ODEs

relating respectively to cases 2(2) and 4 in table 4. Via the ‘independent variable is

missing’ method, we noted equation (4.17) to be reducible to a second order equation

from which exact solutions are not recoverable. Attempting to solve the ODE (4.29) for

special cases of ,0))(( ≠uyf we found exact solutions for ODE (4.29) elusive.

The Lie classical analysis of partition (4.3) of the thin film equation (4.1) led to the

addition of one symmetry-enhancing constraint. This gave rise to steady state solutions

as the only similarity solutions retrievable for system (4.3) and thin film equation (4.1)

under the conditions on ,0)( ≠hf )(hg and )(hj associated with group (VII). This

situation directly resulted from the nature of system (4.3) and the need for consistency in

the invariant surface condition (4.42)3 .

As each of the systems (4.2) and (4.3) produced a different type of solution, we clearly

see that the class of solution generated depends on the form of the individual partition of

the thin film equation (4.1). The inclusion of nontrivial functions in an enlarged system

resulting from the partitioning of thin film equation (4.1) might lead to more diverse

symmetry groups and solutions for thin film equation (4.1). We explore this avenue in

chapter 6 of this thesis.

101

CHAPTER 5

NON-CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE

THIN FILM EQUATION

5.1 INTRODUCTION

Via the method of non-classical symmetry-enhancing constraints, we derive symmetry


[ ] ;0)()()( =++−∂


x (5.1)

where .0)( ≠hf

The technique of non-classical symmetry-enhancing constraints is so termed as it

involves using the non-classical procedure to augment the method of symmetry-

enhancing constraints presented and developed by Goard and Broadbridge [29].

Saccomandi considered this approach [47].

From the viewpoint of this method, we considered two partitions of the thin film

equation (5.1). These yielded new non-classical symmetry groups and generate the

respective systems (4.2) and (4.3), namely

( ) ,0)()(2

=′−+ xxt hhghhjh [ ] ;0)()( =−∂

∂xxxxx hhghhf

x (5.2)

and

,0)()( =+− txxxxxx hhhghhf ( ) .0)()()(2

=+′−′xxxxxx hhjhhghhhf (5.3)

We apply the restrictions

(i) ,0)( ≠hf

(ii) 0)()( =′= hfhg does not occur,

(iii) 0)()( ==′ hjhg does not occur;

to system (5.2) to prevent th and xxxxh vanishing. On system (5.3), we impose the

condition 0)( ≠′ hf with restrictions (i) and (iii) to prevent xxxh and xxxxh disappearing.

102


,x t and ,h namely

( ) ( )( ) ( )( ) ( );,,

,,,

,,,

2

1

2

1

2

1

εεζ

εεη

εεξ

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(5.4)

preserving systems (5.2) and (5.3).

Thus if ),,( txh φ= then from ),,( 111 txh φ= evaluating the expansion of ε∂

∂ 1h at 0=ε

leads to the invariant surface condition

).,,(),,(),,( htxt

hhtx

x

hhtx ζηξ =

∂

∂+

∂

∂ (5.5)

The solutions of invariant surface condition (5.5) are functional forms of the similarity

solutions for systems (5.2) and (5.3).

The following section contains a brief description of the technique of non-classical

symmetry-enhancing constraints. This involves using the non-classical method to obtain

the symmetry groups leaving systems (5.2) and (5.3) invariant. Details of the non-

classical procedure feature in Bluman and Cole [16].

5.2 THE METHOD OF NON-CLASSICAL SYMMETRY-

ENHANCING CONSTRAINTS

Chapter 4 of this thesis includes the description of symmetry-enhancing constraints

presented in Goard and Broadbridge [29]. As new non-classical symmetry groups arise

from systems (5.2) and (5.3), we may consider symmetry-enhancing constraints to be

added to the thin film equation (5.1).

The method of symmetry-enhancing constraints requires dividing the equation of interest

into a larger system of PDEs before applying the classical procedure to this system to

recover its Lie classical groups [29]. Should this system generate Lie groups other than

those admitted by the original single equation (and which are not special cases of groups

admitted by the original equation), we may consider a symmetry-enhancing constraint to

be added to the original equation.

103

The technique of non-classical symmetry-enhancing constraints consists of applying the

non-classical procedure to the system in question to obtain its non-classical groups.

Should this system yield such groups other than those admitted by the original single

equation (and which are not special cases of groups admitted by the original equation), a

symmetry-enhancing constraint may be considered to be added to the original equation.

Accordingly, we apply this procedure to systems (5.2) and (5.3), briefly describing it for

system (5.2). The process is identical for system (5.3).

We begin applying the non-classical method by introducing for transformations (5.4) the

terms ( )htxA ,, and ( )htxB ,, defined as

( ) ( )( )

,,,

,,,,

htx

htxhtxA

η

ζ= ( ) ( )

( );

,,

,,,,

htx

htxhtxB

η

ξ= (5.6)

so that the invariant surface condition (5.5) becomes

.x

hBA

t

h

∂

∂−=

∂

∂ (5.7)

To obtain non-trivial transformations, we require .0),,( ≠htxη

The non-classical method uses relation (5.7) to express all derivatives of h with respect

to t in terms of derivatives of h with respect to .x In the invariance requirements

determined for systems (5.2) and (5.3) via the classical procedure in chapter 4 of this

thesis, these new expressions then replace all derivatives of h with respect to t so that

the left-hand side of each invariance requirement depends on h only via h itself and

derivatives of h with respect to .x

As we express all derivatives of h with respect to t in terms of derivatives of h with

respect to ,x fewer derivatives of h occur in each resulting invariance requirement than

is the case for the classical method. Hence, fewer restrictions apply to the non-classical

group generators ( )htxA ,, and ( )htxB ,, than to the classical group generators ( ),,, htxξ

( )htx ,,η and ( )htx ,,ζ when equating to zero the coefficients of all derivatives of h and

the sum of all remaining terms not involving derivatives of h in each invariance

requirement. This enables the non-classical procedure to generalise the classical method.

Following the replacements mentioned, setting 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 of system (5.2) generates the determining equations

104

,0=hB ,0=+ hhhh AA ηη ,0)( =′hhhg η ,0)( =′

hhhhg η ,0)( =′hhhhhg η

,0)()( =′′hhghf η [ ] [ ] ,0)()( =+++ xtxt hjAAhjA ηηη [ ] ,0)(

2=′

hhg η

[ ]{ } ,0)(2)()( =′+−′xh hghjBhg ηη [ ] [ ] ,0)()()()( =−+− xxxxxxxxxxxx hghfAAhgAhf ηηη

( ) [ ]{ } ,0)()(322)( =′′−−−−+−′ hgAAhjBABhg htxhx ηηηηη

[ ] [ ] [ ] ,0)(2)()()(2)()( =′−+−+′−−−′xxtxtx hgAhjBhjAhgBBhjhjA ηηηη

,0)(

)(4

)(

)(64 =�

�

��

� ′++

′+− xxhxxxxh

hf

hfAA

hf

hfBA ηη

η [ ]Bhj −)( ,0

)(

)(2 =�

�

��

� ′+ hhhhh

hf

hfηη

( ) ��

��

�−

′++−+

′+− xhxxxxxxhxhxxxxxxxxxxxxh

hf

hg

hf

hfAAB

hf

hgA

hf

hfBA ηηη

η )(

)(2

)(

)(42

)(

)(

)(

)(4

,0)(

)()(=�

�

��

�−

−+ xxxxxx

hf

hgBhjηη

η

( ) ��

��

�−

′−�

�

��

� ′++−

′+ xxxxxxxxhxxhhxxxxxhxxhh

hf

hghg

hf

hfABA

hf

hfA ηη

ηηη

η )(

)()(

)(

)(2

33

)(

)(6

,0)(

)(2

)(

)(4

)(=�

�

��

�−

′+

−+ xhxxxxxxh

hf

hg

hf

hfBhjηηη

η

( ) ,0)(

2)(

)(4

)(

)(4 =

−+�

�

��

� ′++−

′+ xxhxhxhhxxxhxhh

Bhj

hf

hfABA

hf

hfA η

ηηη

η (5.8)

��

��

�−

′+

−+�

�

��

� ′++

′+ hhxxhxxhhxhhxhhhxhhxhhh

hf

hg

hf

hfBhj

hf

hfAA

hf

hfA ηηη

ηηη

η )(

)(

)(

)(36

)(

)(

)(34

)(

)(34

,0)(

)(4

)(

)(2

)(=�

�

��

� ′−−

′+ xxxxxxhxh

hf

hf

hf

hghgηηη

η

( ) ,0)(

)(4

)(

)(4

)(

)(

)(

)(

)(4

)(

)(=

′+�

�

��

� ′+

−+

′−�

�

��

� ′+−

′

η

ηηη

ηη

η x

xxh

h

hxhf

hfB

hf

hfBhj

hf

hfA

hf

hf

dh

dAAB

hf

hf

,0)(

)(4

)()(2 =�

�

��

� ′+

−+�

�

��

�′−+ xhxhh

xxhhhh

hhhhf

hfBhjhgAA ηη

ηη

η

η

η [ ] ,0)( =− hhBhj η

,0)(

)(34

)(

)(

)(

)(

)(=�

�

��

� ′+

−+�

�

��

� ′++

′+ xhhxhhhhhhhhhhhhhhhhh

hf

hfBhj

hf

hfAA

hf

hfA ηη

ηηη

η

[ ] [ ] ,04)(

)()()(4

)(

)()( =�

�

��

�+

′′−−+

′− xhxhhh

hf

hfhgBhj

hf

hfBhj ηηηη

[ ]Bhj −)( ,0)(

)(=�

�

��

� ′+ hhhhhhh

hf

hfηη .06

)(

)(

)(

)(246 =+�

�

��

�−−−

η

ηxxhxxxxxxh A

hf

hg

dh

dAB

hf

hgBA

Via a process similar to that for system (5.2) and following the replacements mentioned,

equating to zero the coefficients of all derivatives of h and the sum of all remaining

terms not involving derivatives of h in the invariance requirement of system (5.3) yields

the determining equations

105

,0=hhA ,0=hB ,0)(

)(=�

�

��

�

′

′

hf

hg

dh

dB hη ,0

)(

)(

)(

)(=�

�

��

�−

′

′

hf

hg

hf

hgB hη

( ) ,04)(

)(46

)(

)()()()( 2 =

++

′−−

′++−

η

ηη xx

xhxxxxxxxxt

BBAA

hf

hfAB

hf

hjhfAhfAhgA

( ) ( ) ABhf

hfBABhgBA

hf

hgBAhf txhxxxxxhxxxxxxxh

)(

)(2)(64

)(

)(4)(

′+−−+�

�

��

�−

′

′+−

,0)(

)()(4 =

��

��

��

��

�

′−−−+ xxxh

hf

hj

dh

dhfBBA

Bηηηη

η

,0)(

)(

)(

)(4

)(

)(

)(

)(246 =�

�

��

�

′

′−−�

�

��

�−−−

hf

hg

hf

hgB

hf

hg

dh

dAB

hf

hgBA x

xxxxxxhη

η (5.9)

,0)(

)()(

)(

)()( =

��

��

��

��

�

′+−�

�

��

�

′

′

hf

hj

dh

dhfBB

hf

hg

dh

dhBf hx ηη

( ) ,0)(

)(3

)(

)(

)(

)(=�

�

��

�

′+−

′+

′

′−

hf

hj

dh

dAAB

hf

hjA

hf

hgA hxxxxx

.0)(

)(

)(

)(23 =�

�

��

�

′

′−

′

′−−

hf

hg

dh

dAB

hf

hgBA xxxxxxh

All subscripts in systems (5.8) and (5.9) represent partial differentiation with ,x t and h

as independent variables. Throughout this chapter, primes denote differentiation with

respect to the argument indicated.

Systems (5.8) and (5.9) enable recovery of all symmetries and corresponding conditions

on ,0)( ≠hf )(hg and )(hj for the respective systems (5.2) and (5.3) under

transformations (5.4) via the non-classical procedure.

Inspection of systems (5.8) and (5.9) reveals an interesting departure from conventional

non-classical group construction owing to the presence of the term ( ) 0,, ≠htxη and its

derivatives in these systems. Such terms do not usually feature in the equations

determining the non-classical groups of a system comprising one or more equations. All

distinct symmetries arising from systems (5.8) and (5.9) and extending beyond the

confines of non-classical symmetries derived for the thin film equation (5.1) occur under

specific conditions on ,0),,( ≠htxη rendering these symmetries hybrids of classical and

non-classical symmetries.

System (5.2) generates one new symmetry group extending beyond the confines of

groups retrievable via the non-classical procedure for the thin film equation (5.1). By a

similar process, we obtain twenty-eight new such groups for system (5.3). These new

106

groups enhance the symmetries of the thin film equation (5.1). We may therefore

consider symmetry-enhancing constraints added to the thin film equation (5.1).

The following pages feature a description of each of these groups, a brief mention of the

special cases arising for each such group and the derivation of similarity solutions for

systems (5.2) and (5.3) associated with each group where applicable. Solutions of

systems (5.2) and (5.3) are also solutions to the thin film equation (5.1).

Although groups (XI)-(XVIII) and (XXII)-(XXIV) occurring for system (5.3) yield no

valid similarity solutions, we list them since they extend beyond the confines of groups

derived by applying the non-classical procedure to the thin film equation (5.1).

GROUP (I)


1 ≠= hfefhf 1)( ghg = and ,0)( 1 ≠= jhj system (5.2)

admits symmetry group (I) given by

,0),,( =htxA [ ]

,)(

4)(),,(

0

01

tf

tfjhtxB

α

α += ( ) ;0)(),,( 2 ≠+= chthtx αη (5.10)

where ,00 ≠f ,01 ≠f ,01 ≠j 2c and 1g are arbitrary constants with 02 ≠+ ch while

0)( ≠tα is an arbitrary function of .t

A special case of group (5.10) occurs and is as follows.

Case (1) The case of group (5.10) with 0)( 3 ≠= ctα arises under the conditions

,0)(5

4)(

)(

1 ≠

=−

dscsj

sjc

h

efhf 1)( ghg = and 0)( ≠hj is an arbitrary function of h where

,03 ≠c ,04 ≠c ,01 ≠f 5c and 1g are arbitrary constants with .0)( 5 ≠− chj



,01 =+ xt hjh ( ) ,01010 =−+ xxxxxxxxxx

hfhghhfhef ;01 =xhj (5.11)

where ,00 ≠f ,01 ≠f 01 ≠j and 1g are arbitrary constants.

As 01 ≠j forces 0=xh in equation (5.11)3 , requiring 0=th in equation (5.11)1 ,

system (5.11) admits only the constant solution. Hence under transformations (5.4) and

107

the conditions ,0)( 0

1 ≠= hfefhf 1)( ghg = and ,0)( 1 ≠= jhj the similarity solution of

system (5.2) and the thin film equation (5.1) in connection with group (5.10) is the

constant solution.

GROUP (II)

Under the conditions ,0)( ≠hf 0)()( 1 ≠= hfghg and [ ] )()()( 1

1

0 hfjdssfjhj

h

′��

�

��

�+=

−

are arbitrary functions of ,h system (5.3) admits symmetry group (II), namely

,0),,( =htxA ,0)(),,( ≠= thtxB α [ ]

;0)(),,( 0)()(4

)(

≠= +

′−

jtt

xt

ethtxαα

α

βη (5.12)

where ,01 ≠g 0j and 1j are arbitrary constants while 0)( ≠tα and 0)( ≠tβ are

arbitrary functions of t with .0)( 0 ≠+ jtα Furthermore, 0)( ≠hf is an arbitrary

function of h with .0)( ≠′ hf


Group (5.12), system (5.3) and the invariant surface condition (5.7) give

[ ] ,0)( 1

1

01 =��

�

��

�++−

−jdssfjhghh

h

xxxxx ( ) ,0)( 1 =+− txxxxxx hhghhf ;0)( =+ xt hth α

(5.13)

where ,01 ≠g 0j and 1j are arbitrary constants, 0)( ≠tα is an arbitrary function of t

with 0)( 0 ≠+ jtα and 0)( ≠hf is an arbitrary function of h with .0)( ≠′ hf

Via the method in [24], we find the general solution of equation (5.13)3 to be

;)(),( uytxh = (5.14)

where 0)( ≠tα is an arbitrary function of t and )(uy is an arbitrary function of

.)( −=t

dssxu α

Substituting result (5.14) into equation (5.13)1 gives rise to the cases

(1) ,0)( =′ uy (2) [ ] .0)()()( 1

1

01 =++′−′′′ −

jdssfjuyguy

y

108

Case (1) 0)( =′ uy

Under the constraint ,0)( =′ uy equations (5.13)1 and (5.13)2 vanish and system (5.13)

admits only constant solutions.

Case (2) [ ] 0)()()( 1

1

01 =++′−′′′ −

jdssfjuyguy

y

Together with the constraint on this case, relations (5.14) and (5.13)2 force .0)( =′ uy

Similarly to case (1), only constant solutions arise for system (5.13).

Cases (1) and (2) reflect that under transformations (5.4) and the conditions ,0)( ≠hf

0)()( 1 ≠= hfghg and [ ] )()()( 1

1

0 hfjdssfjhj

h

′��

�

��

�+=

− are arbitrary functions of ,h

the only similarity solution retrievable for system (5.3) and thin film equation (5.1) in

association with group (5.12) is the constant solution.

GROUP (III)


21 ≠+=f

fhfhf ( ) 0)( 0

21 ≠+=f

fhghg and ,0)( =hj

system (5.3) admits symmetry group (III) given by

( ) ,0)(),,( 2 ≠+= fhthtxA α ,0)(),,( 1 ≠= tchtxB α

[ ][ ] ;0)(),,(

21

20

)(4

)()(

≠=

+′−x

tc

tft

ethtxα

αα

βη

(5.15)

where ,01 ≠c ,00 ≠f ,01 ≠f 01 ≠g and 2f are arbitrary constants with 02 ≠+ fh

while 0)( ≠tα and 0)( ≠tβ are arbitrary functions of .t

Two special cases of group (5.15) emerge and are as follows.

Case (1) The case of group (5.15) with 0)( 2 ≠= ctα occurs under the conditions

( ) ,0)( 0

21 ≠+=f

fhfhf ( ) 2210)( gfhghg

f++= and 0)( =hj where ,02 ≠c ,00 ≠f

,01 ≠f ,01 ≠g 2f and 2g are arbitrary constants with .02 ≠+ fh

Case (2) The case of group (5.15) with 01

)(20

≠+

−=ctf

tα appears under the


21 ≠+=f

fhfhf ( ) 0

21)(f

fhghg += and ( ) 0)( 0

21 ≠+=f

fhjhj

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c 2f and 1g are arbitrary constants with

( )( ) .0202 ≠++ ctffh

109



( ) ,00 =− xxxxx hghh ( ) ( ) ,0021

0 =+−+ txxxxxx

fhhghfhf

(5.16)

( ) ;0)()( 21 ≠+=+ fhthtch xt αα

where ,01 ≠c ,00 ≠f ,01 ≠f ,01

1

0 ≠=f

gg 01 ≠g and 2f are arbitrary constants with

02 ≠+ fh while 0)( ≠tα is an arbitrary function of .t

Since 0=xh forces 0=th in equation (5.16)2 , rendering equation (5.16)3 inconsistent,

0≠xh is a requirement. Consequently, equation (5.16)1 forces ,00 ≠= xxxx hgh giving

0=th in equation (5.16)2 . System (5.16) then simplifies to give

,0)(),( ≠= xytxh ,01

)(

)(

12

≠=+

′

cfxy

xy ;0)()( 0 ≠′=′′′ xygxy (5.17)

where ,01 ≠c ,01 ≠f ,01

1

0 ≠=f

gg 01 ≠g and 2f are arbitrary constants while

0)( ≠′ xy owing to the requirement .0≠xh

Directly solving equation (5.17)2 and substituting its general solution into equation

(5.17)3 generates the constraint .02

1

1

1 ≠=c

fg Subsequently, under transformations (5.4)

and the conditions ( ) ,0)( 0

21 ≠+=f

fhfhf ( ) 0)( 0

21 ≠+=f

fhghg and 0)( =hj with

the constraint ,02

1

1

1 ≠=c

fg the similarity solution of system (5.3) and the thin film

equation (5.1) in conjunction with group (5.15) is the steady state solution

;0),( 221 ≠−= fectxh

c

x

(5.18)

where ,01 ≠c ,02 ≠c 01 ≠f and 2f are arbitrary constants.

110

GROUP (IV)

Under conditions ,0)( 0

1 ≠= hfefhf 0)( 0

2

11 ≠= hfecfhg and ,)( 0

1

hfejhj = system

(5.3) admits symmetry group (IV), namely

( ) ,0)(),,( 11

4 ≠−= − xcxcecethtxA α ,0)(),,( 2 ≠= tchtxB α

(5.19)

( )[ ]

;0)(4

)(

4exp)(),,(

2

2

4

21

0 11 ≠��

��

� ′−+= −

xtc

tece

cc

fthtx

xcxc

α

αβη

where ,01 ≠c ,02 ≠c ,00 ≠f ,01 ≠f 4c and 1j are arbitrary constants with

011

4 ≠− − xcxcece while 0)( ≠tα and 0)( ≠tβ are arbitrary functions of .t



,010

12

1 =��

��

�+−

ff

jhchh xxxxx ( ) ,0

2

110 =+− txxxxxx

hfhhchef

(5.20)

( ) ;0)()( 11

42 ≠−=+ − xcxc

xt ecethtch αα


011

4 ≠− − xcxcece while 0)( ≠tα is an arbitrary function of .t


0≠xh is a requirement. Hence 10

12

1ff

jhch xxxx −=− in equation (5.20)1 , causing 0=th

in equation (5.20)2 . System (5.20) reduces to give

,0)(),( ≠= xytxh ,0 )( 11

42 ≠−=′ − xcxcecexyc ;)()(

10

12

1ff

jxycxy −=′−′′′ (5.21)


.011

4 ≠− − xcxcece


(5.21)3 generates the constraint .01 =j Thus under transformations (5.4) and the

conditions ,0)( 0

1 ≠= hfefhf 0)( 0

2

11 ≠= hfecfhg and

hfejhj 0

1)( = with the constraint

111

,01 =j the similarity solution of system (5.3) and the thin film equation (5.1) in

conjunction with group (5.19) is the steady state solution

;0),( 87611 ≠++= −

cecectxhxcxc

(5.22)

where ,01 ≠c ,02 ≠c ,01

21

6 ≠=cc

c ,4c 21

47

cc

cc = and 8c are arbitrary constants such

that .011

4 ≠− − xcxcece

The conditions on )(hg and )(hj associated with groups (5.15) and (5.19) have the

similar form 0)()()( 2 =≠= hjhfghg with 02 ≠g an arbitrary constant. Under such

conditions on )(hg and )(hj where 0),,( ≠htxA in the group concerned, the only

similarity solutions retrievable for system (5.3) and the thin film equation (5.1) are

steady state solutions, of which solution (5.22) is illustrative. Solution (5.18) is a special

case of solution (5.22) with .074 == cc

GROUP (V)

Under conditions ,0)( 0

1 ≠= hfefhf 0)( 0

2

11 ≠= hfecfhg and ,)( 0

1

hfejhj = system

(5.3) yields symmetry group (V), namely

( ) ,0)(),,( 13211 ≠+−= −

cececthtxAxcxcα ,0)(),,( 4 ≠= tchtxB α

(5.23)

( )[ ]

;0)(4

)(

4exp)(),,(

2

4

132

41

0 11 ≠��

��

� ′−++= −

xtc

txcecec

cc

fthtx

xcxc

α

αβη

where ,01 ≠c ,04 ≠c ,00 ≠f ,01 ≠f ,2c 3c and 1j are arbitrary constants with

013211 ≠+− −

cececxcxc



Case (1) The case of group (5.23) with 032 == cc occurs under the conditions

,0)( 0

1 ≠= hfefhf 0)( =hg and 0)( 0

1 ≠= hfejhj where ,00 ≠f 01 ≠f and 01 ≠j are


112



,010

12

1 =��

��

�+−

ff

jhchh xxxxx ( ) ,0

2

110 =+− txxxxxx

hfhhchef

(5.24)

( ) ;0)()( 132411 ≠+−=+ −

cececthtchxcxc

xt αα

where 0)( ≠tα is an arbitrary function of t while ,01 ≠c ,04 ≠c ,00 ≠f ,01 ≠f ,2c

3c and 1j are arbitrary constants such that .013211 ≠+− −

cececxcxc

Since 0=xh leads to 0=th in equation (5.24)2 , rendering equation (5.24)3 inconsistent,

we require .0≠xh Accordingly, 10

12

1ff

jhch xxxx −=− in equation (5.24)1 , causing

0=th in equation (5.24)2 . System (5.24) then simplifies to give

,0)(),( ≠= xytxh ,0)( 132411 ≠+−=′ −

cececxycxcxc

;)()(10

12

1ff

jxycxy −=′−′′′

(5.25)

where ,01 ≠c ,04 ≠c ,00 ≠f ,01 ≠f ,2c 3c and 1j are arbitrary constants with

.013211 ≠+− −

cececxcxc


(5.25)3 yields the constraint .04

3

110

1 ≠=c

cffj Hence under transformations (5.4) and the

conditions ,0)( 0

1 ≠= hfefhf 0)( 0

2

11 ≠= hfecfhg and

hfejhj 0

1)( = with the constraint

,04

3

110

1 ≠=c

cffj the similarity solution of system (5.3) and the thin film equation (5.1)

in connection with group (5.23) is the steady state solution

;0),( 1098711 ≠+++= −

cxcecectxhxcxc

(5.26)

where ,01 ≠c ,04 ≠c ,04

1

9 ≠=c

cc ,2c ,3c ,

41

2

7cc

cc =

41

3

8cc

cc = and 10c are arbitrary

constants with .013211 ≠+− −

cececxcxc

113

GROUP (VI)

Under the conditions ,0)()( 0

21 ≠+= ffhfhf 0)()( 0

2

2

11 ≠+= ffhcfhg and ,0)( =hj

system (5.3) admits symmetry group (VI), namely

( ) ,0)(),,( 21 ≠+= −

fhethtxAxcα ,0

)(),,( 1

1

≠−= − xce

c

thtxB

α

(5.27)

( )

[ ];0

)(4

)(

4

4exp)(),,( 1

2

01 ≠��

��

� ′+

−= xc

et

tx

fcthtx

α

αβη

where 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t while ,01 ≠c ,00 ≠f 01 ≠f

and 2f are arbitrary constants with .02 ≠+ fh



( ) ,02

1 =− xxxxx hchh ( ) ,0)(2

1210 =+−+ txxxxxx

fhhchfhf

(5.28)

( ) ;0)()(

2

1

11 ≠+=− −−fhethe

c

th

xc

x

xc

t αα

where ,01 ≠c ,00 ≠f 01 ≠f and 2f are arbitrary constants with 02 ≠+ fh while


As 0=xh gives 0=th in equation (5.28)2 , rendering equation (5.28)3 inconsistent, we

require .0≠xh Accordingly, 02

1 ≠= xxxx hch in equation (5.28)1 , forcing 0=th in

equation (5.28)2 . System (5.28) then simplifies to give

,0)(),( ≠= xytxh 0,)(

)(1

2

≠−=+

′c

fxy

xy ;0)()(

2

1 ≠′=′′′ xycxy (5.29)

where 01 ≠c and 2f are arbitrary constants while 0)( ≠′ xy as .0≠xh

Directly solving equation (5.29)2 , we find its general solution identically satisfies

equation (5.29)3. Consequently under transformations (5.4) and the conditions

,0)()( 0

21 ≠+= ffhfhf 0)()( 0

2

2

11 ≠+= ffhcfhg and ,0)( =hj the similarity solution

of system (5.3) and the thin film equation (5.1) in connection with group (5.27) is the

steady state solution

114

;0),( 231 ≠−= −

fectxhxc

(5.30)

where ,01 ≠c 03 ≠c and 2f are arbitrary constants.

Solutions (5.30) and (5.18) are equivalent and arise under equivalent conditions on

,0)( ≠hf )(hg and ).(hj

GROUP (VII)


21 ≠+= ffhfhf 0)()( 0

2

2


system (5.3) admits symmetry group (VII), namely

( )( ) ,0)(),,( 221 ≠++= fhcethtxAxcα ( ) ,0

)(),,( 3

1

1 ≠+= cec

thtxB

xcα

(5.31)

( )[ ][ ][ ]

[ ][ ] ;0)(),,(

2

202

3

123

23023

1

)(

)()(

4)(4

)(4)(

2

1

3 ≠��

��

� +=

−′−−−+′

xt

tfct

c

ctc

tcfcct

xc

ec

cethtx α

ααα

αα

βη

where 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t while ,01 ≠c ,03 ≠c ,00 ≠f

,01 ≠f 2c and 2f are arbitrary constants with .02 ≠+ fh



( ) ,02

1 =− xxxxx hchh ( ) ,0)(2

1210 =+−+ txxxxxx

fhhchfhf

(5.32)

( ) ( )( ) ;0)()(

223

1

11 ≠++=++ fhcethcec

th

xc

x

xc

t αα

where 0)( ≠tα is an arbitrary function of t while ,01 ≠c ,03 ≠c ,00 ≠f ,01 ≠f 2c


115


we require .0≠xh Hence 02

1 ≠= xxxx hch in equation (5.32)1 , causing 0=th in

equation (5.32)2 . System (5.32) simplifies to give

,0)(),( ≠= xytxh 0,)(

)(

3

21

21

1

≠+

+=

+

′

ce

cec

fxy

xyxc

xc

;0)()(2

1 ≠′=′′′ xycxy (5.33)

where ,01 ≠c ,03 ≠c 2c and 2f are arbitrary constants while 0)( ≠′ xy owing to the

requirement .0≠xh

By the method in [48], we find the general solution of equation (5.33)3 to be

;0)( 65411 ≠++= −

cececxyxcxc

(5.34)

where ,01 ≠c ,4c 5c and 6c are arbitrary constants with .011


Substituting result (5.34) into equation (5.33)2 generates the cases

(1) ,0524 ==≠ ccc ,2436 fccc −=

(2) ,054 =≠ cc ,23 cc = ,26 fc −=

(3) ,04 ≠c ,23 cc −= ,2

245 ccc = .2 2426 fccc −−=

Case (1) ,0524 ==≠ ccc 2436 fccc −=

Result (5.34) and relations (5.33)1 and (5.33)2 imply that under transformations (5.4) and

the conditions ( ) ,0)( 0

21 ≠+=f

fhfhf ( ) 0)( 0

2

2

11 ≠+=f

fhcfhg and ,0)( =hj the


group (5.31) and the constraints 0524 ==≠ ccc and 2436 fccc −= is the steady state

solution

;0),( 641 ≠+= cectxhxc

(5.35)

where ,01 ≠c ,03 ≠c ,04 ≠c 2436 fccc −= and 2f are arbitrary constants.

Solutions (5.35) and (5.18) are equivalent and occur under equivalent conditions on

,0)( ≠hf )(hg and ).(hj

116

Case (2) ,054 =≠ cc ,23 cc = 26 fc −=

Result (5.34) and equations (5.33)1 and (5.33)2 reflect that under transformations (5.4)

and the conditions ,0)()( 0

21 ≠+= ffhfhf 0)()( 0

2

2


the similarity solution of system (5.3) and the thin film equation (5.1) in association with

group (5.31) and the constraints ,054 =≠ cc 23 cc = and 26 fc −= is the steady state

solution

;0),( 241 ≠−= fectxhxc (5.36)

where ,01 ≠c 04 ≠c and 2f are arbitrary constants.

Solutions (5.36) and (5.18) are equivalent and emerge under equivalent conditions on

,0)( ≠hf )(hg and ).(hj

Case (3) ,04 ≠c ,23 cc −= ,2

245 ccc = 2426 2 fccc −−=

Result (5.34) and relations (5.33)1 and (5.33)2 indicate that under transformations (5.4)

and the conditions ,0)()( 0

21 ≠+= ffhfhf 0)()( 0

2

2


the similarity solution of system (5.3) and the thin film equation (5.1) in conjunction

with group (5.31) and the constraints ,04 ≠c ,23 cc −= 2

245 ccc = and 2426 2 fccc −−=

is the steady state solution

( ) ;0),( 6

2

2411 ≠++= −

cecectxhxcxc

(5.37)

where ,01 ≠c ,04 ≠c ,2c 2426 2 fccc −−= and 2f are arbitrary constants.

Solutions (5.37) and (5.22) are equivalent.

117

GROUP (VIII)

Subject to conditions ,0)()( 0

21 ≠+= ffhfhf 0)( =hg and ( ) ,0)(

1

210 ≠+=

−ffhjhj

system (5.3) yields symmetry group (VIII), namely

( ) ,0)(),,( 2 ≠+= fhthtxA α ( ) ,03

)(),,( 1 ≠+= cx

thtxB

α

(5.38)

( )[ ]

[ ] ;0)(),,( 2

20

)(4

)(3

4)(3

1 ≠+=��

��

��

��

�−+′−

t

tft

cxthtx α

αα

βη

where ,00 ≠f ,01 ≠f ,01 ≠j 1c and 2f are arbitrary constants with ( )( ) 021 ≠++ fhcx




,010

1 =��

��

�+

ff

jhh xxxx ,0)( 0

21 =++ txxxx

fhhfhf

(5.39)

( ) ( ) ;0)(3

)(21 ≠+=++ fhthcx

th xt α

α

where ,00 ≠f ,01 ≠f ,01 ≠j 1c and 2f are arbitrary constants with ( )( ) 021 ≠++ fhcx

while 0)( ≠tα is an arbitrary function of .t


we require .0≠xh Accordingly 010

1 ≠−=ff

jhxxx in equation (5.39)1 , causing 0=th in


,0)(),( ≠= xytxh 0,3

)(

)(

12

≠+

=+

′

cxfxy

xy ;0)(

10

1 ≠−=′′′ff

jxy (5.40)

where ,00 ≠f ,01 ≠f ,01 ≠j 1c and 2f are arbitrary constants with 01 ≠+ cx and

0)( ≠′ xy owing to the requirement .0≠xh


Consequently, under transformations (5.4) and the conditions ,0)()( 0

21 ≠+= ffhfhf

0)( =hg and ( ) ,0)(1

210 ≠+=

−ffhjhj the similarity solution of system (5.3) and the

thin film equation (5.1) in association with group (5.38) is the steady state solution

118

( ) ;06

),( 2

3

1

10

1 ≠−+−= fcxff

jtxh (5.41)

where ,00 ≠f ,01 ≠f ,01 ≠j 1c and 2f are arbitrary constants with .01 ≠+ cx

Bernoff and Witelski also obtained an exact polynomial similarity solution when

studying the special case of the thin film equation (5.1) with hhf =)( and

,0)()( == hjhg using linear stability analysis to demonstrate the linear stability of the

source-type similarity solutions of this equation [9].

GROUP (IX)


21 ≠+= ffhfhf 0)()( 0

2

2


system (5.3) yields symmetry group (IX), namely

( ) ,02

1)(4

)(),,( 22

2

2 111 ≠+��

�

��

��

��

�−+�

��

��

�−= −−

fhec

tec

ethtxAxcxcxc κα

,02

1)(

4

)(),,( 111 2

1

2

22

1

≠��

��

�++�

��

��

�++= −− xcxcxc

ec

c

te

cce

c

thtxB

κα (5.42)

[ ][ ]

( )��

�

�

��

�

�

++

′−′

��

�

�

��

�

�

++

+

=

+′−

)(22)(

)()(2

)()()()(

exp

)(

)(

2

2)(),,(

2

)(4

)()(

2

2

1

2

20

1

1

tcet

tt

tttt

t

tce

ce

thtxxc

t

tft

xc

xc

κα

ακ

καακ

α

κβη

κ

κκ

;02

1)(4

)(4

4

21

2

221

0

111 ≠��

�

��

��

��

�++�

��

��

�++×

−

−−

f

xcxcxce

ctce

ccetc κα

where ,0)( ≠tα 0)( ≠tβ and 0)( ≠tκ are arbitrary functions of t while ,01 ≠c

,00 ≠f ,01 ≠f 2c and 2f are arbitrary constants with .02 ≠+ fh

119



( ) ,02

1 =− xxxxx hchh ( ) ,0)(2

1210 =+−+ txxxxxx

fhhchfhf

(5.43)

=��

�

��

��

��

�++�

��

��

�+++ −−

x

xcxcxc

t hec

c

te

cce

c

th 111

21

)(

4

)( 2

1

2

22

1

κα

( ) ;02

1)(4

)( 22

2

2 111 ≠+��

�

��

��

��

�−+�

��

��

�− −−

fhec

tec

etxcxcxc κα

where ,01 ≠c ,00 ≠f ,01 ≠f 2c and 2f are arbitrary constants with 02 ≠+ fh while

0)( ≠tα and 0)( ≠tκ are arbitrary functions of .t


we require .0≠xh Accordingly 02



,0)(),( ≠= xytxh ,0)()(2

1 ≠′=′′′ xycxy

(5.44)

;0

21)(

4)(

21)(

4)(

)(

)(

111

111

2

2

22

2

2

2

1

2

≠

��

��

�++�

��

��

�++

��

��

�−+�

��

�

��

�−

=+

′

−−

−−

xcxcxc

xcxcxc

ec

tec

cet

ec

tec

et

cfxy

xy

κα

κα

where ,01 ≠c 2c and 2f are arbitrary constants, 0)( ≠tα and 0)( ≠tκ are arbitrary

functions of t and 0)( ≠′ xy owing to the requirement .0≠xh

By the method in [48], we solve equation (5.44)2 , substituting its general solution into

equation (5.44)3. Hence under transformations (5.4) and the conditions

,0)()( 0

21 ≠+= ffhfhf 0)()( 0

2

2

11 ≠+= ffhcfhg and ,0)( =hj the similarity solution

for system (5.3) and the thin film equation (5.1) in association with group (5.42) is the

steady state solution

;0),( 65411 ≠++= −

cecectxhxcxc

(5.45)

120

where ,01 ≠c ,04 ≠c ,2c ,4

2

245

ccc = 2426 fccc −= and 2f are arbitrary constants

with .011



GROUP (X)


21 ≠+= ffhfhf 0)()( 0

2

2


system (5.3) admits symmetry group (X), namely

( ) ,04

)(),,( 2

2

2 11 ≠+��

��

�−= −

fhec

ethtxAxcxcα ,0

4

)(),,( 11

2

22

1

≠��

��

�++= − xcxc

ec

cec

thtxB

α

(5.46)

( )[ ]

[ ];0

)(2

4

2)(4)(

exp2

)(),,(22

22

012

4

2

1

10

1 ≠

��

�

�

��

�

�

��

��

�+

��

��

�+−−′

��

��

�+=

−

tc

e

cextfct

cethtx

xc

xcf

xc

α

αα

βη


0)( ≠tα and 0)( ≠tβ are arbitrary functions of .t



( ) ,02

1 =− xxxxx hchh ( ) ,0)(2

1210 =+−+ txxxxxx

fhhchfhf

(5.47)

( ) ;04

)(4

)(2

2

2

2

22

1

1111 ≠+��

��

�−=�

��

��

�+++ −−

fhec

ethec

cec

th

xcxc

x

xcxc

t αα






equation (5.47)2 . System (5.47) accordingly simplifies to give

,0)(),( ≠= xytxh ,0)()(2

1 ≠′=′′′ xycxy ;0

2

2

)(

)(

22

2

22

2

1

211

11

≠

+

−=

+

′

−

−

xc

xc

xc

xc

ece

ecec

fxy

xy (5.48)

121

where ,01 ≠c 2c and 2f are arbitrary constants and 0)( ≠′ xy since .0≠xh

Via the method in [48], we solve equation (5.48)2 and substitute its general solution into

equation (5.48)3.

Systems (5.48) and (5.44) generate identical solutions for system (5.3) and thin film

equation (5.1) in tandem with the respective groups (5.46) and (5.42). Both groups occur

under identical conditions on ,0)( ≠hf )(hg and ).(hj

Hence under transformations (5.4) and the conditions ,0)()( 0

21 ≠+= ffhfhf

0)()( 0

2

2

11 ≠+= ffhcfhg and ,0)( =hj the similarity solution for system (5.3) and thin

film equation (5.1) in association with group (5.46) is the steady state solution (5.45).

GROUP (XI)

Under conditions ,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj

system (5.3) yields symmetry group (XI), namely

( ) ,0)(),,( 23

4

2

1111 ≠+�

��

��

�−=

−

fhecethtxAxctjc

xcα

,03

2)(),,( 1

3

4

2

1

1111 ≠+�

��

��

�+=

−

jecec

thtxB

xctjcxcα

(5.49)

4/1

13

4

2

1 3

2)()(),,(

1111

−

−

��

�

��

�+��

��

�+= jece

c

tthtx

xctjcxcα

βη

( ) [ ]

( ) [ ]

( ) [ ]

;0

3)()(

9

3)()(

9

113

4

22

2112

1

111

111

111

)(9

)(8

)(3

2)(

111

3

4

2

2

2

112

1

111

3

4

2

2

2

112

1

≠

��

�

�

��

�

�

��

��

�+−

��

��

−

��

��

�++

��

��

−

×

��

��

�

��

−

��

��

�+′

tjc

etcjc

ct

tjctc

xctjc

xctjc

jcetcetc

jcc

jcetcetc

jcc αα

αα

αα

αα

where ,01 ≠c ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants such that

( ) [ ] ( ) 0)(9

23

42

2

2

11 11

≠+��

��

− fhetcjc tjc

α while 0)( ≠tα and 0)( ≠tβ are arbitrary

functions of .t

122



( )

,03

2

21

12

1 =��

��

�

++−

fhf

jhchh xxxxx ( ) ,0)(

2

1

3

21 =+−+ txxxxxx hhchfhf

(5.50)

( ) ;0)(3

2)(2

3

4

213

4

2

1

1111

1111 ≠+�

��

��

�−=

��

�

��

�+��

��

�++

−−

fhecethjecec

th

xctjcxc

x

xctjcxc

t αα

where 0)( ≠tα is an arbitrary function of t while ,01 ≠c ,01 ≠f ,01 ≠j 2c and 2f are

arbitrary constants with ( ) [ ] ( ) .0)(

92

3

42

2

2

11 11

≠+��

��

− fhetcjc tjc

α

Since 0=xh gives 0=th in equation (5.50)2 , rendering equation (5.50)3 inconsistent,

we require ,0≠xh forcing ( )

03

2

21

12

1 ≠+

−=−fhf

jhch xxxx in equation (5.50)1.

Accordingly, system (5.50) simplifies to give

,03

21 ≠−= xt hjh

( ),0

32

21

12

1 ≠+

−=−fhf

jhch xxxx ;0

1111

1111

3

4

2

3

4

21

2

≠

+

−=

+ −

−

xctjcxc

xctjcxc

x

ece

ecec

fh

h

(5.51)

where ,01 ≠c ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants with .0≠xh


(5.51)1 yields a further equation directly solving which generates the result

;0),( 2

3

2

7

3

2

6

1111

≠−+=��

��

�−−�

�

��

�−

fecectxhtjxctjxc

(5.52)

where ,01 ≠c ,06 ≠c ,01 ≠j ,2c 627 ccc = and 2f are arbitrary constants.

Substituting result (5.52) into equation (5.51)2 gives rise to the contradiction .01 =j


21 ≠+= fhfhf

0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj no valid similarity solution arises for

system (5.3) and the thin film equation (5.1) in association with group (5.49).

123

GROUP (XII)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and

,0)( 1 ≠= jhj system (5.3) admits symmetry group (XII), namely

( ) ,0),,( 23

4

23

2

5

1111

11

≠+��

��

�−=

−−

fheceechtxAxctjc

xctjc

,03

2),,( 1

3

4

23

2

1

5 1111

11

≠+��

��

�+=

−−

jeceec

chtxB

xctjcxc

tjc

(5.53)

;03

2)(),,(

4/1

13

4

23

2

1

5 1111

11

≠��

�

��

�+��

��

�+=

−

−−

jeceec

cthtx

xctjcxc

tjc

βη

where 0)( ≠tβ is an arbitrary function of t while ,01 ≠c ,02 ≠c ,05 ≠c ,01 ≠f

01 ≠j and 2f are arbitrary constants with ( )

09 2

2

112

5 ≠=c

jcc and .02 ≠+ fh



( )

,03

2

21

12

1 =��

��

�

++−

fhf


2

1

3


(5.54)

( ) ;03

22

3

4

23

2

513

4

23

2

1

5 1111

111111

11

≠+��

��

�−=

��

�

��

�+��

��

�++

−−−−

fheceechjeceec

ch

xctjcxc

tjc

x

xctjcxc

tjc

t

where ,01 ≠c ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants such that

( )0

9 2

2

112

5 ≠=c

jcc and .02 ≠+ fh

As 0=xh gives 0=th in equation (5.54)2 , rendering equation (5.54)3 inconsistent,

0≠xh is a requirement, forcing ( )

03

2

21

12

1 ≠+

−=−fhf

jhch xxxx in equation (5.54)1.

System (5.54) then simplifies to give

,03

21 ≠−= xt hjh

( ),0

32

21

12

1 ≠+

−=−fhf

jhch xxxx ;0

1111

1111

3

4

2

3

4

21

2

≠

+

−=

+ −

−

xctjcxc

xctjcxc

x

ece

ecec

fh

h

(5.55)

where ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants with .0≠xh

124

System (5.55) is a special case of system (5.51) with 02 ≠c and so ultimately leads to

the contradiction .01 =j Groups (5.53) and (5.49) associated with the respective systems

(5.55) and (5.51) are subject to identical conditions on ,0)( ≠hf )(hg and ).(hj


21 ≠+= fhfhf

0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj no valid similarity solution occurs for

system (5.3) and the thin film equation (5.1) in connection with group (5.53).

GROUP (XIII)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj

system (5.3) yields symmetry group (XIII), namely

( ) ,0)(),,( 23

2

2

111

≠+=−

fhtechtxAxctjc

α ,03

2)(1),,( 1

3

2

1

2 111

≠+��

��

�+−=

−

jtec

chtxB

xctjc

α

(5.56)

;03

2)(1)(),,(

11

11

1111

3

2)()(4

3

2)()()(

13

2

1

23

2)()(4

)(

≠��

�

��

�+�

��

��

�+−=

��

��

�+

��

��

�++′

−

−��

��

�+

′−

jttc

jttct

xctjcjtt

xt

jtec

cethtx

αα

ααα

αα

α

αβη

where ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants with 02 ≠+ fh

while 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t with .03

2)( 1 ≠+ jtα



( )

,03

2

21

12

1 =��

��

�

++−

fhf


2

1

3


(5.57)

( ) ;0)(3

2)(1 2

3

2

213

2

1

2 111111

≠+=��

�

��

�+�

��

��

�+−+

−−

fhtechjtec

ch

xctjc

x

xctjc

t αα

where 0)( ≠tα is an arbitrary function of t while ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f

are arbitrary constants with ( ) .03

2)( 12 ≠��

��

�++ jtfh α

125


0≠xh is a requirement. Thus ( )

03

2

21

12

1 ≠+

−=−fhf

jhch xxxx in equation (5.57)1 and

system (5.57) accordingly simplifies to give

,03

21 ≠−= xt hjh

( ),0

32

21

12

1 ≠+

−=−fhf

jhch xxxx ;0

13

2

2

3

2

21

2 111

111

≠

−

−=

+ −

−

cec

ecc

fh

h

xctjc

xctjc

x

(5.58)


Via the method in [24] we solve equation (5.58)1 , substituting its general solution into

equations (5.58)2 and (5.58)3 and obtaining the relations

,0)(),( ≠= uytxh ,0)(

)(

12

21

21

1

≠−

−=

+

′−

−

cec

ecc

fuy

uyuc

uc

(5.59)

[ ]

;0)(3

)()(2

21

12

1 ≠+

−=′−′′′fuyf

juycuy

where ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants and tjxu 13

2−=



(5.59)3 generates the contradiction .01 =j Hence under transformations (5.4) and the

conditions ,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj no

valid similarity solution arises for system (5.3) and the thin film equation (5.1) in

conjunction with group (5.56).

126

GROUP (XIV)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj

system (5.3) admits symmetry group (XIV), namely

( ) ,0),,( 23

2

2

111

≠+=−

fhechtxAxctjc

,0),,(111

3

2

1

2 ≠−=− xctjc

ec

chtxB

(5.60)

;0)(),,( 4

1

≠=x

c

ethtx βη

where 0)( ≠tβ is an arbitrary function of t while ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f

are arbitrary constants with .02 ≠+ fh



( )

,03

2

21

12

1 =��

��

�

++−

fhf


2

1

3


(5.61)

( ) ;023

2

23

2

1

2 111111

≠+=−−−

fhechec

ch

xctjc

x

xctjc

t

where ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants with .02 ≠+ fh


require ,0≠xh causing ( )

03

2

21

12

1 ≠+

−=−fhf

jhch xxxx in equation (5.61)1 . System

(5.61) then simplifies to give

,03

21 ≠−= xt hjh

( ),0

32

21

12

1 ≠+

−=−fhf

jhch xxxx ;0

3

211

3

2

2

3

2

21

2 111

111

≠

+

−=

+ −

−

jcec

ecc

fh

h

xctjc

xctjc

x

(5.62)




127

,0)(),( ≠= uytxh ,0

3

2)(

)(

112

21

2 1

1

≠

+

−=

+

′

−

−

jcec

ecc

fuy

uy

uc

uc

(5.63)

[ ]

;0)(3

)()(2

21

12

1 ≠+

−=′−′′′fuyf

juycuy

where ,01 ≠c ,02 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants and tjxu 13

2−=



(5.63)3 yields the contradiction .01 =j Hence under transformations (5.4) and the

conditions ,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2


valid similarity solution occurs for system (5.3) and the thin film equation (5.1) in

connection with group (5.60).

GROUP (XV)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj

system (5.3) yields symmetry group (XV), namely

( ) ,0)(),,( 23

2

43

2111111

≠+��

��

�−=

+−−

fhecethtxAtjcxctjcxc

α

,03

2)(

)(),,( 1

2

13

2

43

2

1

111111

≠++��

��

�+=

+−−

jtc

cece

c

thtxB

tjcxctjcxc

αα

(5.64)

4/1

1

2

13

2

43

2

1 3

2)(

)()(),,(

111111

−

+−−

��

�

��

�++�

��

��

�+= jt

c

cece

c

tthtx

tjcxctjcxc

αα

βη

[ ] ( )

[ ] ( )

[ ] ( )

;0

3

2)(2)(

9

4)(

3

4)(

4

3

2)(2)(

9

4)(

3

4)(

4 211

2

3112

22

224

412

1

1

111

111

9

4)(

3

4)(

4)(4

)(

113

2

2

2

11

2

11

2

3

112

2

2

2

24

4

12

1

113

2

2

2

11

2

11

2

3

112

2

2

2

24

4

12

1

≠

��

�

�

��

�

�

��

�

�

��

�

�

��

��

�++

−��

��

++−

��

�

�

��

�

�

��

��

�++

+��

��

++−

×

��

��

�

��

++−

′

−

−

jctc

cjt

c

cccct

tc

tjcxc

tjcxc

jcettc

cc

jctc

cjt

c

cccc

jcettc

cc

jctc

cjt

c

cccc

ααα

α

αα

αα

αα

αα

128

where ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants with 02 ≠+ fh

while 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t with 02)(3 121 ≠+ jctc α and

( )[ ] .04)(12)(492

1

2

2

2

1

3

121

22

24

4

1 ≠++− jcctccjtccc αα


Group (5.64), system (5.3) and the invariant surface condition (5.7) generate relations

( ),0

32

21

12

1 =��

��

�

++−

fhf


2

1

3


(5.65)

( ) ;0)(3

2)(

)(2

3

2

43

2

1

2

13

2

43

2

1

111111111111

≠+��

��

�−=

��

�

��

�++�

��

��

�++

+−−+−−

fhecethjtc

cece

c

th

tjcxctjcxc

x

tjcxctjcxc

t ααα

where ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants with 02 ≠+ fh

while 0)( ≠tα is an arbitrary function of t with .02)(3 121 ≠+ jctc α


0≠xh is a requirement, causing equation (5.65)1 to give

( ).0

32

21

12

1 ≠+

−=−fhf

jhch xxxx System (5.65) then simplifies to give

,03

21 ≠−= xt hjh

( ),0

32

21

12

1 ≠+

−=−fhf

jhch xxxx

(5.66)

;02

13

2

43

2

2

3

2

43

2

7

2 111111

111111

≠

+��

��

�+

��

��

�−

=+ +−−

+−−

cecec

ecec

fh

h

tjcxctjcxc

tjcxctjcxc

x

where ,01 ≠c ,02 ≠c ,0217 ≠= ccc ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants

with .0≠xh

Via the method in [24], we solve equation (5.66)1 , substituting its general solution into

equations (5.66)2 and (5.66)3 and recovering the relations

129

,0)(),( ≠= uytxh ( )

( ),0

)(

)(2

142

47

211

11

≠++

−=

+

′

−

−

cecec

ecec

fuy

uy

ucuc

ucuc

(5.67)

[ ]

;0)(3

)()(2

21

12

1 ≠+

−=′−′′′fuyf

juycuy

where ,01 ≠c ,02 ≠c ,0217 ≠= ccc ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants,

tjxu 13

2−= and 0)( ≠′ uy since .0≠xh


(5.67)3 gives the contradiction .01 =j Hence under transformations (5.4) and the

conditions ,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2


valid similarity solution arises for system (5.3) and the thin film equation (5.1) in tandem

with group (5.64).

GROUP (XVI)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and

,0)( 1 ≠= jhj system (5.3) yields symmetry group (XVI), namely

( ) ,0),,( 23

22

53

2

2

111111

≠+��

��

�−=

+−−

fhecechtxAtjcxctjcxc

,02),,( 53

22

53

2

1

2 111111

≠��

��

�++=

+−−

cecec

chtxB

tjcxctjcxc

(5.68)

;02)(),,(

4/1

53

22

53

2111111

≠��

��

�++=

−+−−

cecethtxtjcxctjcxc

βη

where 0)( ≠tβ is an arbitrary function of t while ,01 ≠c ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j

and 2f are arbitrary constants with ( )( ) 063 52115211 ≠−− ccjcccjc and .02 ≠+ fh



( )

,03

2

21

12

1 =��

��

�

++−

fhf


2

1

3


(5.69)

( ) ;02 23

22

53

2

253

22

53

2

1

2 111111111111

≠+��

��

�−=�

��

��

�+++

+−−+−−

fhecechcecec

ch

tjcxctjcxc

x

tjcxctjcxc

t

130

where ,01 ≠c ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants with

( )( ) 063 52115211 ≠−− ccjcccjc and .02 ≠+ fh


require ,0≠xh causing equation (5.69)1 to give ( )

.03

2

21

12

1 ≠+

−=−fhf

jhch xxxx

Correspondingly, system (5.69) simplifies to give

,03

21 ≠−= xt hjh ( ) ,0

3

2

2

1

12

1 ≠+−=−−

fhf

jhch xxxx

(5.70)

;0

913

22

53

2

2

3

22

53

2

21

2 111111

111111

≠

+��

��

�+

��

��

�−

=+ +−−

+−−

ccecec

ececc

fh

h

tjcxctjcxc

tjcxctjcxc

x

where ,01 ≠c ,02 ≠c ,05 ≠c ( )

,03

32

1

5211

9 ≠−−

=c

ccjcc ,01 ≠f 01 ≠j and 2f are

arbitrary constants with 06 5211 ≠− ccjc and .0≠xh


equations (5.70)2 and (5.70)3 and retrieving the relations

,0)(),( ≠= uytxh ( )

( ),0

)(

)(

91

2

52

2

521

211

11

≠++

−=

+

′

−

−

ccecec

ececc

fuy

uy

ucuc

ucuc

(5.71)

[ ] ;0)(3

)()(2

2

1

12

1 ≠+−=′−′′′ −fuy

f

juycuy

where ,01 ≠c ,02 ≠c ,05 ≠c ( )

,03

32

1

5211

9 ≠−−

=c

ccjcc ,01 ≠f 01 ≠j and 2f are

arbitrary constants with 06 5211 ≠− ccjc while tjxu 13

2−= and 0)( ≠′ uy owing to the

requirement .0≠xh


(5.71)3 gives the contradiction .01 =j Hence under transformations (5.4) and conditions

,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj no valid similarity

solution arises from system (5.3) and the thin film equation (5.1) in association with

group (5.68).

131

GROUP (XVII)


21 ≠+= fhfhf 0)(36)( 3

2

2

1

52

1 ≠+��

��

�= fh

j

ccfhg and

,0)( 1 ≠= jhj system (5.3) admits symmetry group (XVII), namely

( ) ,0),,( 2

3

26

2

5

3

26

2

11

521

1

52

≠+��

�

�

��

�

�−=

��

��

�−�

�

��

�−±

fhecechtxAtjx

j

cctjx

j

cc�

,036

),,( 13

26

2

5

3

26

5

11

1

521

1

52

≠+��

�

�

��

�

�+±=

��

��

�−�

�

��

�−± j

ecec

jhtxB

tjxj

cctjx

j

cc�

(5.72)

;012

1)(),,(

4/1

3

26

2

5

3

26

5

11

521

1

52

≠��

��

�

��

−��

�

�

��

�

�+=

−

��

��

�−�

�

��

�−± tjx

j

cctjx

j

cc

ecec

thtx�

�βη

where 0)( ≠tβ is an arbitrary function of t while ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f

are arbitrary constants with .02 ≠+ fh



( ) ,03

362

2

1

1

2

1

52 =��

�

�

��

�

�++��

�

��

�−

−fh

f

jh

j

cchh xxxxx

,036)(

2

1

523

21 =+��

�

�

��

�

�

��

��

�−+ txxxxxx hh

j

cchfhf (5.73)

( ) ;036

2

3

26

2

5

3

26

213

26

2

5

3

26

5

11

1

521

1

521

1

521

1

52

≠+��

�

�

��

�

�−=

��

��

�

��

+��

�

�

��

�

�+±+

��

��

�−�

�

��

�−±�

�

��

�−�

�

��

�−±

fhecechj

ecec

jh

tjxj

cctjx

j

cc

x

tjxj

cctjx

j

cc

t

��

where ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants with .02 ≠+ fh


0≠xh is a requirement, causing equation (5.73)1 to give

( ).0

336

2

21

1

2

1

52 ≠+

−=��

��

�−

fhf

jh

j

cch xxxx System (5.73) then simplifies to give

132

,03

21 ≠−= xt hjh

( ),0

336

2

21

1

2

1

52 ≠+

−=��

��

�−

fhf

jh

j

cch xxxx

(5.74)

;0

36

13

26

2

5

3

26

5

1

3

26

2

5

3

26

2

2 11

521

1

52

11

521

1

52

≠

−��

�

�

��

�

�+±

��

�

�

��

�

�−

=+ �

�

��

�−�

�

��

�−±

��

��

�−�

�

��

�−±

jece

c

j

ecec

fh

h

tjxj

cctjx

j

cc

tjxj

cctjx

j

cc

x

�

�

where ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants while .0≠xh



,0)(),( ≠= uytxh ,0

36

)(

)(

1

6

2

5

6

5

1

6

2

5

6

2

21

52

1

52

1

52

1

52

≠

−��

�

�

��

�

�+±

��

�

�

��

�

�−

=+

′

±

±

jece

c

j

ecec

fuy

uy

uj

ccu

j

cc

uj

ccu

j

cc

�

�

(5.75)

[ ] ;0)(3

)(36)(2

2

1

1

2

1

52 ≠+−=′��

��

�−′′′ −

fuyf

juy

j

ccuy

where ,02 ≠c ,05 ≠c ,01 ≠f 01 ≠j and 2f are arbitrary constants, tjxu 13

2−= and

0)( ≠′ uy since .0≠xh


(5.75)3 yields the contradiction .01 =j Hence under transformations (5.4) and the

conditions ,0)()( 3

21 ≠+= fhfhf 0)(36)( 3

2

2

1

52

1 ≠+��

��

�= fh

j

ccfhg and ,0)( 1 ≠= jhj

no valid similarity solution is retrievable for system (5.3) and the thin film equation (5.1)

in conjunction with group (5.72).

133

GROUP (XVIII)


21 ≠+= fhfhf 0)()( 3

2

2

11 ≠+= fhcfhg and ,0)( 1 ≠= jhj

system (5.3) admits symmetry group (XVIII), namely

( ) ,0),,( 2

3

2

4

3

2

2

1111

≠+��

�

��

�−=

��

�

�−−�

�

�

�−

fhecechtxAtjxctjxc

,0),,(1111

3

2

4

3

2

1

2 ≠��

�

��

�+=

��

�

�−−�

�

�

�− tjxctjxc

ecec

chtxB (5.76)

;0)(),,(

4/1

4

3

22

411

1

≠��

�

��

�+=

−

��

�

�−

ceethtxtjxcx

c

βη

where 0)( ≠tβ is an arbitrary function of t while ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c




( ) ,03

2

2

1

12

1 =��

��

�++−

−fh

f


2

1

3


(5.77)

( ) ;02

3

2

4

3

2

2

3

2

4

3

2

1

211111111

≠+��

�

��

�−=

��

�

��

�++

��

�

�−−�

�

�

�−�

�

�

�−−�

�

�

�−

fhecechecec

ch

tjxctjxc

x

tjxctjxc

t

where ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants with .02 ≠+ fh


0≠xh is a requirement. System (5.77) accordingly simplifies to give

,03

21 ≠−= xt hjh ( ) ,0

3

2

2

1

12

1 ≠+−=−−

fhf

jhch xxxx

(5.78)

;0

3

21

3

2

4

3

2

1

2

3

2

4

3

2

2

2 1111

1111

≠

−��

�

��

�+

��

�

��

�−

=+

��

�

�−−�

�

�

�−

��

�

�−−�

�

�

�−

jecec

c

ecec

fh

h

tjxctjxc

tjxctjxc

x

where ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants with .0≠xh

134



,0)(),( ≠= uytxh ( )

( ),0

3

2)(

)(

14

1

2

42

2 11

11

≠

−+

−=

+

′

−

−

jecec

c

ecec

fuy

uy

ucuc

ucuc

(5.79)

[ ] ;0)(3

)()(2

2

1

12

1 ≠+−=′−′′′ −fuy

f

juycuy

where ,01 ≠c ,02 ≠c ,01 ≠f ,01 ≠j 4c and 2f are arbitrary constants, tjxu 13

2−=

and 0)( ≠′ uy owing to the requirement .0≠xh


(5.79)3 leads to the contradiction .01 =j Hence under transformations (5.4) and the

conditions ,0)()( 3

21 ≠+= fhfhf 0)()( 3

2

2


valid similarity solution occurs for system (5.3) and the thin film equation (5.1) in

connection with group (5.76).

GROUP (XIX)

Subject to conditions ,0)( 0

1 ≠= hfefhf 0)( =hg and ,0)( 0

1 ≠= hfejhj system (5.3)

yields symmetry group (XIX), namely

,02

)(),,( 43

2

2

3

10

1 ≠��

�

�+++−= cxcxcx

ff

jthtxA α ,0)(),,( ≠= xthtxB α

(5.80)

[ ][ ] ;0)(),,( 4824)(4

)(4)(032023

1

12

2

≠=++−

+′− x

fcx

fcx

f

j

t

tt

exthtx α

αα

βη

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c 3c and 4c are arbitrary constants such that

02

43

2

2

3

10

1 ≠+++− cxcxcxff

j while 0)( ≠tα and 0)( ≠tβ are arbitrary functions of

.t

135


Group (5.80), system (5.3) and the invariant surface condition (5.7) give the relations

,010

1 =��

�

�+

ff

jhh xxxx ,00

1 =+ txxxx

hfhhef

(5.81)

;02

)()( 43

2

2

3

10

1 ≠��

�

�+++−=+ cxcxcx

ff

jtxhth xt αα

where 0)( ≠tα is an arbitrary function of t while ,00 ≠f ,01 ≠f ,01 ≠j ,2c 3c and

4c are arbitrary constants such that .02

43

2

2

3

10

1 ≠+++− cxcxcxff

j

As 0=xh causes 0=th in equation (5.81)2 , rendering equation (5.81)3 inconsistent, we

require .0≠xh Consequently 010

1 ≠−=ff

jhxxx in equation (5.81)1 , forcing 0=th in


,0)(),( ≠= xytxh ,0)(10

1 ≠−=′′′ff

jxy ;0

2)( 4

32

2

10

1 ≠+++−=′x

ccxcx

ff

jxy

(5.82)

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c 3c and 4c are arbitrary constants with

.02

432

2

10

1 ≠��

�

�+++− x

x

ccxcx

ff

j

Differentiating equation (5.82)3 twice with respect to x and substituting the result into

equation (5.82)2 generates the constraint .04 =c We directly solve equation (5.82)3

under this constraint. Therefore under transformations (5.4) and the conditions

,0)( 0

1 ≠= hfefhf 0)( =hg and ,0)( 0

1 ≠= hfejhj the similarity solution for system

(5.3) and thin film equation (5.1) in tandem with group (5.80) and the constraint 04 =c


;026

),( 113

223

10

1 ≠+++−= cxcxc

xff

jtxh (5.83)

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c 3c and 11c are arbitrary constants with

.02

32

2

10

1 ≠++− cxcxff

j Solution (5.83) includes solution (5.41) as a special case

under the restriction .2 1

2

210

3j

cffc −=

136

GROUP (XX)


1 ≠= hfefhf 0)( =hg and ,0)( 0


admits symmetry group (XX), namely

[ ] ,02

)()(),,( 32

2

10

1 ≠��

�

�++−+= cxcx

ff

jtxthtxA γα ,0)()(),,( ≠+= txthtxB γα

(5.84)

[ ] [ ] [ ][ ]

[ ] ;0)()()(),,( 2

22

032023

1

1

)(4

)(4)()()()(4

)()()()(

4824≠+=

+′−+

′−′+++−

t

tttxtt

ttttx

fcx

fcx

f

j

txtethtx α

ααγαα

αγγα

γαβη

where ,0)( ≠tα 0)( ≠tβ and 0)( ≠tγ are arbitrary functions of t while ,00 ≠f

,01 ≠f ,01 ≠j 2c and 3c are arbitrary constants such that .02

32

2

10

1 ≠++− cxcxff

j



,010

1 =��

�

�+

ff

jhh xxxx ,00

1 =+ txxxx

hfhhef

(5.85)

[ ] [ ] ;02

)()()()( 32

2

10

1 ≠��

�

�++−+=++ cxcx

ff

jtxthtxth xt γαγα

where 0)( ≠tα and 0)( ≠tγ are arbitrary functions of t while ,00 ≠f ,01 ≠f ,01 ≠j

2c and 3c denote arbitrary constants with .02

32

2

10

1 ≠++− cxcxff

j


we require .0≠xh Hence 010

1 ≠−=ff

jhxxx in equation (5.85)1 , causing 0=th in

equation (5.85)2 . System (5.85) accordingly simplifies to give

,0)(),( ≠= xytxh ,0)(10

1 ≠−=′′′ff

jxy ;0

2)( 32

2

10

1 ≠++−=′ cxcxff

jxy (5.86)

where ,00 ≠f ,01 ≠f ,01 ≠j 2c and 3c are arbitrary constants such that

.02

32

2

10

1 ≠++− cxcxff

j

Directly solving equation (5.86)3 , we observe that differentiating it twice with respect to

x yields equation (5.86)2. Systems (5.86) and (5.82) (the latter under the constraint

04 =c ) generate identical solutions for system (5.3) and the thin film equation (5.1) in

137

tandem with the respective groups (5.84) and (5.80) (the latter under the constraint

04 =c ). Both groups occur under identical conditions on ,0)( ≠hf )(hg and ).(hj

Consequently under transformations (5.4) and conditions ,0)( 0

1 ≠= hfefhf 0)( =hg

and ,0)( 0

1 ≠= hfejhj the similarity solution for system (5.3) and the thin film equation

(5.1) in connection with group (5.84) is the steady state solution (5.83).

GROUP (XXI)


1 ≠= hfefhf 0)( =hg and ,0)( 0


yields symmetry group (XXI), namely

,02

)(),,( 32

2

10

1 ≠��

�

�++−= cxcx

ff

jthtxA α ,0)(),,( ≠= thtxB α

(5.87)

[ ][ ] ;0)(),,(

2

2032023

1

1

)(4

)()(

824≠=

′−++− x

t

ttfcx

fcx

f

j

ethtx α

αα

βη

where 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t while ,00 ≠f ,01 ≠f ,01 ≠j

2c and 3c are arbitrary constants with .02

32

2

10

1 ≠++− cxcxff

j



,010

1 =��

�

�+

ff

jhh xxxx ,00

1 =+ txxxx

hfhhef

(5.88)

;02

)()( 32

2

10

1 ≠��

�

�++−=+ cxcx

ff

jthth xt αα


02

32

2

10

1 ≠++− cxcxff

j while 0)( ≠tα is an arbitrary function of .t


require .0≠xh Hence 010

1 =≠−= txxx hff

jh in equations (5.88)1 and (5.88)2 , causing

system (5.88) to give

138

,0)(),( ≠= xytxh ,0)(10

1 ≠−=′′′ff

jxy ;0

2)( 32

2

10

1 ≠++−=′ cxcxff

jxy (5.89)


.02

32

2

10

1 ≠++− cxcxff

j

Systems (5.89) and (5.86) are identical, thereby yielding equivalent similarity solutions,

the steady state solution (5.83), for system (5.3) and thin film equation (5.1) in

association with the respective groups (5.87) and (5.84), under transformations (5.4) and

conditions ,0)( 0

1 ≠= hfefhf 0)( =hg and .0)( 0

1 ≠= hfejhj

GROUP (XXII)

Under the conditions ( ) ,0)(3

21 ≠+= fhfhf 0)( =hg and ,0)( 1 ≠= jhj system (5.3)

admits symmetry group (XXII), namely

( ) ,0)(),,( 2 ≠+= fhthtxA α ,03

2

3

2)(),,( 121 ≠+�

�

�

�+−= jctjxthtxB α

(5.90)

[ ][ ]

;03

2

3

2)()(),,(

2

2

)(4

)()(

121 ≠��

��

�+��

�

�+−=

+′−

t

tt

jctjxtthtxα

αα

αβη

where 0)( ≠tα and 0)( ≠tβ are arbitrary functions of t while ,01 ≠f ,02 ≠f 01 ≠j

and 2c are arbitrary constants with .02 ≠+ fh



( ) ,03

2

2

1

1 =��

��

�++

−fh

f

jhh xxxx ( ) ,0

3

21 =++ txxxx hhfhf

(5.91)

( ) ;0)(3

2

3

2)( 2121 ≠+=�

�

��

�+��

�

�+−+ fhthjctjxth xt αα

where ,01 ≠f ,02 ≠f 01 ≠j and 2c are arbitrary constants with 02 ≠+ fh while



0≠xh is a requirement. System (5.91) accordingly gives

139

( )

,03

2

21

1 ≠+

−=fhf

jhxxx ,0

3

21 ≠−= xt hjh ;0

3

2

1

212

≠

+−

=+

ctjxfh

hx (5.92)

where ,01 ≠f ,02 ≠f 01 ≠j and 2c are arbitrary constants and .0≠xh

Using the method in [24], we solve equation (5.92)2 , substituting its general solution

into equations (5.92)1 and (5.92)3 and recovering the relations

,0)(),( ≠= uytxh ,01

)(

)(

22

≠+

=+

′

cufuy

uy

[ ];0

)(3)(

2

21

1 ≠+

−=′′′fuyf

juy (5.93)

where ,01 ≠f ,02 ≠f 01 ≠j and 2c are arbitrary constants, tjxu 13

2−= and 0)( ≠′ uy



(5.93)3 gives the contradiction .01 =j Thus under transformations (5.4) and conditions

( ) ,0)(3

21 ≠+= fhfhf 0)( =hg and ,0)( 1 ≠= jhj no valid similarity solution exists

for system (5.3) and the thin film equation (5.1) in tandem with group (5.90).

GROUP (XXIII)



yields symmetry group (XXIII), namely

( ) ,03

2)(),,( 221 ≠+�

�

�

�+−= fhctjxthtxA α

,03

2

3

2

3

2

2

1)(),,( 1312

2

1 ≠+��

�

��

�+��

�

�−+�

�

�

�−= jctjxctjxthtxB α (5.94)

4/1

1312

2

13

2

3

2

3

2

2

1)()(),,(

−

�

��

�

��

+��

�

��

�+��

�

�−+�

�

�

�−= jctjxctjxtthtx αβη

( )

( )

[ ]

( )

;0

3

4)(2

3

2)(

3

4)(2

3

2)( 1

223

2/3

3

4)(24

)()(

1

2

2321

1

2

2321

≠

��

�

�

��

�

�

+−+��

�

�+−−

+−+��

�

�+−

×

+−

′−

−

jtcc

tti

jtccictjxt

jtccictjxtα

αα

αα

αα

where ,01 ≠f ,01 ≠j ,2c 3c and 2f are arbitrary constants such that 02 ≠+ fh while

0)( ≠tα and 0)( ≠tβ are arbitrary functions of t with ( ) .03

4)(2 1

2

23 ≠+− jtcc α

140



( ) ,03

2

2

1

1 =��

��

�++

−fh

f

jhh xxxx ( ) ,0

3

21 =++ txxxx hhfhf

(5.95)

( ) ;03

2)(

3

2

3

2

3

2

2

1)( 2211312

2

1 ≠+��

�

�+−=

�

��

�

��

+��

�

��

�+��

�

�−+�

�

�

�−+ fhctjxthjctjxctjxth xt αα

where ,01 ≠f ,01 ≠j ,2c 3c and 2f are arbitrary constants while 0)( ≠tα is an

arbitrary function of t such that ( ) ( ) .03

4)(2 21

2

23 ≠+��

��

�+− fhjtcc α


require .0≠xh System (5.95) then reduces to give

( )

,03

2

21

1 ≠+

−=fhf

jhxxx ,0

3

21 ≠−= xt hjh

(5.96)

;0

3

2

3

2

2

1

3

2

312

2

1

21

2

≠

+��

�

�−+�

�

�

�−

+−=

+ctjxctjx

ctjx

fh

hx

where ,01 ≠f ,01 ≠j ,2c 3c and 2f are arbitrary constants with .0≠xh



,0)(),( ≠= uytxh ,0

2

1)(

)(

32

2

2

2

≠

++

+=

+

′

cucu

cu

fuy

uy

[ ];0

)(3)(

2

21

1 ≠+

−=′′′fuyf

juy

(5.97)

where ,01 ≠f ,01 ≠j ,2c 3c and 2f are arbitrary constants, tjxu 13

2−= and 0)( ≠′ uy



(5.97)3 gives the contradiction .01 =j Thus under transformations (5.4) and conditions

( ) ,0)(3

21 ≠+= fhfhf 0)( =hg and ,0)( 1 ≠= jhj no valid similarity solution occurs

for system (5.3) and the thin film equation (5.1) in tandem with group (5.94).

141

GROUP (XXIV)



admits symmetry group (XXIV), namely

( ) ,03

2),,( 2311 ≠+�

�

�

�+−= fhctjxchtxA ,0

3

2

2),,(

2

311 ≠�

�

�

�+−= ctjx

chtxB

(5.98)

;03

2

2)(),,(

4/12

311 ≠

��

�

��

��

�

�+−=

−

ctjxc

thtx βη

where ,01 ≠c ,01 ≠f ,01 ≠j 3c and 2f are arbitrary constants with 02 ≠+ fh while

0)( ≠tβ is an arbitrary function of .t



( ) ,03

2

2

1

1 =��

��

�++

−fh

f

jhh xxxx ( ) ,0

3

21 =++ txxxx hhfhf

(5.99)

( ) ;03

2

3

2

22311

2

311 ≠+�

�

�

�+−=�

�

�

�+−+ fhctjxchctjx

ch xt

where ,01 ≠c ,01 ≠f ,01 ≠j 3c and 2f are arbitrary constants while .02 ≠+ fh


0≠xh is a requirement. System (5.99) accordingly simplifies to give

( )

,03

2

21

1 ≠+

−=fhf

jhxxx ,0

3

21 ≠−= xt hjh

(5.100)

;0

3

2

3

2

2

1

3

2

513

2

1

31

2

≠

+��

�

�−+�

�

�

�−

+−=

+ctjxctjx

ctjx

fh

hx

where ,01 ≠c ,01 ≠f ,01 ≠j ,3c 1

1

2

315

6

43

c

jccc

−= and 2f are arbitrary constants with

.0≠xh

142


equations (5.100)1 and (5.100)3 and deriving the relations

,0)(),( ≠= uytxh ,0

2

1)(

)(

53

2

3

2

≠

++

+=

+

′

cucu

cu

fuy

uy

[ ];0

)(3)(

2

21

1 ≠+

−=′′′fuyf

juy

(5.101)

where ,01 ≠c ,01 ≠f ,01 ≠j ,3c 1

1

2

315

6

43

c

jccc

−= and 2f are arbitrary constants,

tjxu 13

2−= and 0)( ≠′ uy since .0≠xh


(5.101)3 leads to the contradiction .01 =j Consequently under transformations (5.4) and


21 ≠+= fhfhf 0)( =hg and ,0)( 1 ≠= jhj no valid similarity

solution arises for system (5.3) and the thin film equation (5.1) in tandem with group

(5.98).

GROUP (XXV)


1 ≠= hfefhf 0)( =hg and ,0)( 1 ≠= jhj system (5.3) yields

symmetry group (XXV), namely

,0)(),,( ≠= thtxA α ( ) ,0)(3

),,( 111

0 ≠++−= jctjxtf

htxB α

(5.102)

( )

[ ][ ]

;0)(3

)(),,(2

0

20

)(4

)()(3

111

0 ≠��

��

�++−=

+′−

tf

tft

jctjxtf

thtxα

αα

αβη

where ,00 ≠f ,01 ≠f 01 ≠j and 1c are arbitrary constants while 0)( ≠tα and

0)( ≠tβ are arbitrary functions of .t

A special case of group (5.102) occurs and is as follows.

Case (1) The case of group (5.102) with 02

3)(

30

≠+

=ctf

tα arises under the

conditions ,0)( 0

1 ≠= hfefhf 1)( ghg = and 0)( 1 ≠= jhj where ,00 ≠f ,01 ≠f

,01 ≠j 3c and 1g are arbitrary constants and .02 30 ≠+ ctf

143



,00

10

1 =��

�

�+ − hf

xxxx eff

jhh ,00

1 =+ txxxx

hfhhef

(5.103)

( ) ;0)()(3

111

0 ≠=��

��

�++−+ thjctjxt

fh xt αα

where ,00 ≠f ,01 ≠f 01 ≠j and 1c are arbitrary constants while 0)( ≠tα is an

arbitrary function of .t


we require .0≠xh System (5.103) correspondingly reduces to give

,00

10

1 ≠−= − hf

xxx eff

jh ,01 ≠−= xt hjh

( );0

3

110

≠+−

=ctjxf

hx (5.104)

where ,00 ≠f ,01 ≠f 01 ≠j and 1c are arbitrary constants and .0≠xh


equations (5.104)1 and (5.104)3 and obtaining the equations

,0)(),( ≠= uytxh ( )

,03

)(10

≠+

=′cuf

uy ;0)()(

10

1 0 ≠−=′′′ − uyfe

ff

juy (5.105)

where ,00 ≠f ,01 ≠f 01 ≠j and 1c are arbitrary constants and tjxu 1−= with

.01 ≠+ cu

We directly solve equation (5.105)2 and substitute its general solution into equation

(5.105)3. Consequently, we find that under transformations (5.4) and the conditions

,0)( 0

1 ≠= hfefhf 0)( =hg and ,0)( 1 ≠= jhj the similarity solution for system (5.3)

and thin film equation (5.1) in connection with group (5.102) is the travelling wave of

velocity ,01 ≠j namely

;06

ln1

ln3

),(1

1

0

11

0

≠++−=f

j

fctjx

ftxh (5.106)

where ,00 ≠f ,01 ≠f 01 ≠j and 1c are arbitrary constants.

144

GROUP (XXVI)


21 ≠+=f

fhfhf 0)( =hg and ,0)( 1 ≠= jhj system (5.3)

admits symmetry group (XXVI), namely

( ) ,0)(),,( 2 ≠+= fhthtxA α ,011

)(3

),,( 1

0

0

21

0

00 ≠−

+��

�

�+

−−= j

f

fctj

f

fxt

fhtxB α

(5.107)

[ ][ ]

;011

)(3

)(),,(

20

20

)(4

)()(3

1

0

021

0

00 ≠��

��

� −+��

�

�+

−−=

+′−

tf

tft

jf

fctj

f

fxt

fthtx

α

αα

αβη

where ( ),3,1,00 ∉f ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants with 02 ≠+ fh




( ) ,001

2

10

1 =��

��

�++

− f

xxxx fhff

jhh ( ) ,00

21 =++ txxxx

fhhfhf

(5.108)

( ) ;0)(11

)(3

21

0

021

0

00 ≠+=��

��

� −+��

�

�+

−−+ fhthj

f

fctj

f

fxt

fh xt αα

where ( ),3,1,00 ∉f ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants with 02 ≠+ fh

while 0)( ≠tα is an arbitrary function of .t

As 0=xh causes 0=th in equation (5.108)2 , rendering equation (5.108)3 inconsistent,

0≠xh is a requirement. System (5.108) accordingly gives

( )

,01

210

1

0

≠+

−=−fxxx

fhff

jh ,0

11

0

0 ≠−

−= xt hjf

fh

(5.109)

( )

;01

3

201002

≠+−−

=+ cftjfxffh

hx

where ( ),3,1,00 ∉f ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants and .0≠xh



145

,0)(),( ≠= uytxh ( )

,03

)(

)(

202

≠+

=+

′

cuffuy

uy

[ ];0

)()(

1

210

1

0

≠+

−=′′′−f

fuyff

juy

(5.110)

where ( ),3,1,00 ∉f ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants and

tjf

fxu 1

0

0 1−−= with .02 ≠+ cu


(5.110)3 yields the constraint .2

30 ≠f Hence under transformations (5.4) and the


21 ≠+=f

fhfhf 0)( =hg and ,0)( 1 ≠= jhj the similarity solution

for system (5.3) and the thin film equation (5.1) in tandem with group (5.107) and the

constraint 2

30 ≠f is the travelling wave of velocity ,0

11

0

0 ≠−

jf

f namely

( )( )

;01

2333),( 2

3

21

0

0

1

001

2

0100

≠−��

�

�+

−−

��

�

��

�

−−−= fctj

f

fx

fff

fjtxh

ff

(5.111)

where ,3,2

3,1,00 �

�

�

�∉f ,01 ≠f ,01 ≠j 2c and 2f are arbitrary constants.

GROUP (XXVII)


1 ≠= hfefhf 1)( ghg = and ,0)( 0


yields symmetry group (XXVII), namely

,02

23

2),,(

2

612

10

113

2

2

3

10

1

≠+

++��

�

�−++−

=ft

ctgcxtff

jgcxcx

ff

j

htxA

,02

),,(2

7 ≠+

+=

ft

cxhtxB (5.112)

( ) ( )[ ]

( )( )

( );0)(),,( 2

12

8

32

7

48

36612362

21

1710211

112

71723102

171023

1

≠+=−+

+��

�� −−−+++−

ftf

jcffcgf

xtjgcjcccffxjcffcxj

cxethtx βη

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c ,3c ,6c ,7c 2f and 1g are arbitrary constants with

( )( ) 02 27 ≠++ ftcx while 0)( ≠tβ is an arbitrary function of .t

146



,010

1 =��

�

�+

ff

jhh xxxx ,011

0 =+− txxxxxx

hfhhghef

(5.113)

;02

23

2

2 2

612

10

113

2

2

3

10

1

2

7 ≠+

++��

�

�−++−

=+

++

ft

ctgcxtff

jgcxcx

ff

j

hft

cxh xt

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c ,3c ,6c ,7c 2f and 1g are arbitrary constants such

that ( )( ) .02 27 ≠++ ftcx


we require .0≠xh Equation (5.113)1 consequently yields ,010

1 ≠−=ff

jhxxx implying

;0)()()(6

),( 32

2

1

3

10

1 ≠+++−= txtxtxff

jtxh ααα (5.114)

where ,00 ≠f 01 ≠f and 01 ≠j are arbitrary constants while ),(1 tα )(2 tα and )(3 tα

are arbitrary functions of t with 0)()(22

21

2

10

1 ≠++− txtxff

jαα owing to the

requirement .0≠xh

Substituting result (5.114) into equations (5.113)2 and (5.113)3 , we find that under

transformations (5.4) and conditions ,0)( 0

1 ≠= hfefhf 1)( ghg = and ,0)( 0

1 ≠= hfejhj

the similarity solution for system (5.3) and the thin film equation (5.1) in tandem with

group (5.112) and constraint 7723

2

7

10

1217

10

126

22

3ccccc

ff

jfgc

ff

jcc ��

�

�

�−+−+��

�

�

�+= is

;06

),( 181716

10

112

14

3

10

1 ≠++��

�

�+−++−= ctcxct

ff

jgxcx

ff

jtxh (5.115)

where ,00 ≠f ,01 ≠f ,01 ≠j ,2c ,3c ,7c ,4

2

10

17102

14ff

jcffcc

+=

( ) ( ),

2

22

10

72310

2

712116

ff

cccffcgfjc

−+−=

( ),

2

2

10

171021

17ff

jcffcgc

+= ,18c 2f and 1g are

arbitrary constants with .022

16

10

1114

2

10

1 ≠+−+− ctff

jgxcx

ff

j

147

GROUP (XXVIII)

Subject to conditions ( ) ,0)(4

21 ≠+= fhfhf ( ) 0)(4

2

2

11 ≠+= fhcfhg and ,0)( =hj

system (5.3) admits symmetry group (XXVIII), namely

( )( ) ,0),,( 23211 ≠+−= −

fhecechtxAxcxc

,0),,( 6

1

3211

≠++

=−

cc

ecechtxB

xcxc

(5.116)

;0)(),,( ≠= thtx βη

where ,01 ≠c ,01 ≠f ,2c ,3c 6c and 2f are arbitrary constants with 011


and 02 ≠+ fh while 0)( ≠tβ is an arbitrary function of .t



( ) ,02

1 =− xxxxx hchh ( ) ( ) ,02

1

4


(5.117)

( )( ) ;02326

1

32 11

11

≠+−=��

�

�+

++ −

−

fhecechcc

ecech

xcxc

x

xcxc

t

where ,01 ≠c ,01 ≠f ,2c ,3c 6c and 2f are arbitrary constants with 011


and .02 ≠+ fh



1 =≠= txxxx hhch in equations (5.117)1 and (5.117)2 ,

simplifying system (5.117) to give

,0)(),( ≠= xytxh ,0)()(2

1 ≠′=′′′ xycxy ( )

;0)(

)(

6132

321

211

11

≠++

−=

+

′−

−

ccecec

ececc

fxy

xyxcxc

xcxc

(5.118)

where ,01 ≠c ,2c ,3c 6c and 2f are arbitrary constants with .011


We directly solve equation (5.118)3 , finding its general solution to identically satisfy

equation (5.118)2. Hence under transformations (5.4) and the conditions

( ) ,0)(4

21 ≠+= fhfhf ( ) 0)(4

2

2

11 ≠+= fhcfhg and ,0)( =hj the similarity solution

arising for system (5.3) and the thin film equation (5.1) in connection with group (5.116)


148

;0),( 98711 ≠++= −

cecectxhxcxc

(5.119)

where ,01 ≠c ,010 ≠c ,2c ,3c ,6c ,1

102

7c

ccc = ,

1

103

8c

ccc = 21069 fccc −= and 2f are

arbitrary constants such that .011


Solution (5.22) is a special case of solution (5.119) with .02 ≠c

GROUP (XXIX)


21 ≠+= fhfhf ( ) 0)(4

2

2

11 ≠+= fhcfhg and ,0)( =hj

system (5.3) yields symmetry group (XXIX), namely

( )( ) ,014

1),,( 243

2

11 ≠+−−+

= −fhecec

cthtxA

xcxc

(5.120)

,4

1),,( 7

1

43

2

11

��

�

�+

+

+=

−

cc

ecec

cthtxB

xcxc

;0)(),,( ≠= thtx βη

where 0)( ≠tβ is an arbitrary function of t while ,01 ≠c ,01 ≠f ,2c ,3c ,4c 7c and

2f are arbitrary constants with 0111

43 ≠−− − xcxcecec and ( )( ) .04 22 ≠++ fhct



( ) ,02

1 =− xxxxx hchh ( ) ( ) ,02

1

4


(5.121)

( )( ) ;014

1

4

1243

2

7

1

43

2

11

11

≠+−−+

=��

�

�+

+

++ −

−

fhececct

hcc

ecec

cth

xcxc

x

xcxc

t

where ,01 ≠c ,01 ≠f ,2c ,3c ,4c 7c and 2f are arbitrary constants with

0111

43 ≠−− − xcxcecec and ( )( ) .04 22 ≠++ fhct


we require .0≠xh Thus 02

1 =≠= txxxx hhch in equations (5.121)1 and (5.121)2 ,

reducing system (5.121) to give

,0)(),( ≠= xytxh ,0)()(2

1 ≠′=′′′ xycxy ( )

;01

)(

)(

7143

431

211

11

≠++

−−=

+

′−

−

ccecec

ececc

fxy

xyxcxc

xcxc

(5.122)

149

where 0)( ≠′ xy since 0≠xh while ,01 ≠c ,3c ,4c 7c and 2f are arbitrary constants

with 0111

43 ≠−− − xcxcecec and .07143

11 ≠++ −ccecec

xcxc

By the method in [48], we find the general solution of equation (5.122)2 to be

;0)( 109811 ≠++= −

cececxyxcxc

(5.123)

where ,01 ≠c ,2c ,1

28

c

cc = 9c and 10c are arbitrary constants with .011

912 ≠− − xcxceccec

Substituting result (5.123) into equation (5.122)3 , we see that the only cases generating

valid similarity solutions for system (5.3) and the thin film equation (5.1) are

(1) ,03 ≠c (2) ,034 =≠ cc ,01

1

7 ≠=c

c

(3) ,034 =≠ cc ,01

1

7 ≠−=c

c (4) .0437 ==≠ ccc

Case (1) 03 ≠c

From result (5.123) and equation (5.122)3 , the constraint ( )

3

2

714

4

1

c

ccc

−= arises.

Therefore under transformations (5.4) and the conditions ( ) ,0)(4

21 ≠+= fhfhf

( ) 0)(4

2

2

11 ≠+= fhcfhg and ,0)( =hj the similarity solution of system (5.3) and the

thin film equation (5.1) in association with group (5.120) and the constraints 03 ≠c and

( )

3

2

714

4

1

c

ccc

−= is the steady state solution

;0),( 1211811 ≠++= −

cecectxhxcxc

(5.124)

where ,01 ≠c ,02 ≠c ,03 ≠c ,01

28 ≠=

c

cc ,7c

( ),

4

12

31

2

712

11cc

cccc

+=

( )

31

231712

12

1

cc

fcccccc

−+= and 2f are arbitrary constants.


Case (2) ,034 =≠ cc 01

1

7 ≠=c

c

Result (5.123) and relations (5.122)1 and (5.122)3 imply that under transformations (5.4)


21 ≠+= fhfhf ( ) 0)(4

2

2


150


group (5.120) and the constraints 034 =≠ cc and 01

1

7 ≠=c

c is the steady state solution

;0),( 291 ≠−= −

fectxhxc

(5.125)

where ,01 ≠c 09 ≠c and 2f are arbitrary constants. Solutions (5.125) and (5.18) are

equivalent.

Case (3) ,034 =≠ cc 01

1

7 ≠−=c

c

Result (5.123) and relations (5.122)1 and (5.122)3 imply that under transformations (5.4)


21 ≠+= fhfhf ( ) 0)(4

2

2


similarity solution of system (5.3) and the thin film equation (5.1) in tandem with group

(5.120) and the constraints 034 =≠ cc and 01

1

7 ≠−=c

c is the steady state solution

;0),( 1291111 ≠++= −

cecectxhxcxc

(5.126)

where ,01 ≠c ,04 ≠c ,09 ≠c ,02

4

9

11 ≠=c

cc

4

249

12

2

c

fccc

+−= and 2f are arbitrary

constants. Solution (5.126) is a special case of solution (5.22) with .074 ≠cc

Case (4) 0437 ==≠ ccc

From result (5.123) and relations (5.122)1 and (5.122)3 , two similarity solutions with

their accompanying constraints arise under transformations (5.4) for system (5.3) and the

thin film equation (5.1) in tandem with group (5.120) and its associated conditions on

,0)( ≠hf )(hg and ).(hj The first such solution with its corresponding constraint is

equivalent to solution (5.125) with the constraint .171 =cc

With respect to the second solution, result (5.123) with relations (5.122)1 and (5.122)3

gives rise to the condition .171 −=cc Hence under transformations (5.4) and the


21 ≠+= fhfhf ( ) 0)(4

2

2



group (5.120) and the constraints 171 −=cc and 043 == cc is the steady state solution

;0),( 281 ≠−= fectxhxc

(5.127)

151

where ,01 ≠c ,02 ≠c 01

28 ≠=

c

cc and 2f are arbitrary constants. Solutions (5.127) and

(5.18) are equivalent.

The infinitesimal generators 2921 ,...,, VVV denote the algebras for the symmetry groups

(I), (II),…, (XXIX) respectively; (see Gandarias [28]). These generators are as follows.

A List of Infinitesimal Generators for Groups (I)-(XXIX)

The generators 2921 ,...,, VVV for the respective symmetry groups (I), (II),…, (XXIX) are

[ ],

)(

4)(

0

01

1txtf

tfjV

∂

∂+

∂

∂+=

α

α ,)(2

txtV

∂

∂+

∂

∂= α

( ) ,)()( 213h

fhttx

tcV∂

∂++

∂

∂+

∂

∂= αα ( ) ,)()( 11

424h

ecettx

tcVxcxc

∂

∂−+

∂

∂+

∂

∂= −αα

( ) ,)()( 1324511

hcecect

txtcV

xcxc

∂

∂+−+

∂

∂+

∂

∂= −αα

( ) ,)()(

2

1

611

hfhet

txe

c

tV

xcxc

∂

∂++

∂

∂+

∂

∂−= −− α

α

( ) ( )( ) ,)()(

223

1

711

hfhcet

txce

c

tV

xcxc

∂

∂+++

∂

∂+

∂

∂+= α

α

( ) ( ) ,)(3

)(218

hfht

txcx

tV

∂

∂++

∂

∂+

∂

∂+= α

α

+∂

∂+

∂

∂

��

�

��

��

�

�++�

��

�

�++= −−

txe

c

c

te

cce

c

tV

xcxcxc 111

21

)(

4

)( 2

1

2

22

1

9

κα

( ) ,2

1)(4

)( 22

2

2 111

hfhe

cte

cet

xcxcxc

∂

∂+

��

�

��

��

�

�−+�

��

�

�− −− κα

( ) ,4

)(4

)(2

2

2

2

22

1

101111

hfhe

cet

txe

cce

c

tV

xcxcxcxc

∂

∂+�

��

�

�−+

∂

∂+

∂

∂��

�

�++= −− α

α

( ) ,)(3

2)(2

3

4

213

4

2

1

11

1111

1111

hfhecet

txjece

c

tV

xctjcxc

xctjcxc

∂

∂+�

��

�

�−+

∂

∂+

∂

∂

��

�

��

�+��

�

�+=

−−

αα

( ) ,3

22

3

4

23

2

513

4

23

2

1

5

12

1111

111111

11

hfheceec

txjecee

c

cV

xctjcxc

tjcxctjcxc

tjc

∂

∂+�

��

�

�−+

∂

∂+

∂

∂

��

�

��

�+��

�

�+=

−−−−

( ) ,)(3

2)(1 2

3

2

213

2

1

213

111111

hfhtec

txjte

c

cV

xctjcxctjc

∂

∂++

∂

∂+

∂

∂

��

�

��

�+�

��

�

�+−=

−−

αα

152

( ) ,23

2

23

2

1

214

111111

hfhec

txe

c

cV

xctjcxctjc

∂

∂++

∂

∂+

∂

∂−=

−−

+∂

∂+

∂

∂

��

�

��

�++�

��

�

�+=

+−−

txjt

c

cece

c

tV

tjcxctjcxc

1

2

13

2

43

2

1

153

2)(

)( 111111

αα

( ) ,)( 23

2

43

2111111

hfhecet

tjcxctjcxc

∂

∂+�

��

�

�−

+−−

α

( ) ,2 23

22

53

2

253

22

53

2

1

216

111111111111

hfhecec

txcece

c

cV

tjcxctjcxctjcxctjcxc

∂

∂+�

��

�

�−+

∂

∂+

∂

∂��

�

�++=

+−−+−−

+∂

∂+

∂

∂

�

��

�

��

+��

�

�

��

�

�+±=

��

�

�−�

�

�

�−±

tx

jece

c

jV

tjxj

cctjx

j

cc

36

13

26

2

5

3

26

5

117

11

521

1

52�

( ) ,2

3

26

2

5

3

26

2

11

521

1

52

hfhecec

tjxj

cctjx

j

cc

∂

∂+

��

�

�

��

�

�−

��

�

�−�

�

�

�−± �

( ) ,2

3

2

4

3

2

2

3

2

4

3

2

1

218

11111111

hfhecec

txece

c

cV

tjxctjxctjxctjxc

∂

∂+

��

�

��

�−+

∂

∂+

∂

∂

��

�

��

�+=

��

�

�−−�

�

�

�−�

�

�

�−−�

�

�

�−

,2

)()( 43

2

2

3

10

119

hcxcxcx

ff

jt

txxtV

∂

∂��

�

�+++−+

∂

∂+

∂

∂= αα

[ ] [ ] ,2

)()()()( 32

2

10

120

hcxcx

ff

jtxt

txtxtV

∂

∂��

�

�++−++

∂

∂+

∂

∂+= γαγα

,2

)()( 32

2

10

121

hcxcx

ff

jt

txtV

∂

∂��

�

�++−+

∂

∂+

∂

∂= αα

( ) ,)(3

2

3

2)( 212122

hfht

txjctjxtV

∂

∂++

∂

∂+

∂

∂��

��

�+��

�

�+−= αα

( ) ,3

2)(

3

2

3

2

3

2

2

1)( 2211312

2

123h

fhctjxttx

jctjxctjxtV∂

∂+�

�

�

�+−+

∂

∂+

∂

∂

�

��

�

��

+��

�

��

�+��

�

�−+�

�

�

�−= αα

( ) ,3

2

3

2

22311

2

311

24h

fhctjxctx

ctjxc

V∂

∂+�

�

�

�+−+

∂

∂+

∂

∂��

�

�+−=

( ) ,)()(3

1110

25h

ttx

jctjxtf

V∂

∂+

∂

∂+

∂

∂��

��

�++−= αα

( ) ,)(11

)(3

21

0

021

0

0026

hfht

txj

f

fctj

f

fxt

fV

∂

∂++

∂

∂+

∂

∂��

��

� −+��

�

�+

−−= αα

153

,2

23

2

2 2

612

10

113

2

2

3

10

1

2

727

hft

ctgcxtff

jgcxcx

ff

j

txft

cxV

∂

∂

+

++��

�

�−++−

+∂

∂+

∂

∂

+

+=

( )( ) ,2326

1

32

2811

11

hfhecec

txc

c

ececV

xcxc

xcxc

∂

∂+−+

∂

∂+

∂

∂��

�

�+

+= −

−

( )( ) ;14

1

4

1243

2

7

1

43

2

2911

11

hfhecec

cttxc

c

ecec

ctV

xcxc

xcxc

∂

∂+−−

++

∂

∂+

∂

∂��

�

�+

+

+= −

−

where the details of each iV for all i ∈ }29,...,2,1{ relate to the respective groups (I) –

(XXIX).

We next present four tables of results. Table 1 features the functions ),(hf )(hg and

)(hj (distinguishing enhanced symmetries of thin film equation (5.1)) with their

associated infinitesimal generators .iV Tables 2a, 2b and 2c outline a dimensional

classification of the mathematical structure of groups (I)-(XXIX) and the corresponding

.iV As spatial limitations preclude the inclusion of ( ),,, htxA ( )htxB ,, and ( )htx ,,η in

the same table, the component ( )htx ,,η for each of the groups (I)-(XXIX) and the

corresponding iV are separately tabulated. Table 3 displays the similarity solutions

),( txh for systems (5.2) and (5.3) in tandem with groups (I)-(XXIX) where applicable.

154


Table 1. Each row lists the functions ),(hf )(hg and )(hj (distinguishing the enhanced

symmetries of thin film equation (5.1)) with the associated infinitesimal generators .iV

)(hf )(hg )(hj iV

00

1 ≠hfef 1g 01 ≠j 1V

arbitrary 0≠ 0)(1 ≠hfg [ ] )()( 1

1

0 hfjdssfj

h

′��

�

��

�+�

−

2V

( ) 00

21 ≠+f

fhf ( ) 00

21 ≠+f

fhg 0 3V

00

1 ≠hfef 00

2

11 ≠hfecf

hfej 0

1 4V

00

1 ≠hfef 00

2

11 ≠hfecf

hfej 0

1 5V

( ) 00

21 ≠+f

fhf ( ) 00

2

2

11 ≠+f

fhcf 0 6V

( ) 00

21 ≠+f

fhf ( ) 00

2

2

11 ≠+f

fhcf 0 7V

( ) 00

21 ≠+f

fhf 0 ( ) 01

210 ≠+

−ffhj 8V

( ) 00

21 ≠+f

fhf ( ) 00

2

2

11 ≠+f

fhcf 0 9V

( ) 00

21 ≠+f

fhf ( ) 00

2

2

11 ≠+f

fhcf 0 10V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 11V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 12V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 13V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 14V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 15V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 16V

( ) 03

21 ≠+ fhf 0)(36 3

2

2

1

52

1 ≠+��

�

�fh

j

ccf

01 ≠j 17V

( ) 03

21 ≠+ fhf ( ) 03

2

2

11 ≠+ fhcf 01 ≠j 18V

00

1 ≠hfef 0 00

1 ≠hfej 19V

00

1 ≠hfef 0 00

1 ≠hfej 20V

00

1 ≠hfef 0 00

1 ≠hfej 21V

( ) 03

21 ≠+ fhf 0 01 ≠j 22V

( ) 03

21 ≠+ fhf 0 01 ≠j 23V

( ) 03

21 ≠+ fhf 0 01 ≠j 24V

00

1 ≠hfef 0 01 ≠j 25V

( ) 00

21 ≠+f

fhf 0 01 ≠j 26V

00

1 ≠hfef 1g 00

1 ≠hfej 27V

( ) 04

21 ≠+ fhf ( ) 04

2

2

11 ≠+ fhcf 0 28V

( ) 04

21 ≠+ fhf ( ) 04

2

2

11 ≠+ fhcf 0 29V

155

The entries for ),(hf )(hg and )(hj in each of rows 1-29 in table 1 respectively

correspond to the symmetry groups (I)-(XXIX).

Table 2a. A dimensional classification of the mathematical structure of groups (I)-(XIV)

(the enhanced symmetries of the thin film equation (5.1)) with their associated

infinitesimal generators .iV 02

1)(

4

)(111 2

1

2

22

1

≠��

�

�++�

��

�

�++ −− xcxcxc

ec

c

te

cce

c

t κα

),,( htxA ),,( htxB iV

0 [ ])(

4)(

0

01

tf

tfj

α

α +

1V

0 0)( ≠tα 2V

( ) 0)( 2 ≠+ fhtα 0)(1 ≠tc α 3V

( ) 0)( 11

4 ≠− − xcxcecetα 0)(2 ≠tc α 4V

( ) 0)( 13211 ≠+− −

cecectxcxcα 0)(4 ≠tc α 5V

( ) 0)( 21 ≠+−

fhetxcα

0)(

1

1

≠− − xce

c

tα 6V

( )( ) 0)( 221 ≠++ fhcetxcα ( ) 0

)(3

1

1 ≠+ cec

t xcα 7V

( ) 0)( 2 ≠+ fhtα ( ) 03

)(1 ≠+ cx

tα 8V

( ) 02

1)(4

)( 22

2

2 111 ≠+��

�

��

��

�

�−+�

��

�

�− −−

fhec

tec

etxcxcxc κα

0

21

)(

4

)(111 2

1

2

22

1

≠��

�

�++�

��

�

�++ −− xcxcxc

ec

c

te

cce

c

t κα 9V

( ) 04

)( 2

2

2 11 ≠+��

�

�− −

fhec

etxcxcα 0

4

)(11

2

22

1

≠��

�

�++ − xcxc

ec

cec

tα

10V

( ) 0)( 23

4

2

1111 ≠+�

��

�

�−

−

fhecetxctjc

xcα 03

2)(1

3

4

2

1

1111 ≠+�

��

�

�+

−

jecec

t xctjcxcα

11V

( ) 023

4

23

2

5

1111

11

≠+��

�

�−

−−

fheceecxctjc

xctjc

03

21

3

4

23

2

1

5 1111

11

≠+��

�

�+

−−

jeceec

c xctjcxc

tjc

12V

( ) 0)( 23

2

2

111

≠+−

fhtecxctjc

α 03

2)(1 1

3

2

1

2 111

≠+��

�

�+−

−

jtec

c xctjc

α 13V

( ) 023

2

2

111

≠+−

fhecxctjc

0111

3

2

1

2 ≠−− xctjc

ec

c

14V

156

Table 2a. Continued.

),,( htxη iV

( ) 0)( 2 ≠+ chtα 1V

[ ]0)( 0)()(4

)(

≠+

′−

jtt

xt

etαα

α

β 2V

[ ][ ] 0)(

21

20

)(4

)()(

≠

+′−x

tc

tft

etα

αα

β 3V

( )[ ]

0)(4

)(

4exp)(

2

2

4

21

0 11 ≠��

��

� ′−+ −

xtc

tece

cc

ft

xcxc

α

αβ

4V

( )[ ]

0)(4

)(

4exp)(

2

4

132

41

0 11 ≠��

��

� ′−++ −

xtc

txcecec

cc

ft

xcxc

α

αβ

5V

( )[ ]

0)(4

)(

4

4exp)( 1

2

01 ≠��

��

� ′+

− xce

t

tx

fct

α

αβ

6V

( )[ ][ ][ ]

[ ][ ] 0)(

2

202

3

123

23023

1

)(

)()(

4)(4

)(4)(

2

1

3 ≠��

�

� +−′−

−−+′

xt

tfct

c

ctc

tcfcct

xc

ec

cet α

ααα

αα

β

7V

( )[ ]

[ ] 0)( 2

20

)(4

)(3

4)(3

1 ≠+��

��

��

�

�−+′−

t

tft

cxt α

αα

β

8V

[ ][ ]

( )��

�

�

��

�

�

++

′−′

��

�

�

��

�

�

++

+

+′−

)(22)(

)()(2

)()()()(

exp

)(

)(

2

2)(

2

)(4

)()(

2

2

1

2

20

1

1

tcet

tt

tttt

t

tce

ce

txc

t

tft

xc

xc

κα

ακ

καακ

α

κβ

κ

κκ

02

1)(4

)(4

4

21

2

221

0

111 ≠��

�

��

��

�

�++�

��

�

�++×

−

−−

f

xcxcxce

ctce

ccetc κα

9V

( )[ ]

[ ]0

)(2

4

2)(4)(

exp2

)(22

22

012

4

2

1

10

1 ≠

��

�

�

��

�

�

��

�

�+

��

�

�+−−′

��

�

�+

−

tc

e

cextfct

cet

xc

xcf

xc

α

αα

β

10V

4/1

13

4

2

1 3

2)()(

1111

−

−

��

�

��

�+��

�

�

�+ jece

c

tt

xctjcxcα

β

( )[ ]

( )[ ]

( )[ ]

0

3)()(

9

3)()(

9

113

4

22

2112

1

111

111

111

)(9

)(8

)(3

2)(

11

13

4

2

2

2

112

1

11

13

4

2

2

2

112

1

≠

��

�

�

��

�

�

��

��

�+−

�

��

�

��

−

��

��

�++

�

��

�

��

−

×

�

�

�

�

�

�−

��

��

�+′

tjc

etcjc

ct

tjctc

xctjc

xctjc

jcetcetc

jcc

jcetcetc

jcc αα

αα

αα

αα

11V

157

Table 2a. Continued.

),,( htxη iV

03

2)(

4/1

13

4

23

2

1

5 1111

11

≠��

�

��

�+��

�

�+

−

−−

jeceec

ct

xctjcxc

tjc

β 12V

03

2)(1)(

11

11

1111

3

2)()(4

3

2)()()(

13

2

1

23

2)()(4

)(

≠��

�

��

�+�

��

�

�+−

��

��

�+

��

��

�++′

−

−��

��

�+

′−

jttc

jttct

xctjcjtt

xt

jtec

cet

αα

ααα

αα

α

αβ

13V

0)( 4

1

≠x

c

etβ 14V

Table 2b. A dimensional classification of the mathematical structure of groups (XV)-

(XXIII) (the enhanced symmetries of the thin film equation (5.1)) with their associated



( ) 0)( 23

2

43

2111111

≠+��

�

�−

+−−

fhecettjcxctjcxc

α 03

2)(

)(1

2

13

2

43

2

1

111111

≠++��

�

�+

+−−

jtc

cece

c

t tjcxctjcxc

αα 15V

( ) 023

22

53

2

2

111111

≠+��

�

�−

+−−

fhecectjcxctjcxc

02 53

22

53

2

1

2 111111

≠��

�

�++

+−−

cecec

c tjcxctjcxc

16V

( ) 02

3

26

2

5

3

26

2

11

521

1

52

≠+��

�

�

��

�

�−

��

�

�−�

�

�

�−±

fhecectjx

j

cctjx

j

cc�

036

13

26

2

5

3

26

5

11

1

521

1

52

≠+��

�

�

��

�

�+±

��

�

�−�

�

�

�−± j

ecec

j tjxj

cctjx

j

cc�

17V

( ) 02

3

2

4

3

2

2

1111

≠+��

�

��

�−

��

�

�−−�

�

�

�−

fhecectjxctjxc

01111

3

2

4

3

2

1

2 ≠��

�

��

�+

��

�

�−−�

�

�

�− tjxctjxc

ecec

c

18V

02

)( 43

2

2

3

10

1 ≠��

�

�+++− cxcxcx

ff

jtα

0)( ≠xtα 19V

[ ] 02

)()( 32

2

10

1 ≠��

�

�++−+ cxcx

ff

jtxt γα

0)()( ≠+ txt γα 20V

02

)( 32

2

10

1 ≠��

�

�++− cxcx

ff

jtα

0)( ≠tα 21V

( ) 0)( 2 ≠+ fhtα 0

3

2

3

2)( 121 ≠+�

�

�

�+− jctjxtα 22V

( ) 03

2)( 221 ≠+�

�

�

�+− fhctjxtα 0

3

2

3

2

3

2

2

1)( 1312

2

1 ≠+��

�

��

�+��

�

�−+�

�

�

�− jctjxctjxtα 23V

158

Table 2b. Continued.

),,( htxη iV

4/1

1

2

13

2

43

2

1 3

2)(

)()(

111111

−

+−−

��

�

��

�++�

��

�

�+ jt

c

cece

c

tt

tjcxctjcxc

αα

β

[ ] ( )

[ ] ( )

[ ] ( )

0

3

2)(2)(

9

4)(

3

4)(

4

3

2)(2)(

9

4)(

3

4)(

4 211

2

3112

22

224

412

1

1

111

111

9

4)(

3

4)(

4)(4

)(

113

2

2

2

11

2

11

2

3

112

2

2

2

24

4

12

1

113

2

2

2

11

2

11

2

3

112

2

2

2

24

4

12

1

≠

��

�

�

��

�

�

��

�

�

�

��

��

�++

−��

��

++−

��

�

�

�

��

��

�++

+��

��

++−

×

�

��

�

��

++−

′

−

−

jctc

cjt

c

cccct

tc

tjcxc

tjcxc

jcettc

cc

jctc

cjt

c

cccc

jcettc

cc

jctc

cjt

c

cccc

ααα

α

αα

αα

αα

αα

15V

02)(

4/1

53

22

53

2111111

≠��

�

�++

−+−−

cecettjcxctjcxc

β 16V

012

1)(

4/1

3

26

2

5

3

26

5

1

1

521

1

52

≠ �

��

�

��

−��

�

�

��

�

�+

−

��

�

�−�

�

�

�−± tjx

j

cctjx

j

cc

ecec

t�

�β 17V

0)(

4/1

4

3

22

411

1

≠��

�

��

�+

−

��

�

�−

ceettjxcx

c

β 18V

[ ][ ] 0)( 4824)(4

)(4)(032023

1

12

2

≠++−

+′− x

fcx

fcx

f

j

t

tt

ext α

αα

β 19V

[ ] [ ] [ ][ ]

[ ] 0)()()( 2

22

032023

1

1

)(4

)(4)()()()(4

)()()()(

4824≠+

+′−+

′−′+++−

t

tttxtt

ttttx

fcx

fcx

f

j

txtet α

ααγαα

αγγα

γαβ 20V

[ ][ ] 0)(

2

2032023

1

1

)(4

)()(

824≠

′−++− x

t

ttfcx

fcx

f

j

et α

αα

β 21V

[ ][ ]

03

2

3

2)()(

2

2

)(4

)()(

121 ≠��

��

�+��

�

�+−

+′−

t

tt

jctjxttα

αα

αβ

22V

4/1

1312

2

13

2

3

2

3

2

2

1)()(

−

�

��

�

��

+��

�

��

�+��

�

�−+�

�

�

�− jctjxctjxtt αβ

( )

( )

[ ]

( )

0

3

4)(2

3

2)(

3

4)(2

3

2)( 1

223

2/3

3

4)(24

)()(

1

2

2321

1

2

2321

≠

��

�

�

��

�

�

+−+��

�

�+−−

+−+��

�

�+−

×

+−

′−

−

jtcc

tti

jtccictjxt

jtccictjxtα

αα

αα

αα

23V

159

Table 2c. A dimensional classification of the mathematical structure of groups (XXIV)-

(XXIX) (the enhanced symmetries of the thin film equation (5.1)) with their associated



( ) 03

22311 ≠+�

�

�

�+− fhctjxc 0

3

2

2

2

311 ≠�

�

�

�+− ctjx

c

24V

0)( ≠tα ( ) 0)(

3111

0 ≠++− jctjxtf

α 25V

( ) 0)( 2 ≠+ fhtα 0

11)(

31

0

0

21

0

00 ≠−

+��

�

�+

−− j

f

fctj

f

fxt

fα 26V

02

23

2

2

612

10

113

2

2

3

10

1

≠+

++��

�

�−++−

ft

ctgcxtff

jgcxcx

ff

j

0

2 2

7 ≠+

+

ft

cx

27V

( )( ) 023211 ≠+− −

fhececxcxc

06

1

3211

≠++ −

cc

ececxcxc

28V

( )( ) 014

1243

2

11 ≠+−−+

−fhecec

ct

xcxc

��

�

�+

+

+

−

7

1

43

2

11

4

1c

c

ecec

ct

xcxc

29V

Table 2c. Continued.

),,( htxη iV

03

2

2)(

4/12

311 ≠

��

�

��

��

�

�+−

−

ctjxc

tβ 24V

( )

[ ][ ]

0)(3

)(2

0

20

)(4

)()(3

1110 ≠�

�

��

�++−

+′−

tf

tft

jctjxtf

tα

αα

αβ

25V

[ ][ ]

011

)(3

)(

20

20

)(4

)()(3

1

0

021

0

00 ≠��

��

� −+��

�

�+

−−

+′−

tf

tft

jf

fctj

f

fxt

ft

α

αα

αβ

26V

( ) ( )[ ]

( )( )

( )0)( 2

12

8

32

7

48

36612362

2

1

1710211

112

71723102

171023

1

≠+−+

+��

�� −−−+++−

ftf

jcffcgf

xtjgcjcccffxjcffcxj

cxetβ

27V

0)( ≠tβ 28V

0)( ≠tβ 29V

160

Table 3. Row 1 features the similarity solution ),( txh for system (5.2) in association

with group (I). The remaining rows list the similarity solutions ),( txh for system (5.3) in

connection with groups (II)-(XXIX) where applicable.

Group ),( txh

(I) constant

(II) constant

(III) 022

1 ≠− fecc

x

under the constraint 02

1

11 ≠=

c

fg

(IV) 087611 ≠++ −

cececxcxc

under the constraint 01 =j

(V) 010987

11 ≠+++ −cxcecec

xcxc under the constraint 0

4

3

110

1 ≠=c

cffj

(VI) 0231 ≠−−

fecxc

(VII) 0641 ≠+ cecxc

under the constraints 0524 ==≠ ccc and ,2436 fccc −=

0241 ≠− fecxc under the constraints ,054 =≠ cc 23 cc = and ,26 fc −=

( ) 06

2

2411 ≠++ −

cececxcxc


,04 ≠c ,23 cc −= 2

245 ccc = and 2426 2 fccc −−=

(VIII) ( ) 0

62

3

1

10

1 ≠−+− fcxff

j

(IX), (X) 065411 ≠++ −

cececxcxc

(XIX) 0

26113

223

10

1 ≠+++− cxcxc

xff

j under the constraint 04 =c

(XX), (XXI) 0

26113

223

10

1 ≠+++− cxcxc

xff

j

(XXV) 0

6ln

1ln

3

1

1

0

11

0

≠++−f

j

fctjx

f

(XXVI)

( )( )0

1

23332

3

21

0

0

1

001

2

0100

≠−��

�

�+

−−

��

�

��

�

−−− fctj

f

fx

fff

fj ff

under the constraint 2

30 ≠f

(XXVII) 0

6181716

10

112

14

3

10

1 ≠++��

�

�+−++− ctcxct

ff

jgxcx

ff

j under the

constraint 7723

2

7

10

1

217

10

1

2622

3ccccc

ff

jfgc

ff

jcc ��

�

�

�−+−+��

�

�

�+=

(XXVIII) 098711 ≠++ −

cececxcxc

161

Table 3. Continued.

Group ),( txh

(XXIX) 012118

11 ≠++ −cecec

xcxc under constraints 03 ≠c and

( ),

4

1

3

2

71

4c

ccc

−=

0291 ≠−−

fecxc

under constraints 03 =c and ,171 =cc

01291111 ≠++ −

cececxcxc

under constraints 034 =≠ cc and ,171 −=cc

0281 ≠− fecxc

under constraints 171 −=cc and 043 == cc


Non-classical symmetry analysis of partitions (5.2) and (5.3) of thin film equation (5.1)

generated symmetries occurring under specific conditions on ,0),,( ≠htxη rendering

these symmetries hybrids of classical and non-classical symmetries. For system (5.2), we

derived one new symmetry group (namely group (I)) extending beyond the range of

groups obtainable via the non-classical treatment of the thin film equation (5.1). By a

similar process, twenty-eight new symmetry groups (namely groups (II)-(XXIX)) arose

for system (5.3), extending beyond the range of groups retrievable via the non-classical

approach to the thin film equation (5.1). Groups (I)-(XXIX) enhance the symmetries of

the thin film equation (5.1) and therefore we may consider twenty-nine symmetry-

enhancing constraints added to the latter equation.

Systems (5.2) and (5.3) are respectively identical to systems (4.2) and (4.3). A

comparison of symmetry group (I) obtained for system (5.2) with the classical Lie

groups (I)-(VI) arising for system (4.2) showed that in view of the forms of 0),,( ≠htxη

occurring in both sets of groups, neither set encompasses the other. Comparing

symmetry groups (II)-(XXIX) arising for system (5.3) with classical Lie group (VII)

derived for system (4.3) also indicates that neither set of groups includes the other as a

special case.

We derived similarity solutions for systems (5.2) and (5.3) in association with the

symmetry groups for each system. The constant solution is the only similarity solution

occurring for systems (5.2) and (5.3) (and hence for the thin film equation (5.1)) in

connection with the respective groups (I) and (II) and their associated conditions. The

reason for this is set out in the derivations of similarity solutions for systems (5.2) and

(5.3) in conjunction with groups (I) and (II) respectively.

162

Valid similarity solutions do not arise for system (5.3) in tandem with groups (XI)-

(XVIII) and (XXII)-(XXIV) under the condition 0)( 1 ≠= jhj since the contradiction

01 =j arises. Steady state solutions, travelling wave solutions and cubic solutions are

the only valid similarity solutions occurring for system (5.3) and the thin film equation

(5.1) in association with groups (III)-(X), (XIX)-(XXI) and (XXV)-(XXIX) and their

accompanying conditions. Clearly, a greater variety of similarity solutions for system

(5.3) is available via the non-classical approach than via the Lie classical analysis

applied to the equivalent system (4.3) in Chapter 4.

163

CHAPTER 6

CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE

THIN FILM EQUATION

INVOLVING ARBITRARY FUNCTIONS

6.1 INTRODUCTION

We use the method of classical symmetry-enhancing constraints to derive symmetry


[ ] ;0)()()( =++−∂


x (6.1)

where the restriction 0)( ≠hf applies.

The technique of classical symmetry-enhancing constraints, outlined in chapter 4 of this

thesis, is identical to the method of symmetry-enhancing constraints [29]. The method of

classical symmetry-enhancing constraints is thus termed to distinguish it from the

method of symmetry-enhancing constraints [29] which is augmented by the non-classical

procedure and features in chapter 5 of this thesis. A detailed account of the method of

symmetry-enhancing constraints occurs in [29].

Applying the method of classical symmetry-enhancing constraints, we consider the

enlarged system (4.2) arising from partitioning equation (6.1). System (4.2) engenders

new Lie symmetry groups. Here, augmenting system (4.2) with the arbitrary nontrivial

functions )(xa and )(tb generates the system

( ) ,0)()()()(2

≠=′−+ tbxahhghhjh xxt

(6.2)

[ ] ;0)()()()( ≠−=−∂

∂tbxahhghhf

xxxxxx

where )(xa and )(tb denote arbitrary nontrivial functions of x and t respectively.

164

Introducing nontrivial terms into an enlarged system of equations resulting from

partitioning a target equation admits more classical symmetries than the original target

equation possesses and is studied by Goard and Broadbridge [29].


,x t and ,h namely

( ) ( ),,, 2

1 εεξ Ohtxxx ++=

( ) ( ),,, 2

1 εεη Ohtxtt ++= (6.3)

( ) ( );,, 2

1 εεζ Ohtxhh ++=

which preserves system (6.2).

Thus if ),( txh φ= , then from ),( 111 txh φ= , evaluating the expansion of ε∂

∂ 1h at 0=ε

generates the invariant surface condition

).,,(),,(),,( htxt

hhtx

x

hhtx ζηξ =

∂

∂+

∂

∂ (6.4)

The solutions of equation (6.4) correspond to functional forms of the similarity solutions

for system (6.2).

In the following section, we use the Lie classical method to obtain the symmetry groups

leaving system (6.2) invariant.

6.2 CLASSICAL SYMMETRY-ENHANCING CONSTRAINTS

By the method of classical symmetry-enhancing constraints, system (6.2) remains

invariant under group transformations (6.3) provided the group generators ),,( htxξ ,

),,( htxη and ),,( htxζ satisfy the determining equations

165

,0=hξ ,0== xh ηη ,0=hhζ ( ) ,03)( =−′xxxxxhhf ξζ

[ ] ,0),()()()()()()()()()( =′−′−′−++ txtbxattbxattbxahj hxt ξηηζζζ

[ ] ,0)()(2)( =′′−−′−′ hgthg hx ζζηξ ,0)(

)(=�

�

��

� ′ζ

hf

hf

dh

d ( ) ,0)( =−′

xxxhhf ξζ

(6.5)

[ ] ,0)(2)()()( =′−−−′+′xtx hgthjhj ζξξηζ ,0

)(

)(64 =

′+− xxxxh

hf

hfζξζ

( ) ,02)(

)(

)(

)(4 =−+

′+− xhxxxxxxxxxxxxh

hf

hg

hf

hfζξζξζ

,0)(

)(2

)(

)(46 =−�

�

��

�−− xxxxxxh

hf

hg

hf

hg

dh

dξζξζ

.0),()()()()()()(

)(4)()()()( =′+′+�

�

��

� ′−−+− txtbxattbxa

hf

hftbxahghf hxxxxxxx ξηζζξζζ


terms not involving derivatives of h within the invariance requirements for system (6.2)

generates system (6.5). All subscripts in system (6.5) denote partial differentiation with

,x t and h regarded as independent variables. Primes throughout this chapter denote

differentiation with respect to the argument indicated.

System (6.5) enables the derivation of all the Lie classical symmetries and corresponding

conditions on ,0)( ≠hf )(hg and )(hj for system (6.2) under transformations (6.3).

System (6.2) admits seventeen new Lie classical groups extending beyond the confines

of groups derived via the classical method for the thin film equation (6.1). These new

groups enhance the symmetries of the thin film equation (6.1). One may therefore

consider symmetry-enhancing constraints to be added to thin film equation (6.1).

The following pages contain a description of each of these groups, a brief mention of the

special cases arising for the group in question and the derivation of similarity solutions

of system (6.2) pertaining to groups (III)-(XVII), where applicable. The solutions of

system (6.2) are also solutions to the thin film equation (6.1).

166

GROUP (I)

Subject to the conditions ,0)( 1 ≠= fhf 0)( =hg and ,)( 1jhj = system (6.2) admits

classical group (I) given by

( ) ( ) ,)(),,( 4113

2

15 atjtjxatjxahtx ++−+−= αξ ),(),,( thtx αη =

(6.6)

( )[ ] ;),(3),,( 215 txdhatjxahtx ++−=ζ

such that ),( txd satisfies the equations

( )[ ] ,48)()()( 31511 atjxattbxaddjdf txxxxx −−−′=++ α

(6.7)

( ) ( )[ ]4113

2

151 )()()()()()( atjtjxatjxatbxattbxaddj tx ++−+−′+′=+ αα

( )[ ] ;3)()()( 215 atjxattbxa −−−′+ α

where ,01 ≠f ,2a ,3a ,4a 5a and 1j are arbitrary constants while ,0)( ≠xa 0)( ≠tb

and )(tα are arbitrary functions of their respective arguments.

GROUP (II)

Under the conditions ,0)( 1 ≠= fhf 1)( ghg = and ,)( 1jhj = system (6.2) admits

classical group (II) given by

,)(),,( 41 atjhtx += αξ ),(),,( thtx αη = ;),(),,( 2 txdhahtx +=ζ (6.8)

such that ),( txd satisfies the equations

),()()(111 ttbxaddjdgdf txxxxxxx α ′=++−

(6.9)

[ ] [ ] ;)()()()()()()()()( 2411 attbxaatjtbxattbxaddj tx −′++′+′=+ ααα

where 01 ≠f , 1g , 1j , 2a and 4a are arbitrary constants while ,0)( ≠xa 0)( ≠tb and

)(tα are arbitrary functions of their respective arguments.

167

GROUP (III)

Subject to the conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( =hj ( ) 0)( 37 ≠+= exexa and

,0)(exp)( 21 ≠��

�

��

�= �

t

dsseetb γ system (6.2) admits classical group (III) given by

( ),),,( 35 exehtx +=ξ ,0)(

1),,( ≠=

thtx

γη

(6.10)

( ) ( ) ;0)(exp4)(

)(5),,( 113102527152 ≠++

�

��

��

� �

+��

�

��

��

��

�+

′−++= � � eexedsdrree

s

seeheehtx

t s

γγ

γζ

where ,01 ≠f ,01 ≠e ,07 ≠e ,2e ,3e ,5e 10e and 11e are arbitrary constants while

0)( ≠tγ is an arbitrary function of t with .0)(4)( 2

5 ≠+′ tet γγ



( ) ,0)(exp 2371 ≠��

�

��

�+= �

t

t dsseexeeh γ ( ) ,0)(exp 23711 ≠��

�

��

�+−= �

t

xxxx dsseexeehf γ

(6.11)

( ) ( ) ×�

��

��

� �

+��

�

��

��

��

�+

′−++=++ � �

t s

tx edsdrrees

seeheeh

thexe 10252715235 )(exp4

)(

)(5

)(

1γ

γ

γ

γ

( ) ;0113 ≠++ eex

where ,01 ≠f ,01 ≠e ,07 ≠e ,2e ,3e ,5e 10e and 11e are arbitrary constants while

0)( ≠tγ is an arbitrary function of t with .0)(4)( 2

5 ≠+′ tet γγ

Directly solving equation (6.11)2 and substituting its general solution into equations

(6.11)1 and (6.11)3 gives rise to the constraints .0111052 ==== eeee

Hence under transformations (6.3) and conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( =hj

( ) 0)( 37 ≠+= exexa and 0)( 1 ≠= etb with constraints ,0111052 ==== eeee the

similarity solution for system (6.2) and thus for the thin film equation (6.1) in

conjunction with group (6.10) is

( ) ;0),( 15181471

2

13

3

12

4

17

5

16 ≠+++++++= etexeteexexexexetxh (6.12)

168

where ,01 ≠e ,07 ≠e ,0120 1

71

16 ≠−=f

eee ,01 ≠f ,3e ,12e ,13e ,14e ,15e

1

731

1724 f

eeee −=

and 73118 eeee = are arbitrary constants while .03 ≠+ ex

GROUP (IV)

With respect to the conditions ,0)( 1 ≠= fhf ,)( 1ghg = ,0)( 10 ≠+= jhjhj

0)( 8 ≠= axa and ,0)(2

1 ≠+

=at

atb system (6.2) admits classical group (IV) given by

( ) ( ) ( )[ ] ,0ln),,( 40517323273 ≠++−++−+−= atjajaaataataahtxξ

(6.13)

( ) ,0),,( 23 ≠+−= atahtxη ( ) ;0ln),,( 523103 ≠++−+= aataahahtxζ

where ,01 ≠f ,00 ≠j ,01 ≠a ,03 ≠a ,07 ≠a ,08 ≠a ,010 ≠a ,2a ,4a ,5a 1g and 1j

are arbitrary constants with .02 ≠+ at



( ) ,02

81

10 ≠+

=++at

aahjhjh xt ,0

2

81

11 ≠+

−=−at

aahghf xxxxxx

(6.14)

( ) ( ) ( )[ ]{ } ( ) =+−++−++−+− tx hatahatjajaaataataa 2340517323273 ln

( ) ;0ln 523103 ≠++−+ aataaha

where ,01 ≠f ,00 ≠j ,01 ≠a ,03 ≠a ,07 ≠a ,08 ≠a ,010 ≠a ,2a ,4a ,5a 1g and 1j

are arbitrary constants and .02 ≠+ at

Consistency in equation (6.14)2 requires 0≠xxh since the contradiction 081 =aa

otherwise arises.

Solving equation (6.14)2 by the methods of Lagrange [24] and differential operators

gives rise to the cases to be considered, namely (1) ,01 ≠g (2) .01 =g

Substituting the general solution of equation (6.14)2 for these cases into equation 6.14)1

leads to the contradiction .0081 =jaa Hence under transformations (6.3), there is no

valid similarity solution for system (6.2) and thus for the thin film equation (6.1) in

conjunction with group (6.13) and its associated conditions.

169

GROUP (V)

Under the conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( 10 ≠+= jhjhj 0)( 6 ≠= axa and

( ) ,042

42)(

2

119

2

10109

119

2

101092/5

1110

2

91 ≠��

�

�

��

�

�

−+−−

−++++=

−

a

aaaata

aaaataatataatb system (6.2) admits

group (V) given by

( ) ( ) ,02

)(2),,( 416

21912

0

1914919059 ≠+++

��

�

��

�+−+++= � � atat

jadsas

j

jadrrbaraajxatahtx

t s

ξ

,0),,( 1110

2

9 ≠++= atatahtxη (6.15)

( ) ( ) ( ) ;)(2),,( 12149191

0

9

917 adssbasaatjxj

ahtaahtx

t

+++−+−= �ζ

where 10a and 11a are constants such that the only cases applying are

(i) 010 ≠a and 11a are arbitrary constants, (ii) 01011 =≠ aa .

Furthermore, ,01 ≠a ,06 ≠a ,09 ≠a ,019 ≠a ,01 ≠f ,00 ≠j ,2a ,4a ,5a ,12a ,14a

,16a 17a and 1j are arbitrary constants such that ,059 ≠+ ata 01110

2

9 ≠++ atata and

( ) .024 109119

2

10 ≠+±− ataaaa



( ) ( ) ,042

422

119

2

10109

119

2

101092/5

1110

2

96110 ≠��

�

�

��

�

�

−+−−

−++++=++

−

a

xt

aaaata

aaaataatataaahjhjh

( ) ,042

422

119

2

10109

119

2

101092/5

1110

2

9611 ≠��

�

�

��

�

�

−+−−

−++++−=

−

a

xxxx

aaaata

aaaataatataaahf (6.16)

( ) ( ) +�

��

��

� �

+++��

�

��

�+−+++ � � x

t s

hatatja

dsasj

jadrrbaraajxata 416

219

12

0

19

149190592

)(2

( ) ( ) ( ) ( ) ;)(2 12149191

0

99171110

2

9 adssbasaatjxj

ahtaahatata

t

t +++−+−=++ �

where 10a and 11a are constants such that the only cases applying are

(i) 010 ≠a and 11a are arbitrary constants, (ii) 01011 =≠ aa .

170

In addition, ,01 ≠a ,06 ≠a ,09 ≠a ,019 ≠a ,01 ≠f ,00 ≠j ,2a ,4a ,5a ,12a ,14a

,16a 17a and 1j are arbitrary constants such that ,059 ≠+ ata 01110

2

9 ≠++ atata and

( ) .024 109119

2

10 ≠+±− ataaaa


(6.16)1 results in the contradiction .0)(06 =tbja Hence under transformations (6.3), there

is no valid similarity solution for system (6.2) and thus for thin film equation (6.1) in

connection with group (6.15) and its associated conditions.

GROUP (VI)


,0exp)( 25

1 ≠��

��

�= −

t

atatb system (6.2) admits group (VI) given by

( ) ,0exp),,( 13211

1084

2

359 ≠+��

��

��

��

�++++++= a

t

a

t

aatatataxatahtxξ

,0),,( 2

9 ≠= tahtxη (6.17)

( ) ( ) ;exp),,( 1223

14

2

1516

191

0

9917 a

t

atata

t

aatjx

j

ahtaahtx +�

�

��

��

��

�++++−+−= −−ζ

where ,01 ≠a ,02 ≠a ,06 ≠a ,09 ≠a ,01 ≠f ,00 ≠j ia and 1j are arbitrary constants

for all i ∈ }19,17,16,15,14,13,12,11,10,8,5,4,3{ while ( ) .059

2 ≠+ atat



( ) ,0exp 25

6110 ≠��

��

�=++ −

t

ataahjhjh xt ,0exp 25

611 ≠��

��

�−= −

t

ataahf xxxx

(6.18)

( )tx htaha

t

a

t

aatatataxata

2

913211

1084

2

359 exp +�

� �

+��

��

��

��

�++++++

( ) ( ) ;exp 1223

14

2

1516

191

0

9917 a

t

atata

t

aatjx

j

ahtaa +�

�

��

��

��

�++++−+−= −−


for all i ∈ }19,17,16,15,14,13,12,11,10,8,5,4,3{ while ( ) .059

2 ≠+ atat

171


(6.18)1 generates the contradiction .0)(06 =tbja Hence under transformations (6.3), no

valid similarity solution exists for system (6.2) and thus for thin film equation (6.1) in

connection with group (6.17) and its associated conditions.

GROUP (VII)

Subject to the conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( 10 ≠+= jhjhj 0)( 6 ≠= axa

and ,0)( 5

1 ≠= −tatb system (6.2) admits group (VII) given by

( ) ,0),,( 3

4

2

8103

2

259 ≠++++++= −−tataatataxatahtxξ ,0),,( 2

9 ≠= tahtxη

(6.19)

( ) ( ) ;),,( 4

13

3

11121

0

9

914

−− +++−+−= tataatjxj

ahtaahtxζ

where ,01 ≠a ,06 ≠a ,08 ≠a ,09 ≠a ,011 ≠a ,01 ≠f ,00 ≠j ia and 1j are arbitrary

constants for all i ∈ }14,13,12,10,5,4,3,2{ while ( ) .059

2 ≠+ atat



( ) ,05

6110 ≠=++ −taahjhjh xt ,05

611 ≠−= −taahf xxxx

(6.20)

( )[ ]tx htahtataatataxata

2

9

3

4

2

8103

2

259 +++++++ −−

( ) ( ) ;4

13

3

11121

0

9914

−− +++−+−= tataatjxj

ahtaa


constants for all i ∈ }14,13,12,10,5,4,3,2{ while ( ) .059

2 ≠+ atat


(6.20)1 gives rise to the contradiction .0061 =jaa Therefore under transformations (6.3),

there is no valid similarity solution for system (6.2) and thus for thin film equation (6.1)

in tandem with group (6.19) and its accompanying conditions.

172

GROUP (VIII)


( ) ,0)( 2

11101 ≠+=a

ataatb system (6.2) admits group (VIII) given by

( ) ,0),,( 7

2

11104352 ≠++++=

+aataataxahtx

aξ

,0),,( 1110 ≠+= atahtxη (6.21)

( ) ;),,( 12

1

1110892 aataahahtx

a+++=

+ζ

where ,01 ≠a ,05 ≠a ,06 ≠a ,010 ≠a ,01 ≠f ,00 ≠j ,1,22 −−≠a ia and 1j are

arbitrary constants for all i ∈ }12,11,9,8,7,4,3{ while .01110 ≠+ ata



( ) ( ) ,02

11106110 ≠+=++a

xt ataaahjhjh ( ) ,02

1110611 ≠+−=a

xxxx ataaahf

(6.22)

( )[ ] ( ) tx

ahatahaataataxa 11107

2

11104352 ++++++

+ ( ) ;12

1

1110892 aataaha

a+++=

+

where ,01 ≠a ,05 ≠a ,06 ≠a ,010 ≠a ,01 ≠f ,00 ≠j ,1,22 −−≠a ia and 1j are

arbitrary constants for all i ∈ }12,11,9,8,7,4,3{ while .01110 ≠+ ata


(6.22)1 gives rise to the contradiction .0061 =jaa Therefore under transformations (6.3),

there is no valid similarity solution for system (6.2) and thus for thin film equation (6.1)

in association with group (6.21) and its accompanying conditions.

173

GROUP (IX)

Subject to the conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( 10 ≠+= jhjhj 0)( 6 ≠= axa

and ( ) ,0)(2

11101 ≠+=−

ataatb system (6.2) admits group (IX) given by

,0ln),,( 41110325 ≠++++= aataataxahtxξ

(6.23)

,0),,( 1110 ≠+= atahtxη ;),,( 7

1110

8

9 aata

ahahtx +

++=ζ

where ,01 ≠a ,05 ≠a ,06 ≠a ,010 ≠a ,01 ≠f ,00 ≠j ia and 1j denote arbitrary

constants for all i ∈ }11,9,8,7,4,3,2{ with .01110 ≠+ ata


Group (6.23), system (6.2) and the invariant surface condition (6.4) yield the relations

( ) ( ) ,02

11106110 ≠+=++−

ataaahjhjh xt ( ) ,02

1110611 ≠+−=−

ataaahf xxxx

(6.24)

[ ] ( )tx hatahaataataxa 111041110325 ln ++++++ ;7

1110

8

9 aata

aha +

++=

where ,01 ≠a ,05 ≠a ,06 ≠a ,010 ≠a ,01 ≠f ,00 ≠j ia and 1j are arbitrary

constants for all i ∈ }11,9,8,7,4,3,2{ while .01110 ≠+ ata


(6.24)1 forces the contradiction .0061 =jaa Consequently under transformations (6.3),

no valid similarity solution exists for system (6.2) and thus for the thin film equation

(6.1) in conjunction with group (6.23) and its associated conditions.

174

GROUP (X)


,0)(1110

1 ≠+

=ata

atb system (6.2) admits group (X) given by

( ) ,0ln),,( 411101110325 ≠+++++= aataataataxahtxξ

(6.25)

,0),,( 1110 ≠+= atahtxη ;ln),,( 7111089 aataahahtx +++=ζ


constants for all i ∈ }11,9,8,7,4,3,2{ with .01110 ≠+ ata


Group (6.25), system (6.2) and the invariant surface condition (6.4) generate

( ) ,01110

61

10 ≠+

=++ata

aahjhjh xt ,0

1110

61

1 ≠+

−=ata

aahf xxxx

(6.26)

( )[ ] ( )tx hatahaataataataxa 1110411101110325 ln +++++++ ;ln 7111089 aataaha +++=


constants for all i ∈ }11,9,8,7,4,3,2{ while .01110 ≠+ ata


(6.26)1 gives rise to the contradiction .0061 =jaa Hence under transformations (6.3), no

valid similarity solution exists for system (6.2) and thus for the thin film equation (6.1)

in connection with group (6.25) and its associated conditions.

175

GROUP (XI)

With respect to the conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( 10 ≠+= jhjhj

0)( 6 ≠= axa and ,0)( 2

1 ≠= taeatb system (6.2) admits group (XI) given by

,0),,( 74352 ≠+++= aeataxahtxtaξ

(6.27)

,0),,( 11 ≠= ahtxη ;),,( 89102 aeahahtxta ++=ζ


constants for all i ∈ }.10,9,8,7,4,3{


Group (6.27), system (6.2) and the invariant surface condition (6.4) generate

( ) ,02

6110 ≠=++ ta

xt eaahjhjh ,02

611 ≠−= ta

xxxx eaahf

(6.28)

( )tx

tahahaeataxa 117435

2 ++++ ;89102 aeahata ++=


constants for all i ∈ }.10,9,8,7,4,3{


(6.28)1 yields the contradiction .0061 =jaa Hence under transformations (6.3), no valid

similarity solution exists for system (6.2) and thus for the thin film equation (6.1) in

conjunction with group (6.27) and its accompanying conditions.

176

GROUP (XII)


,0)( 1 ≠= atb system (6.2) admits group (XII) given by

,0),,( 43

2

25 ≠+++= atataxahtxξ

(6.29)

,0),,( 7 ≠= ahtxη ;),,( 8910 atahahtx ++=ζ


for all i ∈ }.10,9,8,4,3,2{



( ) ,06110 ≠=++ aahjhjh xt ,0611 ≠−= aahf xxxx

(6.30)

( ) tx hahatataxa 743

2

25 ++++ ;8910 ataha ++=


for all i ∈ }.10,9,8,4,3,2{


(6.30)1 leads to the contradiction .0061 =jaa Therefore under transformations (6.3),

there is no valid similarity solution for system (6.2) and thus for the thin film equation

(6.1) in tandem with group (6.29) and its associated conditions.

177

GROUP (XIII)

Under conditions ,0)( 1 ≠= fhf ,0)( =hg ,0)( 10 ≠+= jhjhj ( ) 0)( 17 ≠+= axaxa

and [ ]

[ ] ,0)(2exp)(

)(1

33

2 ≠��

�

��

�−= �

−t

dssat

atb γ

γ system (6.2) admits classical group (XIII)

given by

( ) ,02

)(),,( 1

3 ≠++′

= axat

htxγ

ξ ,0)(),,( ≠= thtx γη

(6.31)

++′−′′

+′−

= 10

0

1

0

3 )(22

)(

2

)(),,( at

j

jx

j

th

tahtx γ

γγζ

[ ] [ ] [ ] ;)(2exp2)()(1

33

3

6 � ��

�

��

�−+′ −−

t s

dsdrraassa γγγ

such that 0)( ≠tγ satisfies the equation

[ ]

[ ];0)(

2)()(

4)()()(8)(3)(

3

2

33

2

)4( =′′′+′

+′′−′+′+ t

att

atttatt γ

γγ

γγγγγ (6.32)

where ,02 ≠a ,07 ≠a ,01 ≠f ,00 ≠j ,1a ,3a ,6a 10a and 1j are arbitrary constants

while [ ][ ] .0)(2)( 33 ≠+′+′ atat γγ



( )[ ]

[ ] ,0)(2exp)(

1

33

1

7210 ≠��

�

��

�−

+=++ �

−t

xt dssat

axaahjhjh γ

γ

[ ]

[ ] ,0)(2exp)(

1

33

1

721 ≠��

�

��

�−

+−= �

−t

xxxx dssat

axaahf γ

γ (6.33)

( ) ++′−′′

+′−

=+++′

10

0

1

0

3

1

3 )(22

)(

2

)()(

2

)(at

j

jx

j

th

tahthax

attx γ

γγγ

γ

[ ] [ ] [ ] ;)(2exp2)()(1

33

3

6 � ��

�

��

�−+′ −−

t s

dsdrraassa γγγ

such that 0)( ≠tγ satisfies the equation

[ ]

[ ];0)(

2)()(

4)()()(8)(3)(

3

2

33

2

)4( =′′′+′

+′′−′+′+ t

att

atttatt γ

γγ

γγγγγ (6.34)

178

where ,02 ≠a ,07 ≠a ,01 ≠f ,00 ≠j ,1a ,3a ,6a 10a and 1j are arbitrary constants

while [ ][ ] .0)(2)( 33 ≠+′+′ atat γγ


(6.33)1 leads to the contradiction .0)(07 =tbja Hence under transformations (6.3), there

is no valid similarity solution for system (6.2) and thus for the thin film equation (6.1) in

conjunction with group (6.31) and its associated conditions.

GROUP (XIV)

Subject to the conditions ,0)( 1 ≠= fhf ,0)( =hg ,)( 10 jhjhj += 0)( 4 ≠= exa and

( ) ,0)(4

251 ≠+=−

eteetb system (6.2) admits classical group (XIV) given by

( ),),,( 62

25

3

5 eete

exehtx +

++=ξ ,0),,( 25 ≠+= etehtxη

( );0),,(

3

25

7 ≠+

=ete

ehtxζ

(6.35)

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,01 ≠f ,2e ,3e ,6e 0j and 1j are arbitrary

constants such that .025 ≠+ ete



( ) ( ) ,04

254110 ≠+=++−

eteeehjhjh xt ( ) ,04

25411 ≠+−=−

eteeehf xxxx

(6.36)

( )

( ) tx heteheete

exe 2562

25

3

5 ++��

�

��

�+

++

( );0

3

25

7 ≠+

=ete

e

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,01 ≠f ,2e ,3e ,6e 0j and 1j are arbitrary

constants with .025 ≠+ ete


(6.36)1 forces ,00 =j leading to the contradiction .0541 =eee Therefore under

transformations (6.3), no valid similarity solution exists for system (6.2) and thus for the

thin film equation (6.1) in connection with group (6.35) and its accompanying

conditions.

179

GROUP (XV)


1 ≠= hfefhf ,0)( =hg ,)( 10 jhjhj += 0)( 4 ≠= exa and

,0)(60

1 ≠+

=etf

etb system (6.2) admits classical group (XV) given by

,ln),,( 560

0

02 eetff

jehtx ++−=ξ ,0),,( 2 ≠= ehtxη ;0),,(

60

2 ≠+

−=etf

ehtxζ

(6.37)

where ,01 ≠e ,02 ≠e ,04 ≠e ,00 ≠f ,01 ≠f ,5e ,6e 0j and 1j denote arbitrary

constants such that .060 ≠+ etf



( ) ,060

41

10 ≠+

=++etf

eehjhjh xt ( ) ,0

60

41

010 ≠

+−=+

etf

eehhfhef xxxxxxxx

hf

(6.38)

;0ln60

2

2560

0

02 ≠+

−=+��

��

�++−

etf

eheheetf

f

jetx

where ,01 ≠e ,02 ≠e ,04 ≠e ,00 ≠f ,01 ≠f ,5e ,6e 0j and 1j are arbitrary constants

with .060 ≠+ etf

Consistency of equation (6.38)2 requires 0≠xxxxxx hhh as the contradiction 041 =ee

otherwise occurs. Furthermore if =),( txh constant, the contradictions 041 =ee and

02 =e arise in equations (6.38)1 and (6.38)3 respectively.

Via the method of Lagrange [24], we deduce the general solution of equation (6.38)3 as

;0ln1

)),((),( 60

0

≠+−= etff

txuytxh (6.39)

where ,02 ≠e ,00 ≠f ,5e 6e and 0j are arbitrary constants with 060 ≠+ etf and

0ln2

5

60

0

0 ≠−+e

eetf

f

j while ( )( ).1ln),( 60602

0

0

2

5 −+++−= etfetff

jt

e

extxu

Furthermore, 0)( ≠uy is an arbitrary function of u such that 0)()()( ≠′′′′′′ uyuyuy as

0≠xxxxxx hhh is a requirement for equation (6.38)2 to be consistent.

180

Substituting result (6.39) into equation (6.38)1 generates the relation

;1

)()(60

41

2

5

10etf

eeuy

e

ejuyj

+

+=′�

�

��

�−+ (6.40)

where ,01 ≠e ,02 ≠e ,04 ≠e ,00 ≠f ,5e ,6e 0j and 1j are arbitrary constants with

,060 ≠+ etf ,0ln2

5

60

0

0 ≠−+e

eetf

f

j ( )( )1ln),( 60602

0

0

2


jt

e

extxu

and .0)()()( ≠′′′′′′ uyuyuy

As ,0)( ≠′ uy equation (6.40) gives rise to the following cases for consideration, namely

(1) ,0141 ≠+ee ,0)(2

5

10 ≠−+e

ejuyj (2) ,141 −=ee .)( 1

2

5

0 je

euyj −=

Case (1) ,0141 ≠+ee 0)(2

5

10 ≠−+e

ejuyj

Differentiating equation (6.40) once with respect to x (forcing 00 ≠j ) and integrating

the result once with respect to u yields the equation

;0)()( 11

2

5

10 ≠=′��

��

�−+ euy

e

ejuyj (6.41)

where ,02 ≠e ,011 ≠e ,00 ≠f ,00 ≠j ,5e 6e and 1j are arbitrary constants with

060 ≠+ etf and .0ln2

5

60

0

0 ≠−+e

eetf

f

j In addition,

( )( )1ln),( 60602

0

0

2


jt

e

extxu and 0)( ≠uy is such that

.0)()()( ≠′′′′′′ uyuyuy

Via result (6.41), equation (6.40) forces the contradiction 00 =f , rendering case (1)

invalid.

181

Case (2) ,141 −=ee 1

2

5

0 )( je

euyj −=

It follows that 0)(0 =′ uyj and since ,0)( ≠′ uy we obtain

,141 −=ee ,00 =j ;02

5

1 ≠=e

ej (6.42)

where 02 ≠e and 05 ≠e are arbitrary constants.

Equations (6.38)2 and (6.40) consequently give

,0)()()(1

)(

0

)4(0

≠=′′′′+−

f

euyuyfuy

uyf

,060 >+ etf

(6.43)

,0)()()(1

)(

0

)4(0

≠−=′′′′+−

f

euyuyfuy

uyf

;060 <+ etf

where ,02 ≠e ,05 ≠e ,00 ≠f 01 ≠f and 6e denote arbitrary constants such that

060 ≠+ etf while te

extxu

2

5),( −= and 0)()()( ≠′′′′′′ uyuyuy is a requirement for

equation (6.38)2 to be consistent.


1 ≠= hfefhf ,0)( =hg

,0)(2

5 ≠=e

ehj 0

1)(

1

≠−=e

xa and ,0)(60

1 ≠+

=etf

etb the similarity solution for

system (6.2) and thus for the thin film equation (6.1) in connection with group (6.37)

subject to the constraints ,141 −=ee 00 =j and 02

5

1 ≠=e

ej is

;0ln1

)),((),( 60

0

≠+−= etff

txuytxh (6.44)

such that 0)( ≠uy satisfies equations (6.43) where ,00 ≠f ,01 ≠f ,01 ≠e ,02 ≠e

05 ≠e and 6e are arbitrary constants with 060 ≠+ etf while te

extxu

2

5),( −= and

.0)()()( ≠′′′′′′ uyuyuy

182

GROUP (XVI)

With respect to the conditions ,0)( 0

1 ≠= hfefhf ,0)( =hg ,)( 10 jhjhj +=

0)( 4 ≠= exa and

( ) ( )[ ],0

8

)(

0

3

25

3

725

3

71 ≠+++

=feteeete

eetb system (6.2) admits

classical group (XVI) given by

( ) ( )( ) ( )

++++−

−+++=

−

3/1

0257

3/1

0257

7

3/2

00

5

13

13ln3),,(

fietee

fietee

e

fjixehtxξ

( )

( ) ( ) 3/2

025

3/1

07

2

257

2

3/1

025

7

7

3/2

00

22

2ln

fetefe

etee

fetee

e

fj

++−��

��

�+

��

��

�++−

,12e+

(6.45)

,0),,( 25 ≠+= etehtxη ( )

;08

24),,(

0

3

25

3

7

5 ≠++

=fetee

ehtxζ

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,00 ≠f ,01 ≠f ,2e ,12e 0j and 1j are arbitrary

constants while ,025 ≠+ ete ( ) ,08 0

3

25

3

7 ≠++ fetee ( ) ( ) ,0133/1

0257 ≠++± fietee �

( ) ( ) 022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e and ( ) .0

2

3/1

0257 ≠++ fete

e

183



( )( ) ( )[ ]

,0

8 0

3

25

3

725

3

74110 ≠

+++=++

feteeete

eeehjhjh xt

( )( ) ( )[ ]

,0

8 0

3

25

3

725

3

74101

0 ≠+++

−=+feteeete

eeehhfhef xxxxxxxx

hf

(6.46)

( ) ( )( ) ( )

( )

( ) ( )

xh

e

fetefe

etee

fetee

e

fj

fietee

fietee

e

fjixe

��

��

�

��

�

��

�

+

++−��

��

�+

��

��

�++

++++−

−+++

−

−

12

3/2

025

3/1

07

2

257

2

3/1

025

7

7

3/2

00

3/1

0257

3/1

0257

7

3/2

00

5

22

2ln

13

13ln3

( ) =++ thete 25

( )

;08

24

0

3

25

3

7

5 ≠++ fetee

e

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,00 ≠f ,01 ≠f ,2e ,12e 0j and 1j are arbitrary

constants while ,025 ≠+ ete ( ) ,08 0

3

25

3

7 ≠++ fetee ( ) ( ) ,0133/1

0257 ≠++± fietee �

( ) ( ) 022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e and ( ) .0

2

3/1

0257 ≠++ fete

e

Consistency in equation (6.46)2 requires 0≠xxxxxx hhh as the contradiction 041 =ee

occurs otherwise. Furthermore if =),( txh constant, the contradictions 041 =ee and

05 =e arise in equations (6.46)1 and (6.46)3 respectively.

By the method of Lagrange [24], we obtain the general solution of equation (6.46)3 as

( ) ;08

ln3

1ln

3)),((),(

3

7

03

2525

0

≠��

�

��

�++−++=

e

feteete

ftxuytxh (6.47)

184

where ,05 ≠e ,07 ≠e ,00 ≠f ,2e 12e and 0j are arbitrary constants with ,025 ≠+ ete

( ) ,08 0

3

25

3

7 ≠++ fetee ( ) ( ) ,0133/1

0257 ≠++± fietee � ( ) ,02

3/1

0257 ≠++ fete

e

( ) ( ) 022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e and

( ) ( )( ) ( )

++++−

−+++

−

3/1

0257

3/1

0257

7

3/2

005

13

13ln3

fietee

fietee

e

fjixe

( )

( ) ( )0

22

2ln 12

3/2

025

3/1

07

2

257

2

3/1

025

7

7

3/2

00 ≠+

++−��

��

�+

��

��

�++−

e

fetefe

etee

fetee

e

fj

while

( )( ) ( ) ++++−

+++−

+

+=

3/1

0257

05

0

25

05

0

255

125 13ln13

3ln

2

3),( fietee

fe

j

i

iete

fe

j

etee

exetxu

( ) ( ) +−++−

3/1

0257

05

0 13ln13

3fietee

fe

j

i

i

( )( ) ( )( ) ( ) 3/1

0257

3/1

0257

2575

3/2

00

13

13ln

3

fietee

fietee

eteee

fji

+++−

−++

+

−

+( )

−��

�

��

� −+−

3/1

0

3/1

02571

05

0

3tan

2

3

f

fetee

fe

j

( ) +��

�

��

�++−+

3/1

025

7

25

05

0

2ln2ln3

2fete

eete

fe

j

( )

( )

( ) ( )+

++−��

��

�+

��

��

�++

+

−

3/2

025

3/1

07

2

25

7

2

3/1

025

7

2575

3/2

00

22

2ln

fetefe

etee

fetee

eteee

fj

( ) ( ) .22

ln4

3/2

025

3/1

07

2

25

7

05

0 fetefe

etee

fe

j++−�

�

��

�+ (6.48)

Furthermore, 0)( ≠uy is an arbitrary function of u such that 0)()()( ≠′′′′′′ uyuyuy as

0≠xxxxxx hhh is a requirement for equation (6.46)2 to be consistent.

Substituting result (6.47) into equations (6.46)1 and (6.46)2 gives the relations

185

[ ]

[ ]),,,()(

),,(

),,(

8),,(

),,(ln

1)( 261

3

7

03

3

0

0 htxeuyjhtx

htx

e

fhtx

htx

fuyj ζ

η

ξ

η

η=′

��

��

�

��

�

��

�

+−

��

�

�

��

�

�

+

+

,0)()()()(

1

410

)4( 0 ≠−=′′′′+ − uyfe

f

eeuyuyfuy

[ ]

[ ],0

8),,(

),,(

3

7

03

3

>

+e

fhtx

htx

η

η (6.49)

,0)()()()(

1

410

)4( 0 ≠=′′′′+ − uyfe

f

eeuyuyfuy

[ ]

[ ];0

8),,(

),,(

3

7

03

3

<

+e

fhtx

htx

η

η

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,00 ≠f ,01 ≠f ,2e ,12e ,24

24

5

5

3

74126

e

eeeee

−= 0j

and 1j are arbitrary constants with ,025 ≠+ ete ( ) ,08 0

3

25

3

7 ≠++ fetee

( ) ( ) ,0133/1

0257 ≠++± fietee � ( ) ( ) ,022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e

( ) 02

3/1

0257 ≠++ fete

e and

( ) ( )( ) ( )

++++−

−+++

−

3/1

0257

3/1

0257

7

3/2

005

13

13ln3

fietee

fietee

e

fjixe

( )

( ) ( ).0

22

2ln 12

3/2

025

3/1

07

2

257

2

3/1

025

7

7

3/2

00 ≠+

++−��

��

�+

��

��

�++−

e

fetefe

etee

fetee

e

fj

Furthermore, ),,,( htxξ ),,( htxη and ),,( htxζ are given by results (6.45), ),( txu is

defined by expression (6.48) and 0)( ≠uy is such that .0)()()( ≠′′′′′′ uyuyuy

As ,0)(),,( ≠′ uyhtxζ equation (6.49)1 admits the following two cases, namely

(1) ,026 =e [ ]

[ ],

),,(

),,(

8),,(

),,(ln

1)( 1

3

7

03

3

0

0 jhtx

htx

e

fhtx

htx

fuyj −=

��

��

�

��

�

��

�

+

+η

ξ

η

η

(2) ,026 ≠e [ ]

[ ].0

),,(

),,(

8),,(

),,(ln

1)( 1

3

7

03

3

0

0 ≠+−

��

��

�

��

�

��

�

+

+ jhtx

htx

e

fhtx

htx

fuyj

η

ξ

η

η

186

Case (1) ,026 =e [ ]

[ ]1

3

7

03

3

0

0),,(

),,(

8),,(

),,(ln

1)( j

htx

htx

e

fhtx

htx

fuyj −=

��

��

�

��

�

��

�

+

+η

ξ

η

η

Differentiating [ ]

[ ]1

3

7

03

3

0

0),,(

),,(

8),,(

),,(ln

1)( j

htx

htx

e

fhtx

htx

fuyj −=

��

��

�

��

�

��

�

+

+η

ξ

η

η once with respect to

x forces 00 ≠j and leads to the contradiction ,0)( =′′ uy rendering case (1) invalid.

Case (2) ,026 ≠e [ ]

[ ]0

),,(

),,(

8),,(

),,(ln

1)( 1

3

7

03

3

0

0 ≠+−

��

��

�

��

�

��

�

+

+ jhtx

htx

e

fhtx

htx

fuyj

η

ξ

η

η

We consider the subcases (a) ,00 =j (b) .00 ≠j

Subcase (a) ,0026 =≠ je 0),,(

),,(1 ≠−

htx

htxj

η

ξ

Differentiating equation (6.49)1 with respect to x gives =′′

′

)(

)(5

uy

uye ,0

),,(

),,(1 ≠−

htx

htxj

η

ξ

which result we differentiate once with respect to ,x obtaining the equation

;1)(

)(−=�

�

��

�

′′

′

uy

uy

du

d (6.50)

where ,05 ≠e 2e and 12e are arbitrary constants with 025 ≠+ ete and 0125 ≠+ exe

while ( )

0),(255

125 ≠+

+=

etee

exetxu and 0)( ≠uy is such that .0)()()( ≠′′′′′′ uyuyuy

Directly solving equation (6.50) yields the general solution

;0ln)( 292728 ≠++−= eeueuy (6.51)

where ,05 ≠e 028 ≠e and ,2e ,12e 27e and 29e are arbitrary constants while

,025 ≠+ ete ,0125 ≠+ exe 027 ≠+− eu and ( )

.0),(255

125 ≠+

+=

etee

exetxu

Substituting solution (6.51) into equations (6.49)2 and (6.49)3 forces 04

0

28 ≠=f

e and

leads to the contradiction ,0290 =efe rendering subcase (a) invalid.

187

Subcase (b) ,0026 ≠je [ ]

[ ]0

),,(

),,(

8),,(

),,(ln

1)( 1

3

7

03

3

0

0 ≠+−

��

��

�

��

�

��

�

+

+ jhtx

htx

e

fhtx

htx

fuyj

η

ξ

η

η

Differentiating equation (6.49)1 with respect to x gives

[ ] [ ]

[ ];0

),,(

),,(

8),,(

),,(ln

1)()(

)(

)('1

3

7

03

3

0

050 ≠+−

��

��

�

��

�

��

�

+

+=−′′′

− jhtx

htx

e

fhtx

htx

fuyjeuyj

uy

uy

η

ξ

η

η (6.52)

differentiating which once with respect to x yields the equation

;0)(

)(2

)(

)(

50

50 ≠−′

+′−=�

�

��

�

′′

′

euyj

euyj

uy

uy

du

d (6.53)

where ,05 ≠e ,07 ≠e ,00 ≠f ,00 ≠j 2e and 12e are arbitrary constants with

,025 ≠+ ete ( ) ( ) ,0133/1

0257 ≠++± fietee � ( ) 02

3/1

0257 ≠++ fete

e and

( ) ( ) 022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e while 0)( ≠uy is such that

,0)()()( ≠′′′′′′ uyuyuy 0)( 50 ≠−′ euyj and .0)(2 50 ≠+′− euyj From (6.48),

( )( ) ( ) ++++−

+++−

+

+=

3/1

0257

05

0

25

05

0

255

125 13ln13

3ln

2

3),( fietee

fe

j

i

iete

fe

j

etee

exetxu

( ) ( ) +−++−

3/1

0257

05

0 13ln13

3fietee

fe

j

i

i

( )( ) ( )( ) ( ) 3/1

0257

3/1

0257

2575

3/2

00

13

13ln

3

fietee

fietee

eteee

fji

+++−

−++

+

−

+( )

−��

�

��

� −+−

3/1

0

3/1

02571

05

0

3tan

2

3

f

fetee

fe

j

( ) +��

�

��

�++−+

3/1

0257

25

05

0

2ln2ln3

2fete

eete

fe

j

( )

( )

( ) ( )+

++−��

��

�+

��

��

�++

+

−

3/2

025

3/1

07

2

257

2

3/1

025

7

2575

3/2

00

22

2ln

fetefe

etee

fetee

eteee

fj

( ) ( ) .22

ln4

3/2

025

3/1

07

2

257

05

0 fetefe

etee

fe

j++−�

�

��

�+

188


1 ≠= hfefhf ,0)( =hg

,)( 10 jhjhj += 0)( 4 ≠= exa and

( ) ( )

,0

8

)(

3

7

03

2525

1 ≠

��

�

��

�+++

=

e

feteete

etb the

similarity solution for system (6.2) and thus for the thin film equation (6.1) in

conjunction with group (6.45) subject to the constraint 0026 ≠je is

( ) ( ) ;08

ln3

1ln

3),(),(

3

7

03

2525

0

≠��

�

�

��

�

�++−++=

e

feteete

ftxuytxh (6.54)

such that 0)( ≠uy satisfies equations

,0)(

)(2

)(

)(

50

50 ≠−′

+′−=�

�

��

�

′′

′

euyj

euyj

uy

uy

du

d

,0)()()()(

1

410

)4( 0 ≠−=′′′′+ − uyfe

f

eeuyuyfuy

[ ]

[ ],0

8),,(

),,(

3

7

03

3

>

+e

fhtx

htx

η

η (6.55)

,0)()()()(

1

410

)4( 0 ≠=′′′′+ − uyfe

f

eeuyuyfuy

[ ]

[ ];0

8),,(

),,(

3

7

03

3

<

+e

fhtx

htx

η

η

where ,01 ≠e ,04 ≠e ,05 ≠e ,07 ≠e ,024

24

5

5

3

741

26 ≠−

=e

eeeee ,00 ≠f ,01 ≠f ,00 ≠j

,2e 12e and 1j are arbitrary constants with ,025 ≠+ ete ( ) ,08 0

3

25

3

7 ≠++ fetee

( ) ( ) ,0133/1

0257 ≠++± fietee � ( ) ( ) ,022

3/2

025

3/1

07

2

25

7 ≠++−��

��

�+ fete

feete

e

( ) 02

3/1

0257 ≠++ fete

e and

( ) ( )( ) ( )

++++−

−+++

−

3/1

0257

3/1

0257

7

3/2

005

13

13ln3

fietee

fietee

e

fjixe

( )

( ) ( ).0

22

2ln 12

3/2

025

3/1

07

2

25

7

2

3/1

025

7

7

3/2

00 ≠+

++−��

��

�+

��

��

�++−

e

fetefe

etee

fetee

e

fj

189

Furthermore, ),,( htxξ and ),,( htxη are given by (6.45), ),( txu is defined by (6.48) and

0)( ≠uy is such that ,0)()()( ≠′′′′′′ uyuyuy ,0)( 50 ≠−′ euyj 0)(2 50 ≠+′− euyj and

[ ]

[ ].0

),,(

),,(

8),,(

),,(ln

1)( 1

3

7

03

3

0

0 ≠+−

��

��

�

��

�

��

�

+

+ jhtx

htx

e

fhtx

htx

fuyj

η

ξ

η

η

GROUP (XVII)

Under conditions ,0)( 1 ≠= fhf ,0)( =hg ,)( 1jhj = 0)( 2 ≠= exa and ,0)( 3 ≠= etb

system (6.2) admits classical group (XVII) given by

,)(),,( 11 etjhtx += βξ ,0)(),,( ≠= thtx βη ;0)(),,( 532 ≠+= eteehtx βζ (6.56)

where ,02 ≠e ,03 ≠e ,01 ≠f ,1e 5e and 1j are arbitrary constants while 0)( ≠tβ is an

arbitrary function of t such that 0)( 532 ≠+ etee β and .0)( ≠′ tβ


Case (1) The case of group (6.56) with 0)( 321 ≠+= dedttdβ occurs under

conditions ,0)( 1 ≠= fhf ,0)( =hg ,)( 102 jejhjhj += 0)( 2 ≠= exa and 0)( 3 ≠= etb

where ,01 ≠d ,02 ≠d ,02 ≠e ,03 ≠e ,01 ≠f ,02 ≠j ,3d 0j and 1j are arbitrary

constants such that .0321 ≠+ dedtd



,0321 ≠=+ eehjh xt ,0321 ≠−= eehf xxxx ;51 ehe x = (6.57)

where ,02 ≠e ,03 ≠e ,01 ≠f ,1e 5e and 1j are arbitrary constants.

Directly solving equation (6.57)2 and substituting its general solution into equations

(6.57)1 and (6.57)3 requires 051 == ee since the case 01 ≠e causes 0=xxh in equation

(6.57)3 , forcing the contradiction 032 =ee in equation (6.57)2.

Hence under transformations (6.3) and the conditions ,0)( 1 ≠= fhf ,0)( =hg

,)( 1jhj = 0)( 2 ≠= exa and ,0)( 3 ≠= etb the similarity solution for system (6.2) and

190

thus for the thin film equation (6.1) in association with group (6.56) subject to the

constraint 051 == ee is

( ) ( ) ( ) ( ) ;0),( 103219

2

18

3

17

4

16 ≠++−+−+−+−= eteetjxetjxetjxetjxetxh (6.58)

where ,02 ≠e ,03 ≠e ,024 1

32

6 ≠−=f

eee ,01 ≠f ,7e ,8e ,9e 10e and 1j are arbitrary

constants.

Infinitesimal generators 1721 ,...,, VVV represent the Lie algebras associated with Lie

groups (I), (II), … , (XVII) respectively; (see Gandarias [27]). These generators are as

follows.

A List of Infinitesimal Generators for Groups (I)-(XVII)

The respective generators iV for all i ∈ }17,...,2,1{ for each of groups (I)-(XVII) are

( ) ( )[ ] ( )[ ]{ } ,),(3)()( 2154113

2

151h

txdhatjxat

tx

atjtjxatjxaV∂

∂++−+

∂

∂+

∂

∂++−+−= αα

[ ] [ ] ,),()()( 2412h

txdhat

tx

atjV∂

∂++

∂

∂+

∂

∂+= αα

( ) +∂

∂+

∂

∂+=

ttxexeV

)(

1353

γ

( ) ( ) ,)(exp4)(

)(5 113102527152

heexedsdrree

s

seehee

t s

∂

∂

��

�

�

��

�

�++

�

��

��

� �

+��

�

��

��

��

�+

′−++ � �γγ

γ

( ) ( ) ( )[ ]{ } ( ) +∂

∂+−

∂

∂++−++−+−=

tata

xatjajaaataataaV 23405173232734 ln

( )[ ] ,ln 523103h

aataaha∂

∂++−+

( ) ( ) +∂

∂

�

��

��

� �

+++��

�

��

�+−+++= � � x

atatja

dsasj

jadrrbaraajxataV

t s

416

219

12

0

19

1491905952

)(2

( )t

atata∂

∂++ 1110

2

9 ( ) ( ) ( ) ,)(2 12149191

0

9

917h

adssbasaatjxj

ahtaa

t

∂

∂

��

�

��

�+++−+−+ �

191

( ) +∂

∂+

∂

∂

�

� �

+��

��

��

��

�++++++=

tta

xa

t

a

t

aatatataxataV

2

913

211

1084

2

3596 exp

( ) ( ) ,exp 12

23

14

2

15

16

191

0

9

917h

at

atata

t

aatjx

j

ahtaa

∂

∂

�

� �

+��

��

��

��

�++++−+− −−

( )[ ] +∂

∂+

∂

∂++++++= −−

tta

xtataatataxataV

2

9

3

4

2

8103

2

2597

( ) ( ) ,4

13

3

11121

0

9

914h

tataatjxj

ahtaa

∂

∂��

��

�+++−+− −−

( )[ ] ( ) +∂

∂++

∂

∂++++=

+

tata

xaataataxaV

a

11107

2

111043582

( )[ ] ,12

1

1110892

haataaha

a

∂

∂+++

+

( ) ( ) ,ln 7

1110

8

91110411103259h

aata

aha

tata

xaataataxaV

∂

∂��

��

�+

+++

∂

∂++

∂

∂++++=

( )[ ] ( ) +∂

∂++

∂

∂+++++=

tata

xaataataataxaV 111041110111032510 ln

( ) ,ln 7111089h

aataaha∂

∂+++

( ) ( ) ,89101174351122

haeaha

ta

xaeataxaV

tata

∂

∂+++

∂

∂+

∂

∂+++=

( ) ( ) ,8910743

2

2512h

atahat

ax

atataxaV∂

∂+++

∂

∂+

∂

∂+++=

( ) +∂

∂+

∂

∂+

+′=

tt

xax

atV )(

2

)(1

3

13 γγ

[ ] [ ] [ ],

)(2exp2)()(

)(22

)(

2

)(

1

33

3

6

10

0

1

0

3

hdsdrraassa

atj

jx

j

th

ta

t s ∂

∂

��

��

�

��

�

��

�

��

�

��

�−+′

++′−′′

+′−

� �−− γγγ

γγγ

( )( )

( ),

3

25

7

2562

25

3

514hete

e

tete

xe

ete

exeV

∂

∂

++

∂

∂++

∂

∂

��

�

��

�+

++=

,ln60

2

2560

0

02

15hetf

e

te

xeetf

f

jeV

∂

∂

+−

∂

∂+

∂

∂��

��

�++−=

192

( ) ( )( ) ( )

( )

( ) ( )

+∂

∂

��

��

�

��

�

��

�

+

++−��

��

�+

��

��

�++

++++−

−+++

=−

−

x

e

fetefe

etee

fetee

e

fj

fietee

fietee

e

fjixe

V

12

3/2

025

3/1

07

2

257

2

3/1

0257

7

3/2

00

3/1

0257

3/1

0257

7

3/2

00

5

16

22

2ln

13

13ln3

( )( )

,8

24

0

3

25

3

7

5

25hfetee

e

tete

∂

∂

+++

∂

∂+

[ ] [ ] ;)()()( 5321117h

eteet

tx

etjV∂

∂++

∂

∂+

∂

∂+= βββ

where the details of 1721 ,...,, VVV respectively pertain to groups (I), (II), … , (XVII).

We now present four tables of results. Table 1 lists the functions ),(hf ),(hg ),(hj

)(xa and )(tb (distinguishing the enhanced symmetries of the thin film equation (6.1))

with the associated infinitesimal generators .iV Table 2 is a dimensional classification of

the mathematical structure of groups (I)-(XVII) and their corresponding .iV Table 3

displays the similarity solutions ),( txh and their corresponding similarity variables

),( txu for system (6.2) in association with groups (III)-(XVII) where applicable. Table 4

features the defining ordinary differential equations (ODEs) for the functions 0)( ≠uy

within the functional forms of ),( txh associated with groups (XV) and (XVI) in table 3.

193


Table 1. Each row below shows the functions ),(hf ),(hg ),(hj )(xa and )(tb

(distinguishing the enhanced symmetries of the thin film equation (6.1)) in conjunction

with the associated infinitesimal generators .iV

)(hf )(hg

)(hj )(xa )(tb iV

01 ≠f 0 1j arbitrary 0≠ arbitrary 0≠

1V

01 ≠f 1g 1j arbitrary 0≠ arbitrary 0≠

2V

01 ≠f 0 0 ( ) 037 ≠+ exe

0)(exp 21 ≠

��

�

��

��t

dssee γ 3V

01 ≠f 1g 010 ≠+ jhj 08 ≠a

02

1 ≠+ at

a 4V

01 ≠f 0 010 ≠+ jhj 06 ≠a ( )

042

422

119

2

10109

119

2

10109

2/5

1110

2

91

≠��

�

�

��

�

�

−+−−

−++

×++−

a

aaaata

aaaata

atataa

5V

01 ≠f 0 010 ≠+ jhj 06 ≠a 0exp 25

1 ≠��

��

�−

t

ata 6V

01 ≠f 0 010 ≠+ jhj 06 ≠a 05

1 ≠−ta 7V

01 ≠f 0 010 ≠+ jhj 06 ≠a ( ) 02

11101 ≠+a

ataa 8V

01 ≠f 0 010 ≠+ jhj 06 ≠a ( ) 02

11101 ≠+−

ataa 9V

01 ≠f 0 010 ≠+ jhj 06 ≠a 0

1110

1 ≠+ ata

a 10V

01 ≠f 0 010 ≠+ jhj 06 ≠a 02

1 ≠taea 11V

01 ≠f 0 010 ≠+ jhj 06 ≠a 01 ≠a 12V

01 ≠f 0 010 ≠+ jhj ( ) 017 ≠+ axa

[ ][ ] 0)(2exp

)(

1

33

2 ≠��

�

��

�− �

−t

dssat

aγ

γ

13V

01 ≠f 0 10 jhj + 04 ≠e ( ) 0

4

251 ≠+−

etee 14V

00

1 ≠hfef 0

10 jhj + 04 ≠e 0

60

1 ≠+ etf

e 15V

00

1 ≠hfef 0

10 jhj + 04 ≠e

( ) ( )[ ]0

8 0

3

25

3

725

3

71 ≠+++ feteeete

ee

16V

01 ≠f 0 1j 02 ≠e 03 ≠e 17V

The entries listed for ),(hf ),(hg ),(hj )(xa and )(tb in each of rows 1-17 in the

above table respectively correspond to the Lie classical groups (I)-(XVII).

194

Table 2. A dimensional classification of the mathematical structure of groups (I)-(XVII)

(the enhanced symmetries of the thin film equation (6.1)) in conjunction with their

associated infinitesimal generators iV .


( ) ( ) 4113

2

15 )( atjtjxatjxa ++−+− α )(tα ( )[ ] ),(3 215 txdhatjxa ++− 1V

41 )( atj +α )(tα ),(2 txdha + 2V

( )35 exe + 0

)(

1≠

tγ

( ) ++ hee 52 5

×�

��

��

� �

+��

�

��

��

��

�+

′− � �

t s

edsdrrees

see 1025271 )(exp4

)(

)(γ

γ

γ

( ) 0113 ≠++ eex

3V

( ) ( ) ++−+− 23273 ln ataataa

( )[ ] 0405173 ≠++− atjajaa

( ) 023 ≠+− ata ( ) 0ln 523103 ≠++−+ aataaha 4V

( ) +++++ 416

21959

2atat

jaxata

( )� ��

�

��

�+−+

t s

dsasj

jadrrbaraaj 12

0

19149190 )(2

0≠

01110

2

9 ≠++ atata ( ) ( )+−+− tjx

j

ahtaa 1

0

9917

( ) 1214919 )(2 adssbasaa

t

++�

5V

( ) ++++ tataxata 4

2

359

0exp 13211

108 ≠+��

��

��

��

�++ a

t

a

t

aata

02

9 ≠ta ( ) ( )+−+− tjxj

ahtaa 1

0

9917

1223

14

2

1516

19 exp at

atata

t

aa +��

��

��

��

�+++ −−

6V

( ) 03

4

2

8103

2

259 ≠++++++ −−tataatataxata 02

9 ≠ta ( ) ( ) 4

13

3

11121

0

9914

−− +++−+− tataatjxj

ahtaa

7V

( ) 07

2

11104352 ≠++++

+aataataxa

a 01110 ≠+ ata ( ) 12

1

1110892 aataaha

a+++

+ 8V

0ln 41110325 ≠++++ aataataxa 01110 ≠+ ata 7

1110

89 a

ata

aha +

++ 9V

( ) 0ln 411101110325 ≠+++++ aataataataxa 01110 ≠+ ata 7111089 ln aataaha +++ 10V

074352 ≠+++ aeataxata

011 ≠a 8910

2 aeahata ++ 11V

043

2

25 ≠+++ atataxa 07 ≠a 8910 ataha ++ 12V

( ) 02

)(1

3 ≠++′

axatγ

0)( ≠tγ

++′−′′

+′−

10

0

1

0

3 )(22

)(

2

)(at

j

jx

j

th

taγ

γγ

[ ] [ ] [ ]� ��

�

��

�−+′ −−

t s

dsdrraassa1

33

3

6 )(2exp2)()( γγγ

13V

( ) 62

25

35 e

ete

exe +

++

025 ≠+ ete

( )0

3

25

7 ≠+ ete

e 14V

560

0

02 ln eetff

je++−

02 ≠e 0

60

2 ≠+

−etf

e 15V

195

Table 2. Continued.


++ 125 exe

( ) ( )( ) ( )

++++−

−++−

3/1

0257

3/1

0257

7

3/2

00

13

13ln3

fietee

fietee

e

fji

×−

7

3/2

00

e

fj

( )

( ) ( ) 3/2

025

3/1

07

2

257

2

3/1

025

7

22

2ln

fetefe

etee

fetee

++−��

��

�+

��

��

�++

025 ≠+ ete

( )0

8

24

0

3

25

3

7

5 ≠++ fetee

e

16V

11 )( etj +β 0)( ≠tβ 0)( 532 ≠+ etee β 17V

The entries for ),,,( htxξ ),,( htxη and ),,( htxζ in each of rows 1 – 17 in table 2 relate

to Lie classical groups (I) – (XVII) respectively.

Table 3. Each row lists the similarity solutions ),( txh and any corresponding similarity

variables ),( txu for system (6.2) in association with groups (III)-(XVII) where

applicable. For the entries relating to group XVI in this table, 025 ≠+= eteη .

Group ),( txh ),( txu

III ++++ 2

13

3

12

4

17

5

16 xexexexe

( ) 015181471 ≠+++ etexetee

with

0111052 ==== eeee

XV 0ln

1)),(( 60

0

≠+− etff

txuy

with

41ee 1−= , ,02

5

1 ≠=e

ej 00 =j

te

ex

2

5−

196

Table 3. Continued.

Group ),( txh ),( txu

XVI +)),(( txuy

08

ln3

1ln

33

7

03

0

≠��

�

�

��

�

�+−

e

f

fηη

with 0026 ≠je

+−+

ηη

ln2

3

05

0

5

125

fe

j

e

exe

( ) +++−+

3/1

07

05

0 13ln13

3fie

fe

j

i

iη

( ) +−+−

3/1

07

05

0 13ln13

3fie

fe

j

i

iη

( )( )

+++−

−+−

3/1

07

3/1

07

75

3/2

00

13

13ln

3

fie

fie

ee

fji

η

η

η

−��

�

�

��

�

� −−

3/1

0

3/1

071

05

0

3tan

2

3

f

fe

fe

j η

+��

�

�

��

�

�+−

3/1

07

05

0

2ln2ln3

2f

e

fe

jηη

+

+−��

��

�

��

��

�+−

3/2

0

3/1

07

2

7

2

3/1

0

7

75

3/2

00

22

2ln

ffee

fe

ee

fj

ηη

η

η

3/2

0

3/1

07

2

7

05

0

22ln

4f

fee

fe

j+−�

�

��

�ηη

XVII ( ) ( ) +−+−3

17

4

16 tjxetjxe

( ) ( )+−+− tjxetjxe 19

2

18

01032 ≠+ etee with

051 == ee

197

Table 4. Each row lists the defining ODEs for the functions 0)( ≠uy within the

functional forms of ),( txh associated with groups (XV) and (XVI) in table 3. For the

entry relating to group XVI in this table, .025 ≠+= eteη

Group ( ) 0,,,, )4( =′′′′′′ yyyyyA

XV ,0)()()(

1

)(

0

)4(0

≠=′′′′+−

f

euyuyfuy

uyf

,060 >+ etf

,0)()()(1

)(

0

)4(0

≠−=′′′′+−

f

euyuyfuy

uyf

060 <+ etf

XVI ,0

)(

)(2

)(

)(

50

50 ≠−′

+′−=�

�

��

�

′′

′

euyj

euyj

uy

uy

du

d ,0026 ≠je

,0)()()()(

1

410

)4( 0 ≠−=′′′′+ − uyfe

f

eeuyuyfuy ,0

83

7

03

3

>

+e

fη

η ,0026 ≠je

,0)()()()(

1

410

)4( 0 ≠=′′′′+ − uyfe

f

eeuyuyfuy ,0

83

7

03

3

<

+e

fη

η 0026 ≠je


To conclude, classical symmetry analysis of partition (6.2) of the thin film equation (6.1)

led to the addition of seventeen symmetry-enhancing constraints. This showed that the

inclusion of nontrivial functions into an enlarged system resulting from the partitioning

of equation (6.1) can lead to a greater variety of symmetry groups for equation (6.1).

Adding symmetry-enhancing constraints can result in new solutions not available via the

Lie classical method. As examples of the range of similarity solutions obtainable for

system (6.2) and thus for the thin film equation (6.1) under transformations (6.3), we

deduced via classical symmetry analysis the similarity solutions corresponding to groups

(III)-(XVII) where applicable. We performed many of the required computations via

Mathematica [54]. A completion of the results in this chapter would necessitate

obtaining similarity solutions corresponding to groups (I) and (II). The type of solution

generated clearly depends on the form of the individual partition of the thin film

equation (6.1).

Closer study of the similarity solutions of system (6.2) and therefore of the thin film

equation (6.1) in association with groups (I) and (II) might prove worthwhile and could

form part of a future paper on this subject.

198

CHAPTER 7

LOCATING POTENTIAL SYMMETRIES FOR

THE THIN FILM EQUATION

7.1 INTRODUCTION

We apply a method of finding potential symmetries for the thin film equation (2.1) given

by

;0)()()(3

3

=∂

∂+�

�

��

�+

∂

∂−

∂

∂

∂

∂

t

hhk

x

hhg

x

hhf

x (7.1)

where the restriction 0)( ≠hf holds and ).()( hjhk =′

This method requires expressing the thin film equation (7.1) in the conserved form

;0=∂

∂−

∂

∂

t

G

x

F (7.2)

where

),()()(3

3

hkx

hhg

x

hhfF +

∂

∂−

∂

∂= .hG −= (7.3)

The associated system T is

),()()(3

3

hkx

hhg

x

hhf

t+

∂

∂−

∂

∂=

∂

∂φ ;h

x−=

∂

∂φ (7.4)

where ),( txφ denotes the potential while 0)( ≠hf and ).()( hjhk =′

We consider the one-parameter )(ε Lie group of point transformations in htx ,, and ,φ

given by

( ) ( )( ) ( )( ) ( )( ) ( );,,,

,,,,

,,,,

,,,,

2

1

2

1

2

1

2

1

εφεχφφ

εφεζ

εφεη

εφεξ

Ohtx

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

++=

(7.5)

which leaves system (7.4) invariant.

199

If ( )txh ,α= and ( ),, txβφ = then from ( )111 , txh α= and ( ),, 111 txβφ = evaluating the

expansion of ε∂

∂ 1h and

ε

φ

∂

∂ 1 at 0=ε generates the invariant surface condition

( ) ( ) ( )

( ) ( ) ( ).,,,,,,,,,

,,,,,,,,,,

φχφ

φηφ

φξ

φζφηφξ

htxt

htxx

htx

htxt

hhtx

x

hhtx

=∂

∂+

∂

∂

=∂

∂+

∂

∂

(7.6)

The solutions of equations (7.6) are functional forms of the similarity solutions for

system (7.4). The following section contains a brief description of the method of finding

potential symmetries, introduced and developed by Bluman, Reid and Kumei [17].

7.2 THE METHOD OF OBTAINING POTENTIAL SYMMETRIES

This approach is applicable to a system S of partial differential equations with

independent variables ),...,,( 21 nxxxx = and dependent variables .u (Bluman et al. [17]).

This method requires writing S in a conserved form with respect to some choice of

these variables. Via the conserved form, Bluman et al. [17] introduce potentials .φ The

variables of the resulting system T of partial differential equations are the independent

variables ),,...,,( 21 nxxxx = the dependent variables u of S and the new dependent

variables .φ Bluman et al. [17] then apply the Lie algorithm to determine the one-

parameter )(ε Lie group TG of point transformations, of the enlarged space of variables

( ),,, φux admitted by system .T These authors state, “A transformation in TG is a new

symmetry for S if the infinitesimal of the transformation, corresponding to any of the

variables ),,( ux depends explicitly on .φ ” Bluman et al. [17] add that these new

symmetries are nonlocal symmetries realised as local (point) symmetries in the space

( )φ,,ux and are thus retrievable by Lie’s algorithm. We accordingly apply this technique

to system (7.4).

System (7.4) remains invariant under group transformations (7.5) provided the

infinitesimals ( ),,,, φξ htx ( ),,,, φη htx ),,,( φζ htx and ),,,( φχ htx satisfy the

determining equations

200

,0=== φηηη hx ,0== φξξ h ,0== φζζ hh ,0=hχ

( ) ,03)()()(

)()( =−−−−−

��

��

��

��

�+ xhxttxxx hkhgh

hf

hk

dh

dhf ξζζξχζζ

(7.7)

,0=− xxxh ξζ [ ] ,03)()(

1)( =−

∂

∂+−′

xhfhhf

t ξζχη φ

,0)(

)(2

)(

)(3 =−�

�

��

�−−

hf

hg

hf

hg

dh

dxxxxxxh ξζξζ ( ) ;0=++− xx h χζχξ φ

where 0)( ≠hf and .)()( hjhk =′ Equations (7.7)1 , (7.7)2 and (7.7)3 give

( ) ,)(,,, tahtx =φη

( ) ,),(,,, txbhtx =φξ (7.8)

( ) ;),(),(,,, txdhtxchtx +=φζ

where ,a ,b c and d denote arbitrary functions.


As neither of the infinitesimals ( )φξ ,,, htx and ( )φη ,,, htx nor the infinitesimal

),,,( φζ htx depends explicitly on ,φ we conclude that no new symmetry groups for the

thin film equation (7.1) arise via the method of obtaining potential symmetries presented

by Bluman et al. [17]. The following excerpt from Bluman et al. [17] supports this

statement.

“Now assume that a system T admits a one-parameter )(ε Lie group of point

transformations

( ) ( ) ( ),,,;,, 2* εφεξεφ Ouxxuxfx T ++== (2.11a)

( ) ( ) ( ),,,;,, 2* εφεηεφ Ouxuuxgu T ++== (2.11b)

( ) ( ) ( ),,,;,, 2* εφεζφεφφ Ouxuxh T ++== (2.11c)

where ,Tξ ,Tη and Tζ are the infinitesimals of ,x ,u and ,φ respectively, of the group.

This group maps a solution of T into another solution of T and hence induces a

mapping of a solution of S into another solution of .S Thus the group (2.11) is a

symmetry group of PDE .S This one-parameter symmetry group of PDE S is a new

symmetry group of S if and only if either Tξ or Tη depends explicitly on .φ ”

201

CHAPTER 8

CONCLUSION

Via Lie classical analysis, we obtained eight Lie classical symmetry groups for the thin

film equation (1.1) in chapter 2. We also constructed similarity solutions for this

equation in conjunction with each of these groups. Computer techniques involving the

Maple and Mathematica programs greatly facilitated the construction of these groups and

the recovery of several of these solutions [46, 54].

In chapter 3 we derived nine non-classical symmetries of the thin film equation (1.1)

using the non-classical symmetry method of Bluman and Cole [16]. A comparison of

these non-classical symmetry groups with the classical symmetry groups obtained in

chapter 2 revealed that the thin film equation (1.1) does not admit any non-classical

symmetries arising beyond its classical symmetries. Applying the non-classical

symmetry method of Bluman and Cole to the thin film equation (1.1) does not generate

any similarity solutions which are not retrievable by Lie classical analysis of the same

equation.

In chapter 4 we constructed symmetry groups of the thin film equation (1.1) by the

method of symmetry-enhancing constraints involving equation-splitting [29]. From the

perspective of this method, we examined all the combinations of partitions of the thin

film equation (1.1) involving three terms each. The enlarged systems (4.2) and (4.3)

resulting from two partitions of the thin film equation (1.1) admitted seven new Lie

symmetry groups (six groups arose for system (4.2) and one for system (4.3)).

Accordingly seven symmetry-enhancing constraints are added to equation (1.1). We

therefore showed that for this equation, the method of symmetry-enhancing constraints

successfully enables the recovery of symmetry groups extending beyond the confines of

those derived via Lie classical analysis.

The six new Lie groups for system (4.2) indicate travelling wave solutions to be the only

similarity solutions occurring for system (4.2) and hence for the thin film equation (4.1)

under the conditions on ,0)( ≠hf )(hg and )(hj for these new groups. This situation

arises since 0)()( =′=′ hjhg in the conditions on these groups, causing equation (4.2)1

to generate a travelling wave solution of velocity .01 ≠j The derivations of similarity

202

solutions for system (4.3) and thus for the thin film equation (4.1) in association with the

new Lie group arising for system (4.3) show that under the conditions on this group,

steady state solutions are the only similarity solutions recoverable. The nature of system

(4.3) and the need for consistency in the invariant surface condition (4.42)3 creates this

situation.

In chapter 5 we derived symmetry groups for the thin film equation (1.1) via an approach

combining the method of symmetry-enhancing constraints [29] with the non-classical

procedure [16]. Studying systems (1.4) and (1.5) from the perspective of this combined

approach generated twenty-nine new symmetry groups for these systems. This testifies

that the combined method used in this chapter successfully enables the recovery of

symmetry groups extending beyond the confines of those derived via the non-classical

group method. However, only eighteen of those larger symmetries gave rise to valid

similarity solutions for systems (1.4) and (1.5) and therefore for equation (1.1). The only

valid nontrivial similarity solutions thus obtained are those recoverable for system (1.5)

and thus for the thin film equation (1.1). These solutions are of three forms, namely the

steady state solutions, travelling wave solutions and cubic solutions. A greater variety of

similarity solutions for system (1.5) is therefore available via the approach used in this

chapter than was retrievable by applying Lie classical analysis to the equivalent system

(4.3) in Chapter 4.

Chapter 6 provides a description of the variety of symmetry groups available for the thin

film equation (1.1) by the method of symmetry-enhancing constraints [29] upon

incorporating nontrivial functions into an enlarged system (of which system (4.2) is

illustrative) resulting from the partitioning of equation (1.1). Goard and Broadbridge

discussed the introduction of such functions into an enlarged system arising from the

partitioning of the original target equation(s) with respect to the axisymmetric boundary

layer equations [29].

In chapter 7 we studied the thin film equation (1.1) from the perspective of a method of

obtaining potential symmetries presented by Bluman, Reid and Kumei [17]. We found

no new symmetry groups to arise for this equation via this method.

The studies undertaken in this thesis indicate that for the thin film equation (1.1), the

method of symmetry-enhancing constraints [29] can be successfully extended by the

incorporation of the non-classical procedure [16, 36] in addition to the Lie classical

203

method, as Saccomandi concluded with respect to the steady two-dimensional boundary-

layer equations in the flat and axisymmetric cases [47].

A survey of current literature in this field has demonstrated the viability of this method

in deriving additional symmetry groups and solutions for systems of equations within

disciplines such as physics and biomathematics. This approach has generated new

similarity solutions to equations such as the axisymmetric boundary layer equations [29].

Goard and Broadbridge emphasised the derivation of new symmetry groups by applying

Lie classical analysis to the enlarged system resulting from partitioning the equations of

interest, in keeping with the principles of the method of symmetry-enhancing constraints

[29]. However, these authors mentioned that classical and non-classical symmetry

analysis on different partitions of the axisymmetric boundary layer equations will

produce different solutions [29].

Research conducted in this area has indicated that solutions to practical partial

differential equations are retrievable by classical reductions of a system of equations

enlarged by the addition of symmetry-enhancing constraints chosen to elicit larger

symmetry groups than those of the original equation(s). Correspondingly, new solutions

have emerged hitherto considered unrelated to classical and non-classical symmetries

and which are unobtainable by applying Lie classical analysis and the non-classical

symmetry method of Bluman and Cole [16] to the original equation(s).

204

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symmetry-enhancing for a thin film equation tanya l.m…

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