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SYMMETRY-ENHANCING FOR A THIN FILM EQUATION

TANYA L.M. WALKER

A thesis submitted in fulfilment

of the requirements for the degree of

Doctor of Philosophy - Science

University of Western Sydney

2008


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ABSTRACT

This thesis is concerned with the construction of new one-parameter symmetry groups

and similarity solutions for a generalisation of the one-dimensional thin film equation by

the method of symmetry-enhancing constraints involving judicious equation-splitting.

Firstly by Lie classical analysis we obtain symmetry groups and similarity solutions of

this thin film equation. Via the Bluman-Cole non-classical procedure, we then construct

non-classical symmetry groups of this thin film equation and compare them to the

classical symmetry groups we derive for this equation.

Next we apply the method of symmetry-enhancing constraints to this thin film equation,

obtaining new Lie symmetry groups for this equation. We construct similarity solutions

for this thin film equation in association with these new groups. Subsequently we

retrieve further new symmetry groups for this thin film equation by an approach

combining the method of symmetry-enhancing constraints and the Bluman-Cole non-

classical procedure. We derive similarity solutions for this thin film equation in

connection with these new groups.

Then we incorporate nontrivial functions into a partition (of this thin film equation)

which has previously led to new Lie symmetry groups. The resulting system admits newLie symmetry groups. We recover similarity solutions for this system and hence for the

thin film equation in question.

Finally we attempt to derive potential symmetries for this thin film equation but our

investigations reveal that none occur for this equation.


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PREFACE

In this thesis, the symmetry groups and similarity solutions obtained for the thin film

equation and the systems of equations under consideration form an original contribution.

Where the work of other authors has been used, this has always been specifically

acknowledged in the relevant sections of the text.

Tanya Walker

31stMarch 2008


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ACKNOWLEDGEMENTS

I would like to express my indebtedness to my supervisor Dr. Alec Lee whose

encouragement, enthusiasm, intellectual stimulation and unlimited reserves of patience

have guided my researches since the commencement of this degree.

I wish to thank Professor Broadbridge for discussions leading to the final form of the

generalised thin film equation (1.1) studied in this thesis.

Furthermore I would like to express my deep appreciation of my beloved husband David

for his constant love, tenderness, understanding and confidence in me throughout my

candidature.

Finally I would like to thank my closest friend Karen for the understanding and support

she has always shown me, especially in the undertaking of these studies.

All these factors have combined to make this thesis a reality.


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This thesis is dedicated with deepest love to my husband David.

O how vast the shores of learning,

There are still uncharted seas,

And they call to bold adventure,

Those who turn from sloth and ease

Excerpt from A Students Prayer

Author unknown


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CHAPTER 6: CLASSICAL SYMMETRY-ENHANCING

CONSTRAINTS FOR THE THIN FILM EQUATION

INVOLVING ARBITRARY FUNCTIONS 163

6.1 Introduction 163

6.2 Classical Symmetry-Enhancing Constraints 164

6.3 Tables Of Results 193

6.4 Concluding Remarks 197

CHAPTER 7: LOCATING POTENTIAL SYMMETRIES FOR THE

THIN FILM EQUATION 198

7.1 Introduction 198

7.2 The Method Of Obtaining Potential Symmetries 199

7.3 Concluding Remarks 200

CHAPTER 8: CONCLUSION 201

BIBLIOGRAPHY 204


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CHAPTER 1

INTRODUCTION

We construct new one-parameter symmetry groups and corresponding similarity

solutions for a generalised thin film equation via the method of symmetry-enhancing

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

involves systematic equation-splitting and is restricted to classical symmetries. In

conjunction with this method of symmetry-enhancing constraints, Saccomandi

considered special classes of non-classical symmetries [47]. By similarly augmenting

this method of symmetry-enhancing constraints with the non-classical symmetry method

of Bluman and Cole [16], we retrieve symmetry groups for the enlarged system resulting

from the partitioning of the generalised thin film equation in question.

By means of the symmetry groups obtained for this thin film equation via the method of

symmetry-enhancing constraints, we identify similarity solutions of the latter equation.

Computer techniques involving the Mathematica and Maple programs are instrumental

in the process of deriving these groups and solutions [46, 54].

Applying the method of symmetry-enhancing constraints to solve this generalised thin

film equation does not consistently prove successful in deriving solutions, as is clear

from Chapter 5 of this thesis. However, this method of solving differential equations is

successfully applicable to nonlinear differential equations such as cylindrical boundary-

layer equations, generating new similarity solutions [29].

Other treatments of recovering solutions include the approach developed by Burde to

derive explicit similarity solutions of partial differential equations (PDEs) [20]. His

approach is an extension of the Bluman-Cole non-classical group method [15]. Burdes

method involves directly substituting a similarity form of the solution into the given

PDE and was developed via a variation of the Clarkson-Kruskal technique [22]. Instead

of requiring this given PDE be reduced to an ordinary differential equation (ODE) as in

the Clarkson-Kruskal technique [22], a weaker condition is imposed, namely that this

PDE be reduced to an overdetermined system of ODEs solvable in closed form. The

viability of Burdes approach was justified as it enabled Burde to recover new, exact,

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


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boundary layer problems. Although the solutions thus obtained extend beyond the

confines of those retrievable via classical Lie analysis and the Bluman-Cole non-

classical group method [15], they proved to be merely a special case of solutions derived

within the framework of the method of symmetry-enhancing constraints [29].

The equation under consideration in this thesis is a generalisation of the one-dimensional

thin film equation and is given by

[ ] ;0)()()( =++

txxxxx

hhhjhhghhfx

(1.1)

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

the (one-dimensional) spatial coordinate x parallel to the solid surface. This thesis

assumes the y - independence of ,h namely that the film flows without developing any

structure in the transverse direction [44].

The term )(hf arises from surface tension (which tends to flatten the free surface [44])

between two liquids or between liquid and air and incorporates any slippage at the

liquid/solid interface. This term represents surface tension effects and the viscosity of the

liquid [45].

The term )(hg results from film destabilisation due to thermocapillarity or a density

mismatch between two liquids or physical effects such as evaporation, condensation, the

normal component of gravity to a solid surface and intermolecular forces [2]. This term

can indicate additional forces such as gravity, van der Waals interactions or

thermocapillary effects [45]. If ,0)( hg occurring with repulsive van der Waals

interactions, a long wave instability appears. If ,0)( hg the thin film equation (1.1)

lacks a long wave instability.

The convective term )(hj includes any directed driving forces (such as gravity or

Marangoni stress) corresponding to a dimensionless flux function [14]. In the case of

dominant Marangoni stress, the Burgers flux 2)( hhj = occurs while the compressive

3)( hhj = features in the case of gravitational stress [14]. The Marangoni effect

corresponds to tangential stresses at the gas-liquid interface due to surface tension

gradients while Marangoni flow refers to film flow induced by surface tension

gradients [2].


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The thin film equation (1.1) is a nonlinear degenerate fourth order diffusion equation

describing the flow of thin liquid films of height (or dimensionless thickness) h on an

inclined flat surface under the action of forces of gravity, viscosity and surface tension at

the air/liquid interface [14, 34]. This equation features 0>h in a one-dimensional

geometry so that h depends on one space variable x and time t[18].

The most common derivation of the thin film equation is as a lubrication approximation

(or limit) of the Navier-Stokes equations for incompressible fluids [2, 33, 44]. Thin films

are effectively described by lubrication approximation in which the equation of motion is

given by the thin film equation (1.1) with nhhf =)( and 0)()( == hjhg where 0>h is

a requirement [18].

Grun and Rumpf presented numerical experiments indicating the occurrence of a waiting

time phenomenon for fourth order degenerate parabolic equations [33]. Grun proved

such an occurrence in space dimensions 4n is a parameter [26, 34, 37, 38].

Hastings and Peletier regarded 0>n as a constant dependent on the type of flow

considered [34].

The above case of equation (1.1) with the critical value 3=n features in [6, 38, 43, 53]

and is pronounced most common in physical situations [38] while 4=n is noted as a

critical exponent for the large time behaviour of solutions.


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Bernis, Peletier and Williams considered the critical value2

3=n at which the nature of

the solution near the interface changes [8]. Hulshof studied similarity solutions of the

thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and ,0>n recovering one

such explicit solution via Maple 5 release 2; [37]. Bernis, Hulshof and Quiros studied the

limit of nonnegative, self-similar source-type solutions of this case of the thin film

equation (1.1) as ,0+n consequently obtaining a unique limiting function ,h a

solution of an obstacle-type free boundary problem with constraint ;0h [7].

The thin film equation (1.1) arises in fluid dynamics (hydrodynamics) and material

sciences (cf. the Cahn-Hilliard equation) [1, 31, 32]. The case of equation (1.1) with

nhhf =)( and 0)()( == hjhg (where 0>h is a requirement) occurs in certain fluid

dynamics problems in which inertia is negligible and the dynamics is governed by the

presence of viscosity and capillarity forces [18].

Upon assuming the lubrication approximation with the no-slip condition for the fluid at

the solid surface and the fact that the pressure is entirely due to surface tension, Beretta

and Bertsch derived the above case of thin film equation (1.1) with ;3=n [3]. This case

has great physical significance in lubrication theory in terms of governing the dynamicsof the spreading of a droplet over a solid surface under effects of viscosity and

capillarity. This case is depicted as the height ),( txh of a thin film of slowly flowing

viscous fluid over a horizontal substrate when surface tension is the dominating driving

force [3, 6, 12, 18, 38, 39, 43]. This case corresponding to no-slip boundary conditions

results in infinite viscous dissipation, generating variations on the same problem by

changing boundary conditions at the interface solid fluid [12, 18].

The case of the thin film equation (1.1) with 2)( hhf = and 0)()( == hjhg corresponds

to slip dominated spreading with a Navier slip law [43] and occurs in [4] and [18].

According to Laugesen and Pugh, the case of the thin film equation (1.1) with 0)( =hj

is used to model the dynamics of a thin film of viscous liquid where the air/liquid

interface is at height ),,( tyxhz = and the liquid/solid interface is at ;0=z [45]. These

authors also state that equation (1.1) with 0)( =hj applies if the liquid film is uniform in


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the y direction [45]. An application of equation (1.1) with 0)( =hj lies in its ability to

model the aggregation of aphids on a leaf where h represents population density [45].

The special case of the thin film equation (1.1) with hhf =)( and 0)()( == hjhg is

used to describe the evolution of the interface of a spreading droplet, modelling the

surface tension dominated motion of thin viscous films and spreading droplets,

according to Carrillo and Toscani [21]. This case describes the dynamics of the process

in the gravity-driven Hele-Shaw cell [6, 12, 18, 23, 25, 30, 38, 43, 45]. In this process,

liquid in a fluid droplet is sucked so as to produce a long thin bridge of thickness h

between two masses of fluids, the geometry of which problem being able to be

approximated as one-dimensional under appropriate conditions. This case emerges when

considering a drop on a porous surface [18].

Another of the varied applications of the thin film equation (1.1) is the modelling of

driven contact line experiments involving only one dominant driving force

(corresponding to a convex flux function )(hj ) [14]. In addition, equation (1.1) models

thin film slow viscous flows (viscosity driven flows) such as painting layers [37] and the

drying of a paint film in a specific parameter regime [52]. Equation (1.1) also plays a key

role in plasticity modelling where h represents the density of dislocations. This equation

occurs in the Cahn-Hilliard model of phase separation for binary mixtures where h

denotes the concentration of one component.

The case of the thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and )3,0(n

emerges as a lubrication theory model for the flow of thin viscous films (and spreading

droplets) driven by strong surface tension over a horizontal substrate with ),( txh

denoting the height of the free-surface of the film [7, 9, 26, 31, 32, 33]. The range 0


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profiles for this equation revealed that after initial transients, the flow develops a

travelling wave profile [44].

Via analysis methods (involving a Lyapunov function), Bertozzi and Shearer studied an

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

and the size of the dimensionless parameter governing gravitational, viscous and surface

tension forces as well as the slope of the surface equalling 1; [14]. Experimental and

numerical studies of driven contact lines disclosed that travelling wave solutions of this

equation play a key role in the motion of the film [11, 13, 40, 51]. Travelling wave

solutions also arise in chapters 2 5 of this thesis.

Hulshof and Shishkov [39] examined compactly supported solutions of the case of the

thin film equation (1.1) with ,)( nhhf = 0)()( == hjhg and [ )3,2n on

( ) ( ]{ }TtRRxtxQT ,0,,:),( = with nonnegative initial data and lateral boundary

conditions respectively given by

)()0,( 0 xuxu = with ,00 u ( ) ( ) .0,, == tRutRu xxxx (1.2)

These authors regarded R as a finite positive number. It is also potentially considered as

=R for compactly supported solutions (the Cauchy problem). For the case of zerocontact angle boundary conditions on a finite domain, van den Berg et al. investigated

self-similar solutions of the above case of the thin film equation (1.1) where n is a real

parameter [53].

The outline of the thesis is as follows.

In chapter 2 we obtain the Lie classical symmetry groups of the thin film equation (1.1)

and derive its similarity solutions in association with each of these groups. We use the

one-parameter )( Lie group of general infinitesimal transformations in ,x t and ,h

namely

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

,,,

,,,

2

1

2

1

2

1

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(1.3)

In conjunction with Lie classical analysis discussed in [36], group transformations (1.3)

enable the recovery of the one-parameter Lie classical symmetry groups for the thin filmequation (1.1).


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In chapter 3 we construct non-classical symmetry groups for the thin film equation (1.1)

under the action of group transformations (1.3), using the non-classical symmetry

method of Bluman and Cole [16]. We compare these symmetry groups with those

obtained in chapter 2 and derive for equation (1.1) any similarity solutions not

retrievable by Lie classical analysis. Full details of these solutions occur in chapter 3.

In chapter 4 we apply the method of symmetry-enhancing constraints [29] to the thin

film equation (1.1) in association with group transformations (1.3) with a view to

obtaining new symmetry groups. In line with this method, we studied various partitions

of the thin film equation (1.1).

Two of these partitions lead to new Lie symmetry groups and generate the systems

( ) ,0)()( 2 =+ xxt hhghhjh [ ] ;0)()( =

xxxxx

hhghhfx

(1.4)

and

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

=+ xxxxxx hhjhhghhhf (1.5)

We construct similarity solutions for systems (1.4) and (1.5) and hence for the thin film

equation (1.1) in relation to each of these new groups. A full account of these solutions is

given in chapter 4.

In chapter 5 we derive symmetry groups for the thin film equation (1.1) in association

with group transformations (1.3) by a treatment combining the method of symmetry-

enhancing constraints [29] with the non-classical symmetry method of Bluman and Cole

[16]. Saccomandi considered the combination of these two techniques [47]. Investigating

systems (1.4) and (1.5) from the perspective of this combined approach generates new

symmetry groups for these systems. We retrieve the similarity solutions for systems (1.4)

and (1.5) and thus for the thin film equation (1.1) in connection with these groups.

In chapter 6 we augment system (1.4) with the arbitrary nontrivial functions )(xa and

),(tb obtaining the equations

( ) ,0)()()()( 2 =+ tbxahhghhjh xxt [ ] .0)()()()( =

tbxahhghhf

x xxxxx (1.6)

System (1.6) admits new Lie symmetry groups in association with transformations (1.3).

We derive similarity solutions for system (1.6) and hence for the thin film equation (1.1)

in relation to these groups. We give a full account of these groups and solutions in

chapter 6.


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In chapter 7 we seek potential symmetries for the thin film equation (1.1) by the method

introduced and developed by Bluman, Reid and Kumei [17].

At the end of each chapter, we tabulate all results obtained in the chapter concerned. This

thesis has been written largely in accordance with the guidelines in Higham [35],

Bluman and Kumei [55], Ibragimov [56], Olver [57] and Ovsiannikov [58].


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CHAPTER 2

LIE CLASSICAL SYMMETRIES FOR THE

THIN FILM EQUATION

2.1 INTRODUCTION

By the Lie classical procedure, we determine the Lie classical symmetry groups for the

thin film equation

[ ] [ ] ;0)()()( =++

txxxxx

hhhjhhgx

hhfx

(2.1)

where .0)( hf The restriction 0)( hf applies since the thin film equation (2.1)

generalises the fourth order nonlinear diffusion equation, a special case of equation (2.1)

with .0)()( == hjhg This case of the thin film equation (2.1) occurs in Bernoff and

Witelski [9] and King and Bowen [43]. The term )(hf in the thin film equation (2.1)

represents surface tension effects (Laugesen and Pugh [45]).

We consider the one-parameter )( Lie group of general infinitesimal transformations in

,x tand ,h namely

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

,,,

,,,

2

1

2

1

2

1

Ohtxhh

Ohtxtt

Ohtxxx

++=

++=

++=

(2.2)

preserving the thin film equation (2.1).

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

1h at 0=

gives the invariant surface condition

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

hhtx

x

hhtx =

+

(2.3)

Solutions of the invariant surface condition (2.3) are functional forms of similarity

solutions for the thin film equation (2.1).

The next section contains a brief outline of the Lie classical method, also described in.

Hill [36].


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2.2 THE CLASSICAL PROCEDURE

The classical method requires equating to zero the infinitesimal version of the thin film

equation (2.1) without using the invariant surface condition (2.3). In obtaining the

infinitesimal version of the thin film equation (2.1), we eliminate the highest order

derivative4

4

x

h

in equation (2.1) by expressing it with respect to all the remaining terms

of equation (2.1). Prolongation of the action of group transformations (2.2) on the thin

film equation (2.1) yields the invariance requirement, obtained by equating to zero the

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

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

, and already considered in the coefficient of , the left-hand side of the

invariance requirement.

The thin film equation (2.1) remains invariant under group transformations (2.2)

provided the group generators ),,,( htx ),,( htx and ),,( htx satisfy the determining

equations

,0=h ,0== xh ,0=hh ,0

)(

)(=

hf

hf

dh

d ( ) ,0)( = xxxhhf

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

)()(4 =

hf

hftx

[ ] xxxxxxtx hfhghgdh

d

hf

hj

dh

dhfhj )()()(2

)(

)()()(3 ++

+

( ) ,04)( =+ xxxxxxxhhf (2.4)

,064

)(

)(=+

xxxhx

hf

hf ,0

)(

)(

)(

)(246 =

hf

hg

dh

d

hf

hgxxxxxxh

( ) ( ) .0)(

)(2

)(

)(3

)(

)(=

+

hf

hg

dh

d

hf

hg

hf

hfxhxxxxxh

Equating to zero the coefficients of all derivatives of h and the sum of all remaining

terms not involving derivatives of h within the invariance requirement for the thin film

equation (2.1) produces system (2.4). All subscripts in system (2.4) denote partial

differentiation with ,x t and h as independent variables. Throughout this chapter,

primes represent differentiation with respect to the argument indicated.


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System (2.4) enables the recovery of all Lie classical symmetries and corresponding

conditions on ,0)( hf )(hg and )(hj for the thin film equation (2.1) under group

transformations (2.2).

We now partially solve the determining equations (2.4) to clarify derivations of sets of

conditions on ,0)( hf )(hg and )(hj associated with each Lie classical group we

obtain for the thin film equation (2.1). Subsequently we describe the functional forms of

,0)( hf )(hg and )(hj with the corresponding Lie classical group occurring for the

thin film equation (2.1). Eight such groups arise. Lastly we present the similarity

solutions of the thin film equation (2.1) in connection with each of these groups.

From equations (2.4)1 (2.4)3, it follows that),,(),,( txhtx = ),(),,( thtx = );,(),(),,( txbhtxahtx += (2.5)

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

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

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

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

Case (1) ),(),( txtxa xxx =

It follows that

);(),(),( ttxtxa x += (2.6)

where )(t is an arbitrary function of .t

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

integrating which with respect to x implies

[ ] );,(),()()(2)( htctxbhfhfhfh x =+ (2.7)

where 0)( hf is an arbitrary function of h while ),( htc is an arbitrary function of t

and .h

By results (2.5)-(2.7), equation (2.4)7gives ,)(

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

hf

thfhhtcttx

+== so

),(2

)()(),( tx

tttx

+

+= );()()()(),( thfhthfhtc = (2.8)

where ),(t )(t and )(t are arbitrary functions of .t


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

),,(2

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

ttthtx +

++=

(2.9)

[ ] ).(2

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

ttttthftxbhf

+++=

As equation (2.9)2gives ,0)( = xbhf we obtain the subcases

(a) ,0)( =hf (b) ).(),( tbtxb =

As case (2) includes subcase (a), we need consider only subcase (b).

Subcase (b) )(),( tbtxb =

Results (2.8) and (2.9) yield

),(2

)()(),( tx

tttx

+

+= ),(

2

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

ttththtx +

++==

(2.10)

[ ] );(2

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

ttttthftbhf

+++=

where )(tb is an arbitrary function of .t

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

),()(2)( 1 ttdt = ;)( 2dtb = (2.11)

where 1d and 2d are arbitrary constants.

Results (2.10) and (2.11) give

[ ] ),()(),( 1 txtetx += ),()(2)( 1 ttdt = ,)(),( 21 dhehht +==

(2.12)( ) [ ] );()(2)()( 121 hfdtthfdhe +=+

where .2

11

de =

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

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

where 3d is an arbitrary constant.


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Results (2.12) and (2.13) yield

),(2

)(),( 3 tx

tdtx

+

= ),(2)( 3 tdt = ,)( 21 dheh +=

(2.14)

( ) ).()( 321 hfdhfdhe =+

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

( ) ).(2

)()(

2

)(3)( 321 tx

thj

tdhjdhe

=

+

++ (2.15)

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

;)( 4dt = (2.16)

where 4d is an arbitrary constant.

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

;)( 65 dtdt += (2.17)

where 5d and 6d are arbitrary constants.

Redefining the constants, the determining equations (2.4) and the results for this case are

,),( 654 DtDxDtx ++= ,)( 73 DtDt += ,)( 21 DhDh +=

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

( ) ;)()( 51021 DhjDhjDhD =++

whereiD is an arbitrary constant for all { }10,...,2,1i with ,4 348 DDD =

349 2 DDD = and .4310 DDD =

In view of equations (2.18)4and (2.18)6, we consider the cases

(1) ,021 =+DhD ,0108 ==DD ,05 =D (2) ,021 =+DhD ,0810 =DD

(3) ,021 +DhD ,01012 == DDD (4) ,021 +DhD ,01102 =DDD

(5) ,021 +DhD ,0101 =DD (6) ,021 +DhD .0101 DD

Rewriting cases (1)-(6) above with 348 4 DDD = and 4310 DDD = gives

(a) ,054321 ===== DDDDD (b) ,04 2143 === DDDD

(c) ,04312 == DDDD (d) ,012 =DD ,43 DD

(e) ,0431 = DDD (f) ,01 D .43 DD


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For each of the cases (a) (f), we describe ,0)( hf )(hg and )(hj (obtainable from

the defining equations (2.18)4 (2.18)6) with their associated Lie classical groups (I)

(VI). We also present ,0)( hf )(hg and )(hj with their corresponding Lie classical

groups (VII) (VIII) for case (2). As previously stated, we give the similarity solutions

of the thin film equation (2.1) in conjunction with each of these groups.

GROUP (I)

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

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

,),,( 6Dhtx = ,),,( 7Dhtx = ;0),,( =htx (2.19)

where 6D and 7D are arbitrary constants.

Similarity Solutions

Group (2.19), the invariant surface condition (2.3) and the thin film equation (2.1) give

,076 =+ tx hDhD [ ] ;0)()()( =++

txxxxx hhhjhhghhf

x (2.20)

where 6D and 7D are arbitrary constants while ,0)( hf )(hg and )(hj are arbitrary

functions of .h As 0=xh forces 0=th in equation (2.20)2 , giving =),( txh constant,

we require 0xh for system (2.20) to generate nonconstant similarity solutions.

As no similarity solutions are obtainable for the thin film equation (2.1) when

,076 ==DD we consider only the cases

(1) ,07 D (2) .076 =DD

Case (1) 07 D

By the method in [24], we solve equation (2.20)1and substitute its general solution into

equation (2.20)2 . Therefore under transformations (2.2) and with ,0)( hf )(hg and

)(hj arbitrary functions of ,h the similarity solution of the thin film equation (2.1) in

association with group (2.19) and the constraint 07 D is the travelling wave of

velocity ,11D namely

);(),( uytxh = (2.21)


23/215

16

satisfying

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

( )[ ] .0)()( 11 =+ uyDuyj (2.22)

In relations (2.21)-(2.22), ,07 D 6D and7

611

D

DD = are arbitrary constants while

tDxu 11= and ( ) ,0)( uyf ( ))(uyg and ( ))(uyj are arbitrary functions of ).(uy We

require 0)( uy for solution (2.21) to be nonconstant.

When ,0116 ==DD the travelling wave (2.21) reduces to the steady state solution

satisfying the case of the ordinary differential equation (ODE) (2.22) with .011 =D

Case (2) 076 =DD

Since 06 D forces 0=xh in equation (2.20)1 , giving 0=th in equation (2.20)2 ,

system (2.20) yields only the constant solution. Hence under transformations (2.2) and

with ,0)( hf )(hg and )(hj arbitrary functions of ,h the similarity solution of the

thin film equation (2.1) in connection with group (2.19) and the constraints 076 =DD

is the constant solution.

GROUP (II)

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

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

( ) ,03),,( 614 ++= DtjxDhtx ,04),,( 74 += DtDhtx ;0),,( =htx (2.23)

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


Group (2.23), the invariant surface condition (2.3) and the thin film equation (2.1) imply

( )[ ] ( ) ,043 74614 =++++ tx hDtDhDtjxD [ ] ;0)( 1 =++

txxxx hhjhhf

x (2.24)

where ,04 D ,6D 7D and 1j are arbitrary constants while 0)( hf is an arbitrary

function of .h Since 0=xh causes 0=th in equation (2.24)2, giving =),( txh constant,

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


24/215

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

(2.30)

[ ]{ }20)(2120)4( )()()()(3)( 0 uyguyeDuyuyguy uyg +++

,0)(1

)()(3

1514

1

13

)(3 00 =

+

uyguygeDuyDu

fuyDe .11Dt<

In results (2.29)-(2.30), ,02 D ,01

01

15 =gf

D ,01 f ,00 g ,6D ,7D

,02

711

gDDD = ,

1

112

fgD = ,

1

013

fjD = ,

1

114

fjD = ,

0

016

gjD = ,

02

617

gDDD = ,1g 0j and

1j are arbitrary constants, 0ln 111611

1716

++= DtD

Dt

DtDxu and

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

Case (2) 00 =g

We consider the subcases

(i) ,07 D (ii) .07 =D


26/215

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 0xh while ,02 D

,07 D ,07

211 =

DDD ,0

17

217 =

fDDD ,01 f ,6D ,

2 7

0212

DjDD = ,

7

613

DDD =

,1

114

f

gD = ,

1

015

f

jD = ,

17

61716

fD

DjDD

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

.0132

12 ++= tDtDxu

Subcase (ii) 007 ==gD

We directly solve equation (2.28)1 and substitute its general solution into equation

(2.28)2 , solving the resulting equation using the integrating factor algorithm [48].

Therefore under transformations (2.2) and the conditions ,0)( 1 = fhf 1)( ghg = and

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

group (2.27) and the constraints 007 ==gD is

( );0),(

602

1112

+

+=

DtjD

DtjxDtxh (2.33)

where ,02 D ,6D ,11D 0j and 1j are arbitrary constants such that 0602 +DtjD and

( ) .01112 + DtjxD


27/215


28/215

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)(15 0016 =+ uyfuyfuyD eDuyueDeD .12Dt<

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

,02

D ,04

3

1011

=jfD ( ) ,03

21014

= jfD ,00116

= fjD

( ),0

4 110

1017

=

fjf

jfD

( ),0

4

3

110

18

=fjf

D ,04 10

1021

=

jf

jfD ,01 f ,01 j

,6D ,7D ( ),

4

3

102

712

jfD

DD

= ,

1

113

f

gD = ,

1

015

f

jD = ,

3

10

2119

jf

jjD

=

( ),

3

102

620

jfD

DD

=

( )

( ),

3

102

27622

jfD

jDDD

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

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

( ) .02221221

+= DtjxDtu D

Case (2) 010 = jf

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


,0)(1

1 =

hj

efhf

hj

eghg1

1)( =

and ,)( 201

jejhj

hj+=

the similarity solution of the

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

;0ln1

)(),( 111

= Dtj

uytxh (2.38)

satisfying


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22

[ ]21121)4( )()()()()( uyjuyDuyuyjuy +++

[ ] ,0)( )(15)(1413 11 =+++ uyjuyj eDuyeDD ,11Dt> (2.39)

[ ]

2

1121

)4(

)()()()()( uyjuyDuyuyjuy +++

[ ] ,0)( )(15)(1413 11 =+ uyjuyj eDuyeDD .11Dt<


,02 D ,01

11

15 =jf

D ,01 f ,01 j ,6D ,7D ,12

711

jD

DD = ,

1

112

f

gD =

,1

013

f

jD = ,

112

27614

fjD

jDDD

= ,

12

27616

jD

jDDD

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

011

Dt and ( ) .0ln 1116112 +=

DtDDtjxu

Case (3) 04 10 = jf

We consider the subcases (i) ,07 D (ii) .07 =D

Subcase (i) ,04 10 = jf 07 D

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

equation (2.35)1and substitute its general solution into equation (2.35)2. Therefore under

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

1 = hj

efhf hj

eghg 12

1)( = and

,)( 201 jejhj

hj+= the similarity solution of the thin film equation (2.1) in connection

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

;0)(),( 11 += tDuytxh (2.40)

satisfying

[ ]21)(2

121

)4( )(2)()()(4)( 1 uyjuyeDuyuyjuy uyj +++

[ ] .0)( )(415)(414)(313 111 =+++ uyjuyjuyj eDuyueDeD (2.41)

In relations (2.40)-(2.41), ,02 D ,07 D ,07

211 =

D

DD ,0

17

1214 =

fD

jDD

,017

215 =

fD

DD ,0

7

1216 =

D

jDD ,01 f ,01 j ,6D ,

1

112

f

gD = ,

1

013

f

jD =

,12

27617

jD

jDDD

= ,

12

618

jD

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

and ( ) .017216 += Dtjxeu

tD Furthermore, 0)( uy owing to the requirement

.0xh


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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)( 141 = hjefhf hjeghg 121)( =

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

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

[ ] ;0)(ln1),( 1121

+= tzDtjxj

txh (2.42)

such that

[ ] [ ] [ ] ,0)(2)()()( 513

1

2

0 =++ tzftzgtzjtz ,0112 >+ Dtjx

(2.43)

[ ] [ ] [ ] ,0)(2)()()( 513

1

2

0 =+ tzftzgtzjtz .0112 tz and .0112 + Dtjx

GROUP (V)

Subject to the conditions ( ) ,0)( 0321 += g

fhfhf ( ) 021)( g

fhghg += and

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

namely

( ) ,),,( 6001 DtjxgDhtx ++= ,),,( 701 DtgDhtx += ( ) ;0),,( 21 += fhDhtx

(2.44)

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

.02 + fh


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24



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

(2.45)

( ) ( ) ( )

+++

+++

2

2

021

2

03

2100

3xxx

g

xxxxxxxx

gh

fh

ghfhghh

fh

ghfhf

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

where ,01 D ,01 f ,6D ,7D ,1

611

D

DD = ,

1

712

D

DD = ,2f ,0g ,1g 0j and 1j are

arbitrary constants while .02 + fh As 0=xh forces 0=th in equation (2.45)2 ,

rendering equation (2.45)1inconsistent, we require .0xh

We consider the cases

(1) ,00 g (2) .00 =g

Case (1) 00 g

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

equation (2.45)1 , substituting its general solution into equation (2.45)2 . Hence under

transformations (2.2) and the conditions ( ) ,0)( 0321 += g

fhfhf ( ) 021)( g

fhghg +=

and ,ln)( 120 jfhjhj ++= the similarity solution of the thin film equation (2.1) in

association with group (2.44) and the constraint 00 g is

;0)(),( 2/1

13

0= fDtuytxh

g (2.46)

satisfying

[ ]

[ ]

++

+

)(

)()(

)()(

)()(3)(

2

02

14

0

)4(

0 uy

uyguy

uy

D

uy

uyuyguy

g

[ ] [ ] ,0)(1)()(ln

)(

1

0

103

10

=

+++ uyg

uyujuyjuyf g

,13Dt>

(2.47)

[ ]

[ ]

++

+)(

)()(

)()(

)()(3)(

2

02

140

)4(

0 uy

uyguy

uy

D

uy

uyuyguy

g

[ ] [ ] ,0)(1)()(ln

)(

1

0

103

10

=

++ uyg

uyujuyjuyf g

.13Dt<


32/215

25


,01 D ,01 f ,00 g ,6D ,7D ,01

713

gD

DD = ,

1

114

f

gD = ,

0

015

g

jD = ,

01

616

gD

DD =

,2

f ,1

g 0

j and1

j are arbitrary constants, 0ln1315

13

1615

++= DtD

Dt

DtDxu and

( )( ) .0161513 ++ DtDxDt

Case (2) 00 =g


Subcase (i) 007 =gD

By the method of Lagrange [24], we solve equation (2.45)1 , substituting its general

solution into equation (2.45)2. Therefore under transformations (2.2) and the conditions

,0)( 1 = fhf 1)( ghg = and ,ln)( 120 jfhjhj ++= the similarity solution of the thin

film equation (2.1) in tandem with group (2.44) and the constraints 007 =gD is

;0)(),( 2/ 12 = feuytxh Dt (2.48)

satisfying

[ ] .0)()()(ln)()( 16151413)4(

=++++ uyDuyDuyDuyDuy (2.49)


,01 D ,07 D ,01

712 =

D

DD ,0

17

116 =

fD

DD ,01 f ,6D ,

1

113

f

gD = ,

1

014

f

jD =

,17

61715

fD

DjDD

= ,

2 7

0117

D

jDD = ,

7

618

D

DD = ,2f ,1g 0j and 1j are arbitrary

constants and .0182

17 ++= tDtDxu

Subcase (ii) 007 ==gD


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


33/215

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 .0xh

GROUP (VI)

Under the conditions ( ) ,0)( 021 += f

fhfhf ( ) 32

21

10

)(jf

fhghg+

+= and

( ) ,)( 2201 jfhjhj

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

namely

( )[ ] ,33

),,( 621101 Dtjjxjf

Dhtx += ( ) ,4

3),,( 710

1 DtjfD

htx +=

(2.52)

( ) ;0),,( 21 += fhDhtx

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

while .02 + fh



( ) ( ) ,021514131211 +=++++ fhhDtDhDtDxD tx (2.53)

( ) ( )( )

( ) txxx

jf

xxxxxxxx

fhh

fh

jfhfhghh

fh

fhfhf +

+

+++

+++

+2

2

103

2

21

2

021

3

2100

( )[ ] ;0220 1 =+++ xj hjfhj

where ,01 D ,01 f ,01 j ,6D ,7D ,3

1011

jfD

= ,2112 jjD = ,

1

613

D

DD =

,3

4 1014

jfD

= ,

1

715

D

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

.02 + fh Since 0=xh forces 0=th in equation (2.53)2 , rendering equation (2.53)1

inconsistent, we require .0

xh


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27

We consider the cases

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

Case (1) ( )( ) 04 1010 jfjf



transformations (2.2) and the conditions ( ) ,0)( 021 += f

fhfhf ( ) 32

21

10

)(jf

fhghg+

+=

and ( ) ,)( 2201 jfhjhj

j++= the similarity solution of the thin film equation (2.1) in

association with group (2.52) and the constraints ( )( ) 04 1010 jfjf is

;0)(),( 2/1

1614

= fDtuytxh D

(2.54)

such that

[ ] [ ]

+

++

+)(

)(

3

2)()(

)(

)()()(

2

10170

)4( 18

uy

uyjfuyuyD

uy

uyuyfuy

D

[ ] [ ] [ ] ,0)()()()( 0020 1222119 =+++ ffD

uyDuyuyuDuyD ,16Dt>

(2.55)

[ ] [ ]

++++

)()(

32)()(

)()()()(

2

10170

)4( 18

uyuyjfuyuyD

uyuyuyfuy

D

[ ] [ ] [ ] ,0)()()()( 0020 1222119 =+ ffD

uyDuyuyuDuyD .16Dt<


,01 D ,03

4 1014

=

jfD ( ) ,0

3

21018 = jfD ,00120 = fjD

( ),0

4 110

1021

=

fjfjfD

( ),0

43

110

22

=fjf

D ,04 10

1025

=

jfjfD ,01 f

,01 j ,6D ,7D ( ),

4

3

101

716

jfD

DD

= ,

1

117

f

gD = ,

1

019

f

jD = ,

3

10

2123

jf

jjD

=

( ),

3

101

624

jfD

DD

=

( )

( ),

3

101

27626

jfD

jDDD

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

with ( )( ) ,04 1010 jfjf 02423 ++ DtDx and ( ) .02621625

+= DtjxDtu D


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28

Case (2) 010 = jf

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


( ) ,0)(121 +=

j

fhfhf ( )121)(

j

fhghg += and ( ) ,)( 2201 jfhjhj

j

++= the similarity

solution of the thin film equation (2.1) in tandem with group (2.52) and the constraint

010 = jf is

;0)(),( 2/1

16

1=

fDtuytxh j

(2.56)

satisfying

[ ]

++

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ] )()( 11819 uyuyDD

j++

[ ] ,0)( 1120 =+ j

uyD ,16Dt>

(2.57)

[ ]

++

+)(

)()(

)(

)()()(

2

1171

)4(

uy

uyjuyD

uy

uyuyjuy [ ] )()( 11819 uyuyDD

j+

[ ] ,0)( 1120 = j

uyD .16Dt<


,01

D ,0

1

1120

=jfD ,01

f ,01

j ,6D ,7D ,11

7

16 jD

D

D =

,1

1

17 f

g

D =

,111

27618

fjD

jDDD

= ,

1

019

f

jD = ,

11

27621

jD

jDDD

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

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

Case (3) 04 10 = jf


Subcase (i) ,04 10

= jf 07

D


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

transformations (2.2) and the conditions ( ) ,0)( 1421 += j

fhfhf ( ) 1221)( j

fhghg +=

and ( ) ,)( 2201 jfhjhj

j++= the similarity solution of the thin film equation (2.1) in

connection with group (2.52) and the constraints 04 10 = jf and 07 D is

;0)(),(2

/ 15 = feuytxh Dt

(2.58)

such that


36/215

29

[ ] [ ]

++

+

)(

)(2)()(

)(

)()(4)(

2

1

2

161

)4( 1

uy

uyjuyuyD

uy

uyuyjuy

j

[ ] [ ] )()()( 11 4183

17 uyuyuDuyD jj

++ [ ] .0)( 14119 =+

juyD (2.59)


,01 D ,07 D ,01

715 =

D

DD ,0

17

1118 =

fD

jDD ,0

17

119 =

fD

DD ,0

7

1122 =

D

jDD

,01 f ,01 j ,6D ,1

116

f

gD = ,

1

017

f

jD = ,

11

620

jD

DD = ,

11

27621

jD

jDDD

= ,2f ,1g

0j and 2j are arbitrary constants, 0202 + Dtjx and ( ) .022212 += tDeDtjxu

Subcase (ii) 04 710 == Djf


Hence under transformations (2.2) and the conditions ( ) ,0)( 1421 += j

fhfhf

( ) 1221)( j

fhghg += and ( ) ,)( 2201 jfhjhj

j++= the similarity solution of the thin film

equation (2.1) in tandem with group (2.52) and the constraints 04 710 == Djf is

;0)(),( 2/1

162

1+= fDtjxtytxh

j (2.60)

satisfying

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

)(111 4

19

2

1817 =+++ jjj

tyDtyDtyDty

ty ,0162 >+ Dtjx

(2.61)

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

)(111 4

19

2

1817 =++ jjj

tyDtyDtyDty

ty .0162


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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

),,( 613 Dtjx

Dhtx ++= ,),,( 73 DtDhtx += );,(),,( 1 txbhDhtx += (2.62)

such that

;011 =++ txxxxx bbjbf (2.63)

where ,01 f ,1D ,3D ,6D 7D and 1j are arbitrary constants.

Equation (2.63) admits the travelling wave solution of velocity ,1j namely

( )=

=3

0

1 ;),(n

n

n tjxdtxb (2.64)

where ,0d ,1d ,2d 3d and 1j are arbitrary constants.


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

173613 txbhDhDtDhDtjx

Dtx +=++

++ ;011 =++ txxxxx hhjhf (2.65)

where ,03 D ,01 f ,1D ,6D 7D and 1j are arbitrary constants while ),( txb satisfies

equation (2.63), 073 +DtD and ( ) .034

613

++ DtjxD

As 0=xh forces 0=th in

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

nonconstant similarity solutions.

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


38/215


39/215

32

Similarity Solutions for Case (b)


( ) ( ),)(3

4 812173613 DtjxDhDhDtDhDtjx

D

tx ++=++

++ ;0

11 =++

txxxxx hhjhf

(2.68)

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

073 +DtD and ( ) .034

613

++ DtjxD

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

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

solutions.


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

Subcase (1) ( ) 8121 DtjxDhD ++ 0



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

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

constraints ,031 DD ( ) 08121 ++ DtjxDhD and 812 )(),( DtjxDtxb += is

( ) ,0)(),( 13121119110

++++= DDtjxDDtuytxh D

,4 13 DD

(2.69)

( ) ,0ln)(),( 1614121154/1

142 +++++= DDtDtjxDDtuytxh ;04 13 = DD

satisfying

,0)()(4

1)( 1101

)4(

11 =+ uyDuyuuyf ,0)(4

1)(

4

1)( 1522

)4(

21 =++ uDuyuyuuyf

,9Dt >

(2.70)

,0)()(4

1)( 1101

)4(

11 =+ uyDuyuuyf ,0)(4

1)(

4

1)( 1522

)4(

21 =+ uDuyuyuuyf

.9Dt <


40/215

33

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

110 =

D

DD ,01 f ,1j ,2D ,6D ,7D

,8D ,3

79

D

DD = ,

4

4

13

211

DD

DD

= ,

1

17612

D

jDDD

= ,

1

813

D

DD = ,

4 1

714

D

DD =

,4 1

215

D

DD = ( ) ,2

1

81176216

D

DDjDDDD = ( ) ,4

3

17617

D

jDDD = ( )131

3218

4DDD

DDD

=

and( )

( )131

176219

4

4

DDD

jDDDD

= are arbitrary constants. Furthermore, ( ) 03

461

3++ Dtjx

D

and ( ) .01714/1

9 ++=

DtjxDtu

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

this subcase, we require .0xh Accordingly, 0)( 1114/1

9

10++

DuyDt D

and

.0ln)( 14152 ++ DtDuy In addition, ( ) 0)( 191189110

+++ DtjxDDtuy D

and

( )( ) .04ln)( 14121154/1

142 +++++ DtDtjxDDtuy

Subcase (2) ( ) 08121 =++ DtjxDhD

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

02 =D in equation (2.68)1, system (2.68) yields only the constant solution. Hence under

transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the sole


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 0xh for system (2.71) to admit nonconstant solutions.

The subcases arising are

(1) [ ] ,0),(17

+ txbhDD (2) [ ] ,0),(716

=+ DtxbhDD (3) ,0),(17

=+ txbhDD

(4) ,0),(176 =+= txbhDDD (5) .0),(176 =+== txbhDDD


41/215


42/215

35

[ ] [ ] .0),(),()()( 88 912111019 =+++++++

x

c

D

t

xD

xxxxxx detbDbetxbDbDbDbftyDty

(2.75)

In results (2.74)-(2.75), [ ] 0),()()( + txKtyty and [ ] ,0),(),()( 88 ++ xD

etxbtxKtyD

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

inconsistent for this subcase, we require .0xh Furthermore, ( ) ,,),(8

=

x

c

DdtbetxK

),( txb satisfies equation (2.63) and ,06 D ,01 f ,c ,1D ,6

1

8D

DD =

,4

6

4

11

3

611

9D

DfDDjD

+= ,

6

11

10D

fDD = ,

2

6

2

11

11D

DfD =

3

6

3

11

3

61

12D

DfDjD

+= and 1j are

arbitrary constants.

Similarity Solutions for Case (d)


( ) ,812176 DtjxDhDhDhD tx ++=+ ;011 =++ txxxxx hhjhf (2.76)

where ,01 D ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants. As 0=xh

forces 0=th in equation (2.76)2 , giving =),( txh constant, we require 0xh for

system (2.76) to generate nonconstant solutions.


(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),( 1214

1

11109 ++=

=

DtjxDedetxh

n

tDxc

n

tD n (2.77)


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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 { }.4,3,2,1n

In addition, the travelling waves of velocity ,10D namely( )

,04

1

10 =

n

tDxc

nned are such

that ( ) 04

1

10 =

n

tDxc

nnnedc as the contradiction 01 =D otherwise occurs. Furthermore,

( ).013

4

1

1109 +

=

DedeD

n

tDxc

n

tD n We require ( ) 011

4

1

109 +=

Dedce

n

tDxc

nn

tD n as 0xh is

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

For the scenario ,176 jDD = ,0

4/1

17

11

=

fD

Dc ,0

4/1

17

12

=

fD

Dc

0

4/1

17

13

=

fD

Dic and ,0

4/1

17

14

=

fD

Dic where .1=i

Subcase (2) ,076 =DD ( ) 08121 ++ DtjxDhD

We solve system (2.76) via the method of Lagrange [24] and the integrating factor

algorithm [48]. Hence under transformations (2.2) and the conditions ,0)( 1 = fhf

0)( =hg and ,)( 1jhj = the similarity solution of the thin film equation (2.1) in tandem

with group (2.62) and the constraints ,07361 == DDDD ( ) 08121 ++ DtjxDhD

and ( ) 812),( DtjxDtxb += is

( ) ( ) ;0),( 1311291110 ++=

DtjxDeDtxh

tDxD (2.78)

where ,01 D ,06 D ,09 D ,06

110 = D

DD ,01 f ,2D ,8D ,3

6

3

61

3

1111

D

DjDfD

+

=

,1

212

D

DD =

2

1

628113

D

DDDDD

+= and 1j are arbitrary constants. Furthermore,

( ) 0121091110 +

DeDD

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

this subcase.


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37


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

causes equation (2.76)1to give the scenarios

(i) ,02 =D (ii) .176 jDD =

Scenario (i) 0281 ==+ DDhD

Under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj =

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

the constraints 081321 =+== DhDDDD and 8),( Dtxb = is the constant solution.

Scenario (ii) ,176 jDD = ( ) 08121 =++ DtjxDhD

Under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and ,)( 1jhj =

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

the constraints ,031 =DD ,176 jDD = ( ) 08121 =++ DtjxDhD and

( ) 812),( DtjxDtxb += is the travelling wave of velocity ,1j namely

( ) ;),( 1019 DtjxDtxh += (2.79)

where ,01 D ,2D ,8D ,1

29

DDD =

1

810

DDD = and 1j are arbitrary constants. We

require 09 D for solution (2.79) to be nonconstant.

From the constraint 176 jDD = on this case, it follows that 067 =DD forces ,01 =j

reducing the travelling wave (2.79) to a steady state solution.

Similarity Solutions for Case (e)


( ) ( ) ( ) ,34

81273613 DtjxDhDtDhDtjx

Dtx +=++

++ ;011 =++ txxxxx hhjhf (2.80)

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

073 +DtD and ( ) .034

613

++ DtjxD

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

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

solutions.


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38


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


similarity solution of the thin film equation (2.1) in tandem with group (2.62) and theconstraints 013 =DD and ( ) 812),( DtjxDtxb += is

( ) ;ln)(),( 91211110 DtDDtjxDuytxh ++++= (2.81)

satisfying

,0)(4

1)( 12

)4(

1 =+ Duyuuyf ,9Dt >

(2.82)

,0)(4

1

)( 12)4(

1 =+ Duyuuyf .9Dt <

In relations (2.81)-(2.82), ,03 D ,01 f ,2D ,6D ,7D ,8D ,3

79

D

DD = ,

4

3

210

D

DD =

( ),

4

3

17611

D

jDDD

=

( )2

3

83617212

4

D

DDDjDDD

+= and 1j are arbitrary constants,

( ) 034

613

++ DtjxD

and ( ) .01114/1

9 ++=

DtjxDtu For solution (2.81) to be

nonconstant, we require .0)( 104/1

9 ++

DuyDt

Similarity Solutions for Case (f)


( ) ,81276 DtjxDhDhD tx +=+ ;011 =++ txxxxx hhjhf (2.83)

where ,01 f ,2D ,6D ,7D 8D and 1j are arbitrary constants. As 0=xh forces 0=th

in equation (2.83)2 , forcing =),( txh constant, we require 0xh for system (2.83) to

generate nonconstant solutions.

As no similarity solutions arise for the thin film equation (2.1) when ,076 ==DD we

consider only the subcases

(1) ,07 D (2) .076 =DD

Subcase (1) 07 D

Via the method of Lagrange [24] and the Mathematica program [54], we obtain the

general solution of system (2.83). Hence under transformations (2.2) and the conditions,0)( 1 = fhf 0)( =hg and ,)( 1jhj = the similarity solution of thin film equation (2.1)


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39

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

( ) 812),( DtjxDtxb += is

( ) ( ) ( ) ( ) ,),( 2141312

2

91191021

91 tDtDxDtDxDtDxDeddtxh tDxc

++++++=

,176 jDD

(2.84)

( ) ( )[ ] ,),( 131125

0

1 tDtjxDtjxdtxhn

n

n ++==

.176 jDD =

In solutions (2.84), ,0

3/1

17

6171

=

fD

DjDc ,07 D ,01 f ,2D ,6D ,8D

,7

69

D

D

D = ,176

810

jDD

D

D = ( ) ,2 176

211

jDD

D

D = ,7

212

D

D

D = ,7

813

D

D

D =

( ),

22

7

176214

D

jDDDD

+= ,0d ,1d ,2d ,3d ,

24 17

84

fD

Dd =

17

25

120 fD

Dd = and 1j are


Furthermore, ( ) ( ) 022 9121101191211 +++ tDxc

edcDtDDDxD and

( ) 0125

1

1

1 +=

tDtjxnd

n

n

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

In addition, ( ) ( ) 0291

2110911 ++ tDxcedcDtDxD and ( ) .05

1

11 =

n

nn tjxnd

Subcase (2) 076 =DD


Therefore under transformations (2.2) and the conditions ,0)( 1 = fhf 0)( =hg and

,)( 1jhj = the similarity solution of thin film equation (2.1) in tandem with group (2.62)

and the constraints 07316 === DDDD and ( ) 812),( DtjxDtxb += is the

travelling wave of velocity ,1j namely

( ) ( ) ;),( 111102

19 DtjxDtjxDtxh ++= (2.85)

where ,06 D ,2D ,8D ,2 6

29

D

DD = ,

6

810

D

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

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


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GROUP (VIII)

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

yields Lie classical group (VIII), namely

,),,( 6Dhtx = ,),,( 7Dhtx = );,(),,( 1 txbhDhtx += (2.86)

such that

;0111 =++ txxxxxxx bbjbgbf (2.87)

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

Equation (2.87) admits the travelling wave solution of velocity ,1j namely

( ) ( ) ( );),( 1515 43121tjxdtjxd

ededtjxddtxb

+++= (2.88)

where ,01

15 =

f

gd ,01 f ,01 g ,1d ,2d ,3d 4d and 1j are arbitrary constants.

As equation (2.63) is a special case of equation (2.87) with ,01 =g solution (2.64) of

equation (2.63) is also a solution of equation (2.87) under the restriction .01 =g


We obtain similarity solutions of the thin film equation (2.1) for the cases

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

(b) ,01 D ,)(),( 812 DtjxDtxb += (c) ,01 =D .)(),( 812 DtjxDtxb +=

Similarity Solutions for Case (a)


),,(176 txbhDhDhD tx +=+ ;0111 =++ txxxxxxx hhjhghf (2.89)

where ,01 f ,1D ,6D ,7D 1g and 1j are arbitrary constants while ),( txb is an

arbitrary solution of equation (2.87). As 0=xh gives 0=th in equation (2.89)2, forcing

=),( txh constant, we require 0xh for system (2.89) to generate nonconstant

solutions.

The subcases occurring are

(1) [ ] ,0),(17 + txbhDD (2) [ ] ,0),( 716 =+ DtxbhDD (3) ,0),(17 =+ txbhDD

(4) ,0),(176 =+= txbhDDD (5) .0),(176 =+== txbhDDD


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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

txh tD

(2.90)

satisfying

.0),(),()()()()( 8891)4(1 8 =++++ utKDetxbuyDuyDuyguyf tD (2.91)

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

18

D

DD = ,

7

6179

D

DjDD

=

,7

610

D

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

( ) ,,),( 10108 +=

t

c

DdDtDxbeutK [ ] 0),(),()(88 ++ txbutKuyeD

tDand ),( txb

is an arbitrary solution of equation (2.87).

In addition, 0)( uy is a travelling wave of velocity 10D such that 0)( uy as

equation (2.91) otherwise leads to the contradiction [ ] .0),(),()(88 =++ txbutKuyeD tD

Furthermore, 0)(

+

x

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

for this subcase.

This subcase includes subcase (1) of case (c) in relation to group (2.62) as results (2.72)-

(2.73) and equation (2.63) are a special case of results (2.90)-(2.91) and equation (2.87)

respectively under the restriction .01 =g

Subcase (2) [ ] 0),( 716 =+ DtxbhDD


equation (2.89)1, substituting its general solution into equation (2.89)2. Therefore under

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

similarity solution of the thin film equation (2.1) in tandem with group (2.86) and theconstraints [ ] 0),( 716 =+ DtxbhDD (with ),( txb satisfying equation (2.87)) is


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42

[ ] ;0),()(1

),( 8

6

+= txKtyeD

txh xD (2.92)

such that

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

912111019 =+++++++

x

c

D

t

xD

xxxxxx detbDbetxbDbDbDbftyDty

(2.93)

In results (2.92)-(2.93), [ ] 0),()()( + txKtyty and [ ] ,0),(),()( 88 ++ xD

etxbtxKtyD

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

Furthermore, ( ) ,,),( 8

=

x

c

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

,01 f ,c ,1g ,1j ,1D ,6

18

D

DD =

( ),

4

6

3

611

2

611

4

119

D

DDjDDgDfD

+= ,

6

1110

D

fDD =

2

6

2

61

2

1111

D

DgDfD

= and

3

6

3

61

2

611

3

1112

D

DjDDgDfD

+= are arbitrary constants.

This subcase includes subcase (2) of case (c) in relation to group (2.62) as results (2.74)-

(2.75) and equation (2.63) are a special case of results (2.92)-(2.93) and equation (2.87)

respectively under the restriction .01 =g

Similarity Solutions for Case (b)


( ) ,812176 DtjxDhDhDhD tx ++=+ ;0111 =++ txxxxxxx hhjhghf (2.94)

where ,01 D ,01 f ,2D ,6D ,7D ,8D 1g and 1j are arbitrary constants. As 0=xh

forces 0=th in equation (2.94)2 , giving =),( txh constant, we require 0xh for

system (2.94) to admit nonconstant solutions.


(1) ,07 D ( ) ,08121 ++ DtjxDhD (2) ,076 =DD ( ) ,08121 ++ DtjxDhD

(3) ( ) .08121 =++ DtjxDhD

Subcase (1) ,07 D ( ) 08121 ++ DtjxDhD

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

Mathematica program [54], we solve system (2.94). Hence under transformations (2.2)

and the conditions ,0)( 1 = fhf 1)( ghg = and ,)( 1jhj = the similarity solution of the


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43

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

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

( ) ( ) ;0),( 1214

1

11109 ++=

=

DtjxDedetxh

n

tDxc

n

tD n (2.95)

where ,01 D ,07 D ,07

19 =

D

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

7

610

D

DD =

,1

211

D

DD =

( )2

1

17628112

D

jDDDDDD

+= and

( )

1

176213

D

jDDDD

= are arbitrary

constants for all { }.4,3,2,1n

Furthermore, ( ) 04

1

10 =

n

tDxc

nnnedc as the contradiction 01 =D otherwise arises. In

addition,( )

013

4

1

1109 +

=

DedeD

n

tDxc

n

tD n and as 0xh is necessary for equation (2.94)1

to be consistent for this subcase, ( ) .011

4

1

109 +=

Dedce

n

tDxc

nn

tD n

This subcase includes subcase (1) of case (d) for group (2.62) as solution (2.77) is a

special case of solution (2.95) with .01 =g

Subcase (2) ,076 =DD ( ) 08121 ++ DtjxDhD

Via the method of Lagrange [24] and the integrating factor algorithm [48], we obtain the

general solution of system (2.94). Therefore under transformations (2.2) and the

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

film equation (2.1) in tandem with group (2.86) and the constraints ,0761 =DDD

( ) 08121 ++ DtjxDhD and ( ) 812),( DtjxDtxb += is the sum of two travelling

waves with respective velocities 1j and ,11D namely

( ) ( ) ;0),( 1311291110 ++=

DtjxDeDtxh

tDxD (2.96)

where ,01 D ,06 D ,09 D ,06

110 =

D

DD ,01 f ,2D ,8D

,3

6

3

61

2

611

3

1111

D

DjDDgDfD

+= ,

1

212

D

DD = ,

2

1

628113

D

DDDDD

+= 1g and 1j are



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44

Furthermore, ( ) 0121091110 +

DeDD tDxD

as ( ) 08121 ++ DtjxDhD and as we require

0xh for equation (2.94)1 to be consistent for this subcase. For the case

,02

1011 = Dfg solution (2.96) reduces to a single travelling wave of velocity .1j

This subcase incorporates subcase (2) of case (d) for group (2.62) as solution (2.78) is a



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

causes equation (2.94)1to give the scenarios

(i) ,02 =D (ii) .176 jDD =

Scenario (i) 0281 ==+ DDhD

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

the constant solution is the sole similarity solution of the thin film equation (2.1) in

tandem with group (2.86) and the constraints 08121 =+= DhDDD and .),( 8Dtxb =

Scenario (ii) ( ) ,08121 =++ DtjxDhD 176 jDD

=

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

the similarity solution of the thin film equation (2.1) in connection with group (2.86) and

constraints ,01761 = jDDD ( ) 08121 =++ DtjxDhD and ( ) 812),( DtjxDtxb +=

is the travelling wave of velocity ,1j namely

( ) ;),( 1019 DtjxDtxh += (2.97)

where ,01 D ,2D ,8D ,1

29

D

DD =

1

810

D

DD = and 1j are arbitrary constants. For

solution (2.97) to be nonconstant requires .09 D

Subcase (3) of case (b) for group (2.86) generates results identical to those of subcase (3)

for case (d) in relation to group (2.62).


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45

Similarity Solutions for Case (c)


( ) ,81276 DtjxDhDhD tx +=+ ;0111 =++ txxxxxxx hhjhghf (2.98)

where ,01 f ,2D ,6D ,7D ,8D 1g and 1j are arbitrary constants. As 0=xh gives

0=th in equation (2.98)2 , forcing =),( txh constant, we require 0xh for system

(2.98) to yield nonconstant solutions.

As no similarity solutions occur for the thin film equation (2.1) when ,076 ==DD we

consider only the subcases

(1) ,07 D (2) .076 =DD

Subcase (1) 07 D

By the method of Lagrange [24] and the Mathematica program [54], we solve system

(2.98) for the case .01 g Hence under transformations (2.2) and the conditions

,0)( 1 = fhf 0)( 1 =ghg and ,)( 1jhj = the similarity solution of the thin film

equation (2.1) in tandem with group (2.86) and the constraints 017 =DD and


( ) ( ) ( ) ( ) ,),( 21413122

911910

4

2

19 tDtDxDtDxDtDxDeddtxh

n

tDxc

nn ++++++=

=

,176 jDD

(2.99)

( ) ( ) ( ) ( )[ ] ,),( 131126

5

14

1

1

1115 tDtjxDedtjxdtxh

n

tjxD

n

n

n

n

n

+++= =

=

.176 jDD =

In solutions (2.99), ,07

D ,01

1

15

=

f

gD ,0

1

f ,01

g ,2

c ,3

c ,4

c ,n

d ,1

j ,2

D

,6D ,8D ,7

69

D

DD =

( )

( ),

2

176

172176810

jDD

gDDjDDDD

=

( ),

2 176

211

jDD

DD

= ,

7

212

D

DD =

7

813

D

DD = and

( )2

7

176214

2D

jDDDD

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

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

( ) ( ) ( ) ( ) ( ) 011 126

5

1

15

4

2

2

1115 ++

=

=

tDeDdtjxdn

n

tjxDn

n

n

n

n

n

and


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46

( ) ( ) .022 1011912114

2

9 +++=

DtDDDxDedc

n

tDxc

nnn Nonconstancy of solutions (2.99)1

and (2.99)2 further requires( ) ( ) 02 10911

4

2

9 ++=

DtDxDedc

n

tDxc

nnn and

( ) ( ) ( ) ( ) ( ) 011 6

5

1

15

4

2

2

1115 +

=

=

n

tjxDn

n

n

n

n

n

eDdtjxdn respectively.

Solutions (2.84) are the similarity solutions of the thin film equation (2.1) for this

subcase when 01 =g (in tandem with group (2.86) under transformations (2.2) and the

conditions ,0)( 1 = fhf 0)( =hg and 1)( jhj = ).

Subcase (2) 076 =DD

We directly solve equation (2.98)1, substituting its general solution into equation (2.98)2

and solving the resulting equation. Hence under transformations (2.2) and the conditions

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

(2.1) in association with group (2.86) and the constraints 0716 == DDD and


( ) ( ) ;),( 12111102

19 DtDtjxDtjxDtxh +++= (2.100)

where ,06 D ,2D ,8D ,2 6

29

D

DD = ,

6

810

D

DD = ,

6

1211

D

gDD = ,12D 1g and 1j are

arbitrary constants. For solution (2.100) to be nonconstant requires ( ) .0812 + DtjxD

This subcase includes subcase (2) of case (f) for group (2.62) as solution (2.85) is a


The infinitesimal generators 821 ,...,, VVV denote the Lie algebras for the respective Lie

groups (I), (II),, (VIII); (see Gandarias [27]). These generators are as follows.


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47

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

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

,761 tDxDV

+

=

( )[ ] ( ) ,43 746142t

DtDx

DtjxDV

++

++=

( )[ ] ( ) ,270260023h

Dt

DtgDx

DtjxgDV

+

++

++=

( ) ,433

27102

62110

24h

Dt

DtjfD

xDtjjx

jfDV

+

++

+

=

( )[ ] ( ) ( ) ,2170160015h

fhDt

DtgDx

DtjxgDV

++

++

++=

( )[ ] ( ) ( ) ,43

33

217101

621101

6h

fhDt

DtjfD

xDtjjxjf

DV

++

++

+=

( ) ( ) ( )[ ] ,,34

173613

7h

txbhDt

DtDx

DtjxD

V

++

++

++=

( )[ ] ;,1768h

txbhD

t

D

x

DV

++

+

=

where details of 821 ,...,, VVV relate to the respective groups (I), (II),, (VIII).

Next, we present four tables of results. Table 1 features the functions ),(hf )(hg and

)(hj (distinguishing the Lie classical symmetries of the thin film equation (2.1)) with

their associated infinitesimal generators .iV Table 2 is a dimensional classification of the

mathematical structure of groups (I)-(VIII) and the corresponding iV. Table 3 displays

the similarity solutions ),( txh with their similarity variables (where applicable) for the

thin film equation (2.1) in conjunction with groups (I)-(VIII). Table 4 shows the defining

ordinary differential equations (ODEs) for the functions within the functional forms of

),( txh relating to groups (I)-(VIII) in table 3.


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2.3 TABLES OF RESULTS

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

symmetries of thin film equation (2.1)) with the associated infinitesimal generator .i

V

Group )(hf )(hg )(hj i

V

I arbitrary 0 arbitrary arbitrary1V

II arbitrary 0 01j 2V

III 003

1 hg

ef hg

eg 01 10 jhj + 3V

IV 001 hf

ef hjf

eg 32

1

10+

20

1 jej hj

+ 4V

V ( ) 00321 + g

fhf ( ) 021g

fhg + 120 ln jfhj ++ 5V

VI ( ) 00

21 + f

fhf ( ) 322110

jf

fhg+

+ ( ) 2201 jfhj

j

++ 6V

VII 01 f 0 1j 7V

VIII 01 f 1g 1j 8V

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

(Lie classical symmetries of thin film equation (2.1)) with their associated infinitesimal

generators .iV

Group ),,( htx ),,( htx ),,( htx iV

I6D 7D 0 1V

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

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

IV621

102

3Dtjjx

jfD +

( ) 710

2 43

DtjfD

+ 02 D 4V

V ( ) 6001 DtjxgD ++ 701 DtgD + ( ) 021 + fhD 5V

VI( )[ ] 62110

1 33

DtjjxjfD

+ ( ) 7101 4

3Dtjf

D+

( ) 021 + fhD 6V

VII( ) 61

3 34

DtjxD

++ 73 DtD + ( )txbhD ,1 + 7V

VIII6D 7D ( )txbhD ,1 + 8V


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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 ( ) 01214/1

11 +

DtjxDt

3(1)0ln

1)( 11

0

+ Dt

g

uy

under the constraint 00 g

0ln 111611

1716

++DtD

Dt

DtDx

3(2i) 0)( 11 + tDuy

under the constraints 007 =gD

0132

12 ++ tDtDx

3(2ii) (

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