55

CHAPTER-4

LIE GROUP METHOD FOR SIMILARITY SOLUTION OF

DIFFERENT EQUATIONS

4.0 INTRODUCTION:

The investigations presented in the present chapter are an outgrowth of a

continuing study of ways in which partial differential equations associated with

problems of physical interest may be simplified through transformation of

independent and dependent variables given by Na et al ((1967) ,( 1968),( 1969)).

First it was noted that as it was already discussed in chapter-1, the usual types of

transformation for simplifying partial differential equations of physical problems

were usually of a rather special class. This led to the question of possibly

generalizing the class of transformations noted. A second area of investigation

centered on broadening the definition of similarity to include other transformations

which change a given problem into a simpler problem in some sense or other

instead of the usual interpretation of similarity which implies simply a reduction in

the number of variables of a given problem.

It has been known for some time that Birkhoff (1950), Morgan (1952) and

Hansen (1964) the theory of continuous transformation groups gives promise of

providing a very general method of analysis. The method is by no means new. In

fact, the basic ideas date back to the last century and are found in the work of the

mathematician Sophus Lie (1881). Moreover, the theory of groups has been

applied quite extensively in recent times by investigators in the field of similarity

analysis. Nevertheless, there still exists a need for a truly in depth studies of the

application of Lie’s theory to the similarity analysis of problems other than the

reduction of the number of variables in a partial differential equation given by Von

Muller and Matschat (1962),Cohen (1917) and Bluman and Cole (1969). It

therefore seemed reasonable to find how far Lie’s ideas might be pursued in

formulating a very general approach to similarity analyses.

56

It is well-known fact that the theory of continuous groups was first applied

to the solution of partial differential equations by Birkhoff (1950) who used a one-

parameter group. Morgan (1952) proved a theorem which established the

condition under which the number of variables can be reduced by one. Morgan’s

theorem was later extended by Michal (1952) to similarity transformations which

reduce the number of independent variables by more than one.

One of the difficulties of the approach by Birkhoff and Morgan is that it is

based on particular given groups of transformations. To begin with, a group has to

be arbitrarily assumed and Morgan’s theorem can then be used to establish

whether or not the differential equation transforms conformally under this

particular group. If it does, similarity solutions will then exist and the similarity

variables can then be taken as the functionally independent invariants of this

group. The question which remains is whether or not there are other groups under

which similarity solutions exist.

In this chapter, the idea of Lie’s infinitesimal contact transformation groups

is applied to develop a method for searching possible groups of transformations,

instead of arbitrarily assuming one at the outset. An alternative approach based on

an extended Morgan’s method in searching for possible groups of transformations

has recently been developed by Moran et al. (1968) which starts from a slightly

less general transformation than the contact transformation. It has also been

applied to various types of problems, among which are the reduction of differential

equations to algebraic equations given by Moran et al (1968) and its application to

dimensional analysis given by Moran et al (1968). The method is then applied to a

broader class of similarity analyses: namely, the similarity between partial and

ordinary differential equations, boundary and initial value problems and nonlinear

and linear differential equations.

57

4.1 THE INFINITESIMAL TRANSFORMATION: A group is said to be continuous if, between any two operations of the

group, a series of operations within the group can always be found of which the

effect of any operation in the series differs from the effect of its previous operation

only infinitesimally. The concept of infinitesimal transformations comes as a

natural consequence of the definition of a continuous transformation group. An

infinitesimal transformation is one whose effects differ infinitesimally from the

identical transformation. Thus, any transformation of a finite continuous

transformation group which contains the identical transformation can be obtained

by infinite repetition of an infinitesimal transformation.

Let the identical transformation be

x x, y, a x ; y x, y, a y (4.1)

Where a is a particular value of the general parameter a. Then the

transformation

X x, y, a x ; y x, y, a y (4.2)

Where is an infinitesimal quantity, defines an infinitesimal transformation in a

broad sense. Expanding equation (4.2) in Taylor series, we get

x x, y, a

x, y, a ! a a

! ²

a² a …, (4.3a)

y x, y, a

x, y, a ! a a

! ²

a² a …, (4.3b)

Since is infinitesimal, higher-order terms of can be neglected,

equation (4.3) then becomes

x x ξ x, y , y y η x, y (4.4)

58

Where

ξ a a and η a a

(4.5a, b)

The employment of the infinitesimal transformation, equation (4.4), in

conjunction with the function x, y will be to transform x, y into x , y

which, upon expanding in Taylor series, becomes:

x , y x ξ , y η

x, y ! U

! U …. (4.6)

Where

U x

y (4.7)

is called the group representation and Un means repeating the operator U for n

times.

The function x, y is said to be an invariant function under the

infinitesimal transformation, equation (4.4), if x, y x , y . It can be

shown by Na et al (1967) and Ince (1956) from equation (4.6) that the necessary

and sufficient condition that the function x, y be invariant under the

infinitesimal group of transformation represented by U is U 0 : i. e.

x

y 0 (4.8)

To determine the invariant function, it is necessary to solve equation (4.8)

by the method of Lagrange given in the theories of linear differential equations.

Thus, we solve the related differential equation

dx dy (4.9)

59

If the solution is Ω x, y constant, this function is the invariant function

of the infinitesimal transformation represented by U . Since equation (4.9) has

only one independent solution depending on a single arbitrary constant, a one-

parameter group in two variables has one and only one independent invariant.

In the case of n variables, all the theories corresponding to two variables can

be generalized by following the same pattern. For example, if a function of n

variables x ,… . , xn is invariant under the infinitesimal transformation

x'i xi i x , … . , xn (i = 1, n) (4.10)

then a necessary and sufficient condition is again U 0 which, in its expanded

form is

1 x ,… . , xn x … n x , … . , xn xn

0 (4.11)

Following the same reasoning as in two-dimensional case, the invariant

functions can be obtained by integrating the following equations:

dx dx dxn

n (4.12)

Since there are (n-1) independent solutions to equation (4.12) a one-

parameter group in n variables has (n-1) independent invariants. The invariant

functions are therefore

Ωm x ,… . , xn cm, m 1,… . . , n 1 , (4.13)

and are the solutions to the system of equations given by equation (4.12).

4.2 DIFFERENTIAL EQUATIONS ADMITTING A GIVEN

GROUP OF TRANSFORMATIONS:

Consider now a function

F F x , … . , xm; y , … . , yn; kyx1 k , …… ,

kynxm k (4.14)

60

The argument of which, assumed p in number, contain derivatives of yj up

to order k. Such a function is known as a differential form of the kth order in m

independent variables by Hansen (1964). Designate the arguments by z ,… . , zp,

e.g.

z x , z2 x2, … . . , zp‐ kynxm‐1

k , zp kynxm

k

Equation (4.14) can be written in a simpler form as F F z , … . , p which

is said to admit of a given group represented by

U 1 z ,… . , zp z … p z , … . , zp zp

(4.15)

if it is invariant under this group of transformation. Therefore, the function F

admits of a group if UF 0, or

1Fz

… p Fzp 0 (4.16)

It was shown in the preceding article that there are (p-1) functionally

independent solutions, or invariants, to this equation, namely,

m Ωm z ,… . , zp constant, m 1,… . . , p 1 .

(4.17)

An important theorem will be quoted here without proof. Based on this

theorem, if a differential equation F z , … . , p 0 is invariant under the

infinitesimal transformation, it must be expressible in term of the (p-1)

functionally independent invariants. Thus, we write

F z , … . , p G η ,… . , p‐1 0, (4.18)

Where they s are given by equation (4.17).

61

For a given group of transformations, the transformation functions, i , in

equation (4.10) are known functions. In the original group-theoretic method

developed by Birkhoff (1950), and Morgan (1952), two groups, namely, the

linear and spiral groups were considered. For the linear group i Cizi i

1, … , p whereas for the spiral group 1 C1 and i Cizi i 2, … , p . The

theories outlined above are enough for the determination of possible similarity

transformations given by Na et al (1967). The similarity transformations thus

obtained, however, correspond only to the two particular groups of

transformations. Since there is no proof that these two groups are the only two

possible for similarity solutions to exist for a given partial differential equation, it

is still necessary to raise the question: Given a partial differential equation, what

are all possible groups of transformations that make similarity solution possible?

In other words, are there other groups other than the linear or the spiral groups?

To answer questions of this kind, we shall develop a systematic procedure in

searching for all possible groups of transformations by using Lie’s theories of

infinitesimal contact transformation groups. Although the concepts were

introduced by Lie in the latter part of the nineteenth century, its significance in the

solution of nonlinear differential equations has not been fully explored.

4.3 CONTACT TRANSFORMATIONS:

Before entering into discussion of the contact transformations, the concept

of an extended group has to be introduced. Consider the one-parameter group of

transformation:

x x, y, a , y x, y, a (4.19)

Suppose y is regarded as a function of x, then if the differential coefficient p (=

dy/dx) be considered as a third variable, it will be transformed to p by

p dydx

= / x / y p / x / y p

x, y, p, a (4.20)

62

For the infinitesimal transformation defined by equation (4.4), the

transformed coefficient p can be shown to be

p p ! x, y, p (4.21)

Where

x, y, p x

y

xp

y p (4.22)

Therefore, the infinitesimal transformation given by

x x δε ξ x, y , y y δε η x, y ,

p p δε x, y, p (4.23)

is represented by

UF ξx

y

p (4.24)

Extension of this concept to higher-order derivatives can be made by the

same reasoning.

The group of transformations defined in equation (4.23) where the

transformation functions and η are functions of x and y only is the so-called “the

point transformation” which is not the most general type of transformation. Lie

(1875) and Cohen (1917) defined the so-called “contact transformation” in which

and η are functions of x, y and p, i.e.

x x δε ξ x, y, p , y y δε η x, y, p ,

p p δε x, y, p

(4.25)

The above defined transformation is the most general type of

transformation. The abstractness and the complexity of the theories of contact

transformations prevent any extensive discussion. Here, we merely state some

Lie’s theorems without proof. The detail can be found from the work of Lie

(1875), Cohen (1917) and Na et al (1967). The definition of contact

63

transformation, as given by Lie (1875) is as follows: When Z, X , … . , Xn,

P , … . , Pn are 2n+1 independent functions of the 2n+1 independent quantities

z, x , … . , xn, p , … . , pn

Such that the relation

dZ ‐ Pi dXi dz ‐ pi dxi (4.26)

(Where does not vanish) is identically satisfied then the transformation defined

by

Z Z z, x , … . , xn, p ,… . , pn Xn Xn z, x , … . , xn, p , … . , pn

Pn Pn z, x , … . , xn, p ,… . , pn (4.27)

is called a contact transformation. It will transform a partial differential equation in

z, x , … . , xn, p , … . , pn into one in Z, X , … . , Xn, P , … . , Pn and also the solution

of the first partial differential equation into the solution of the second.

From equation (4.26), the transformation defined by equation (4.27)

satisfies the following relation:

Zz dz Z

xi dxi

Zpi dpi ‐ Pi

Xiz dz Xi

xr dxr

Xipr dpr ρ dz ‐ pidxi

(4.28)

For the infinitesimal transformation

Z z δε ζ z, x , p , Xi xi δε i z, x , p ,

Pi pi δε i z, x , p

(4.29)

64

We get:

ζz‐ pi i

z 0, ζ

pr‐ pi i

pr 0,

ζxr‐ pi i

xr ‐ πr ‐ σpr

(4.30a, b, c)

If a characteristic function, W is defined as W pi i then equation

(4.30) gives

r Wpr, pi

Wpi‐ W, πr ‐ W

xr ‐ pr

Wz

(4.31a, b, c)

To get the transformation functions for higher-order derivatives, we

consider the transformation defined by equation (4.29), adding the following

higher-order terms:

pjk pjk πjk z, x , p , p s (4.32)

And

pjkl pjkl πjkl z, x , p , p s, p st (4.33)

By definition, we write

dpk pjk dXj (4.34)

Which upon substitution of the infinitesimal transformation equations (4.29),

(4.32) and (4.33) and subtracting the quantity j psjk dxs from both sides gives

πjk ddxj πk‐ pjk j (4.35)

Or in terms of j and πk,

πjk πkxj

πkz pj πk

p pµj ‐ pjk

ξixj ξi

z pj ξi

p pµj (4.36) to

express πjk in terms of the characteristic function W, it is necessary to put

expressions for πk and i from equation (4.31) into equation (4.36).

65

Similarly, for the third-order function πjkl, we get

πjkl ddxl πjk‐ pjkl l (4.37)

Or

πjkl πjkxl

πjkz pl πjk

pµ pµl πjk

pbc pbcl-

‐ pjklξtxl ξt

z pl ξt

pµ pµl ξt

pbc pbcl

(4.38)

In order to express πjkl in terms of the characteristic function W, we have to

express πk and i in terms of W, as in equations (4.31) and (4.36) and substitute

into equation (4.38).

4.4 SIMILARITY ANALYSIS OF DIFFUSION EQUATION:

We shall now look at the diffusion equation from another point of view,

namely, the searching for all possible groups of transformation that will reduce the

diffusion equation to an ordinary differential equation. In applying such a

technique to a given differential equation, it may turn out that for some or all of

the groups other than the linear and spiral groups, the boundary condition cannot

be transformed although the partial differential equation can be transformed into

an ordinary differential equation. For such cases, we are at least assured that the

groups of transformations that remain are the groups possible for the given

boundary value problems. A similarity analysis of the diffusion equation from this

point of view is apparently not covered in the literature. The one-dimensional form

of the diffusion equation in rectangular coordinate is chosen because of its

simplicity. Extension of analyses to equations expressed in other coordinates can

readily be made.

66

4.4.1 LIE GROUPANALYSIS:

Consider the diffusion equation

ut – υ u

y 0 (4.39)

On which an infinitesimal transformation is to be made on the dependent

and independent variables and derivatives of the dependent variable with respect

to the independent variable. The infinitesimal transformation is

t' t δε ξ t, y, u, p, q

y' y δε η t, y, u, p, q

u' u δε ζ t, y, u, p, q

p' p δε t, y, u, p, q

q' q δε t, y, u, p, q

p' p δε t, y, u, p, q, p , p , p (4.40)

Where in terms of the characteristic function W,

Wp

Wq

p Wp

q Wq – W

‐ Wt

p Wu

‐ Wy

q Wu

(4.41)

‐ Wy²

2q Wy u

q Wu² 2p W

y pq W

u p

67

2p Wy q

q Wu q

p Wp²

2p p Wp q

p Wq²

p Wu

The characteristic function W, is a function of t, y, u, p and q. We note that

p ut , q u

y , p u

y² , p u

t y (4.42)

Now, the necessary and sufficient condition that a partial differential

equation F t, y, u, p, q, p ,p 0 invariant under the group of transformation

represented by Uf is UF = 0 which for the diffusion equation is

U p ‐ υp 0 (4.43)

Or

Expanding the above expression by employing the operator U:

t

y

u

p

q

p

p

p 0 (4.44)

Where the parenthesis represents the differential equation

p ‐ υp 0 (4.45)

Carrying out the operation in equation (4.44) yields:

‐ υ 0 (4.46)

Upon substituting expressions from equation (4.41) into equation (4.46) yields

‐ Wt – p W

u υ W

y² 2υq W

y u υq W

u²

2p Wy p

υ 2p q Wu p

υ 2p Wy q

2pq Wu q

68

υ p Wp² 2p p W

p q pυ

Wq²

p Wu 0

(4.47)

Equation (4.47) is seen to be a linear partial differential equation in

W (t, y, u, p, q ).

Since W is not a function of p , the coefficients of the terms involving p

and p should be zero. We then get

Wp² 0 (4.48)

Wy p

q Wu p

pυ

Wp q

0 (4.49)

Equation (4.48) indicates that W is a linear function of p. Thus we can write

W W t, y, u, q p W t, y, u, q (4.50)

Substituting this form of W into equation (4.49), we get

Wy q W

u p

υ W

q 0 (4.51)

Since W is not a function of p, the coefficient of p in equation (4.51) must

be zero. We therefore obtain two equations namely,

Wq 0 (4.52)

Wy q W

u 0 (4.53)

Equation (4.52) indicates that W is not a function of q i.e. W

W t, y, u and so equation (4.53) can be broken into the two equations,

Wy 0 (4.54a)

And

69

Wu 0 (4.54b)

This means W is independent of both y and u and, as a result,

W W t (4.55)

And the characteristic function now takes the form

W W t, y, u, q p W t (4.56)

Putting this form of W into equation (4.47), we get

‐ Wt – p W

t υ W

y² 2υq W

y u υq W

u²

2p Wy q

2pq Wu q

pυ

Wq² 0

(4.57)

Since both W and W are independent of p, equation (4.57) can be

separated into three equations, corresponding to the coefficients of p , p and p .

We then get

p ‐ Wt υ W

y² 2υq W

y u υq W

u² 0

(4.58)

p ‐ Wt 2 W

y q 2q W

u q 0 (4.59)

p Wq² 0 (4.6o)

From equation (4.60), W is linearly dependent on q, therefore, it becomes

W W t, y, u W t, y, u q (4.61)

Putting this form of W into equation (4.59), we get

‐ Wt 2 W

y 2q W

u 0 (4.62)

70

Both W and W are independent of q, therefore, equation (4.62) becomes

‐ Wt 2 W

y 0 (4.63)

Wu 0 (4.64)

From equation (4.64), W is independent of u. Also, since W is a function

of t only, equation (4.63) shows that W depends linearly on y i.e.

W W t W t y (4.65)

Equation (4.63) then becomes

‐ dWdt

2W 0 (4.66)

We will make use of this equation later.

The characteristic function W, now becomes

W W t, y, u W t W t y q W t p (4.67)

Putting equation (4.61) into equation (4.58), we get

‐ Wt ‐ dW

dtdWdt

y q υ Wy²

2υq Wy u

υq Wu²

0

(4.68)

Since W , W and W are independent of q terms with different powers

of q are grouped and their coefficients are put equal to zero. Three equations are

obtained:

q ‐ Wt

υ Wy²

0 (4.69)

q ‐ dWdt

‐ dWdt

y 2υ Wy u

0 (4.70)

q Wu²

0 (4.71)

71

Equation (4.71) shows that W is linearly dependent on u. Therefore,

W W t, y W t, y u (4.72)

Equation (4.70) then gives

‐ dWdt

‐ dWdt

y 2υ dWy 0 (4.73)

Therefore, W can be written as

W W t W t y W t y (4.74)

Equation (4.73), then become

‐ dWdt

‐ dWdt

y 2υW 4υyW 0 (4.75)

Since all the W’s in equation (4.75) are independent of y, we get

‐ dWdt

2υW 0 (4.76)

‐ dWdt

4υW 0 (4.77)

Putting W into equation (4.69), we get

‐ Wt

υ Wy²

‐ dWdt

dWdt

y dWdt

y u 2υW u 0

(4.78)

Equation (4.78) can be separated into:

u ‐ Wt

υ Wy²

0 (4.79)

u ‐ dWdt

2υW 0 (4.80)

dWdt

0 (4.81)

dWdt

0 (4.82)

72

From equations (4.81) and (4.82),

W c (4.83a)

And

W c (4.83b)

From equation (4.80),

W 2υc t c (4.84)

From equations (4.76) and (4.77),

W 2υc t c (4.85)

W 4υc t c (4.86)

From equation (4.66),

W 4υc t 2c t c (4.87)

The final form of the characteristic function is therefore

W t, y, u, p, q W t, y 2υc t c c y c y u

2υc t c 4υc t c y q 4υc t 2c c p

(4.88)

Where W t, y is any function satisfying equation (4.79), i.e.

Wt ‐ υ W

y² 0 (4.89)

The characteristic function, W given in equation (4.88) will now be used to

determine the absolute invariants.

73

4.4.2 ABSOLUTE INVARIANTS AND THE TRANSFORMED

EQUATIONS:

With the characteristic function W, obtained as in equation (4.88), we now

make use of the general theory to find the absolute invariants. From equation

(4.12), the following relations are obtained:

dt dy du (4.90)

Where the transformation functions , and ζ can be obtained by putting

into equation (4.41), the characteristic function W given by equation (4.88).

Equation (4.90) then becomes

dt 4υc t 2c t c

dy2υc t c 4υc t c y

du‐ W t, y ‐ 2υc t c c y c y u

(4.91)

The number of possible groups is large, due to the fact that all c’ s are

arbitrary and W t, y is an arbitrary function satisfying equation (4.79).

Therefore, we investigate a few special cases of the parameters. Other groups can

be obtained in a similar manner.

CASE 1: W t, y c c c c

Equation (4.91) becomes

dt

2c t dy

c y du

‐ c u (4.92)

The two independent solutions to equation (4.92) are

74

y√t constant

And

ut

constant; α ‐ c3c5

According to the theories in the preceding articles, the diffusion equation

can be expressed in terms of these two invariants i.e.

y√t

(4.93a)

And

f η ut

(4.93b)

The diffusion equation is then transformed into an ordinary differential equation

υf" αf ‐ f' (4.94)

The transformation is seen to be the linear group of transformations.

CASE 2: W t, y c c c c

Equation (4.91) then becomes

dt c

dy 0 ‐ du

c u (4.95)

And

y = constant

Following the same arguments as in case 1, we get absolute invariants as

y and f η ue; ‐ c3c6

(4.96a, b)

And the diffusion equation is transformed to

75

υf" ‐ βf 0 (4.97)

The transformations are seen to be the spiral group.

CASE 3: W t, y c c c c c

Equation (4.91) then becomes

dt 0

dy υ c t

du ‐ c yu

And

t = constant (4.98a, b)

The absolute invariants are

t And f η u

e yυt

(4.99a, b)

And the transformed equation is

2 f' f 0 (4.100)

CASE 4: W t, y c c c

Equation (4.91) then becomes

dt c

dy c

du ‐ c u

(4.101)

The absolute invariants are

y ‐ c4c6 t And f u e

c3c6t (4.102a, b)

The diffusion equation becomes

υf" c4c6 f' c3c6

f 0 (4.103)

76

The above four cases are examples of cases where the solution of equation

(4.91) is straightforward, i.e., the two independent solutions can be solved by

simple pairing of equations. The following cases are those in which the two

solutions have to be solved in a sequence of steps.

CASE 5: W t, y c c c c c

Equation (4.91) now becomes

dt

4υt dy

υty du

– 2υt y u (4.104)

The first of the two equations gives the solution

yt

k (4.105)

Next, replacing t in the last two terms of equation (4.104) by k y [based on

equation (4.105)], we get

‐ 2υk1y y 4υk1y

dy du

(4.106)

The solution of equation (4.106) is

uy / ey/ υk constant

Or using equation (4.105) again,

uy / ey / υt k (4.106a)

According to the applicable theorems, the absolute invariants are, from equations

(4.105) and (4.106a)

yt And f η uy / ey / υt (4.107a, b)

The diffusion equation is then transformed into

f" – η f' f 0 (4.108)

77

CASE 6: W t, y c c c c

Equation (4.91) becomes

dt c

dy υ c t

du ‐ c yu

(4.109)

The first two terms give

y ‐ υc c

t k (4.110)

Where k is the constant of integration. Combining the first and the third term and

making use of equation (4.110), we get

uec1c6 k

υc1 c6

t k

Where k is the constant of integration. Using equation (4.110), we get

uec1c6 yt ‐

υc1 c6

t k (4.111)

Equations (4.110) and (4.111) give the invariants,

y ‐ υc1 c6 t

And

f η uec1c6 yt ‐

υc1 c6

t (4.112a, b)

The diffusion equation is transformed to

f" c1c6υ f 0 (4.113)

CASE 7: W t, y c c c c

Equation (4.91) becomes

dt

2c t dy υ c t c y

du ‐ c yu

(4.114)

78

By following the same steps as in the two previous cases, the invariants are found

to be

y√t ‐ 2υ c1 c5

√t

And

f η uecc y ‐

υc12

c52 t (4.115a, b)

The diffusion equation is then transformed to

υf" f' 0 (4.116)

In all the above cases, W t, y was taken to be zero. This is not necessary

as we shall show in the following two cases.

CASE 8: W t, y υa t a y

c c c c c 0

It can be shown easily that this functional form of W t, y satisfies

equation (4.89). For this case, equation (4.91) becomes

dt

2c t dy

c y du

υa1t ‐ 12 a1y2

(4.117)

Using the same method as in previous cases, we find

y√t

And

f η u a1 2υt y2

c (4.118a, b)

The diffusion equation is transformed to

79

υf" f' 0 (4.119)

CASE 9: W t, y a υa t a y

c c c c c 0

The only change made in this case is the addition of a constant term a , to

W t, y . The invariants in this case are

y√t

And

f η u a υt yc

ac lnt (4.120a, b)

The diffusion equation becomes

υf" f' ac 0 (4.121)

It is seen that as a result of the additional constant term a , one more term is

added to f η and the transformed equation, as compared with Case 8.

4.5 CONCLUDING REMARKS:

The method given in this chapter can be summarized as follows: Consider a

partial differential equation

F z , … , zp 0 (4.122)

Where

z x

z x

…

80

…

zp‐ ky

xm‐1

k

zp ky

xmk

This equation is said to be invariant under the infinitesimal contact transformation

z'i zi ξi ; i 1, …., p

If the following condition is satisfied

UF Fz

… . . pFzp 0 (4.123)

Since the function in the transformation are expressed in terms of a

characteristic function W, equation (4.123) is used to predict the form of W. The

invariants can then be obtained by solving the following system of equations:

dz … dzp

p (4.124)

Finally, using the theorems given in article 4.2, the number of variables can

be reduced by one using the invariants as new dependent and independent

variables.

For simultaneous differential equations, the functions in the infinitesimal

contact transformation are expressed in terms of characteristic functions, Wi where

i = 1. …. m and m is the number of dependent variables.

The present method is seen to be a systematic way of searching for all

possible groups of transformation which will reduce the number of variables by

one. For reducing more variables, the same steps have to be repeated. The more

applications on this method are found in the recent work of Heena Barad (2010).