Thermoelastic problem of an orthotropic elastic plane containing a cruciform crack

In?. 1. En@18Sci. Vol. 30, No. 8, pp. 1049-1059,1992 Printed in Great Britain. All rights reserved

0020-7225/92 $5.00+ 0.00 Copyright @ 1992 Pergamon Press Ltd

ON THE ISOGROUPS OF THE SEMILINEAR HYPERBOLIC EQUATION u,t =f(u) AND ITS EXACT SOLUTIONS FOR THE CASE OF THE LIOUVILLE EQUATION 0. P. BHUTAN1 Department

of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110 016, India

K. VIJAYAKUMAR Department

of Mathematics, Punjab University, Chandigarh, India

(Communicated

by D. G. B. EDELEN)

Ah&ad-Herein the physically and mathematically significant semilinear hyperbolic equation u,, =f(u), where f(u) is an arbitrary smooth function of u and which encompasses Liouville equation as its particular form has been analysed via the isovector approach. Following the procedure given in [I], the components of isovector fields have been tabulated for nine different forms of f(u). The corresponding orbital equations have been solved only for five physically interesting choices of f(u). The solutions of the orbital equations under certain conditions yield invariant groups of transformation which reduce the given partial differential equation to a non-linear ordinary differential equation (NLODE), which has either been solved exactly or reduced to standard forms. However, for the case when f(u) = en”, n 20 which corresponds to the case of Liouville equation, the NLODE has been solved via Noether’s theorem, and it has resulted in a closed form exact solution that doesn’t seem to have been reported earlier. Further, two more solutions to Liouville equation have been arrived at via standard techniques.

1.

INTRODUCTION

is in sequel to our earlier work (Bhutani and Vijayakumar [l]) wherein we have emphasized the importance and efficacy of the isovector approach over the other group theoretic techniques utilized for obtaining exact solutions of the generalized diffusion equation. In fact, the said approach when carried over to the Boussinesq equation in hydrology resulted in its comparison with classical and non-classial approaches due to Bluman and Cole [2]. Herein, we have utilized the isovector approach to generate new similarity solutions of another important class of equations-semilinear hyperbolic equations. This has resulted in, for physically realizable forms of the function of dependent variable involved, a number of new solutions of the corresponding differential equations, either by reducing them to standard (Painleve) forms or by solving them exactly. Of special mention is the first integral of the corresponding ordinary differential equation of the Liouville equation. More specifically, after establishing the existence and hence formulating the alternate potential principle for NLODE corresponding to the Liouville equation, the invariance identities of Rund [3] involving the Lagrangian and the generators of the infinitesimal Lie groups have been utilized for writing down the first integral via Noether’s theorem. By repeated application of invariance transformations, a new exact solution of Liouville equation has been obtained. This

2. SEMILINEAR

HYPERBOLIC

EQUATION

We consider the semilinear hyperbolic equation of the form

(2.1)

1050

0. P. BHUTAN1and K. VIJAYAKUMAR

where f(u) is an arbitrary function of u. For the case f(u) = k0 sin u, where k0 is a constant. equation (2.1) finds applications in many areas of physics and mathematical sciences including acoustics with special reference to propagation in ferromagnetic materials of waves carrying rotations of direction of magnetization, propagation of ultra short optical pulses, propagation in a large Josephian junctions and propagation of crystal dislocation, magnetic flux (Barone et al. [4]). For f(u) = u3 - u, equation (2.1) serves as a model of nonlinear meson theory of nuclear forces (Shiff [5]) and in nonlinear theory of elementary particles (Perring and Skyrme [6]). For the case when f(u) = enu, n # 0, equation (2.1) becomes a field theoretic model (Calogero [7]). Further, equation (2.1) has been studied extensively for geometric properties by Eisenhart [8], for Backlund transformations by Shadwick [9] and for Painleve analysis by many authors in the recent past, and particularly so by Clarkson et al. [lo] for the case f (4 = cle zsw + c,eSoU + c,e@@ + c,e-2Bou , where c,, c2, c3, c4 and PO are arbitrary constants.

3. ISOVECTOR

METHOD

Define a five dimensional space Es, whose coordinates are f, x, u, y, z. Writing equation (2.1) in the language of exterior differential forms, we get cu=du-ydr-zdt,

(3. I)

da = -dy A dx - dz A dt, p = -dz A dt -f(u) where cu, da; /3 are 1,2,2-forms,

respectively.

(3.2)

dx A dt,

(3.3)

Further, y and z are given by

y=u,,

(3.4)

2 = UI,

(3.5)

when (Y= 0, and symbols A and d used above have the same meaning as given in [ 11. Closed ideal Let I be the ideal of A(E,)

generated by the exterior differential forms (a; da, /3). The necessary and sufficient condition for Z to be a closed ideal of the algebra of exterior differential forms is that dZ c I. Alternatively, the exterior differential of a form in Z is either contained in I or expressible as linear combination of forms in Z [ll, 161. Symbolically, d&i = $$ Qf: A ai,

(i = 1,2,3

and (Ye= (Y,a2 = da, a3 = @)

j=l

where QTs are some arbitrary 1 or 2-forms. In the light of this definition, one can easily check that da = 0 mod I. For db = 0 mod Z, we find that d/!?=(YhQ, where 51= -f’(u)

dx

A

dt, a two form. Hence Z is closed.

Zsovector field

Define a vector field V over the space E5 with components directions of t, x, u, y, and z respectively; that is, V=V’d,+V”d,+V”d,+VYdy+VLa,,

V’,

V”, V”,

Vy, V” in the

(3.6)

where VL, V”, VU, Vy and V” are functions of t, x, u, y, z and d, = d/at, d, = a/ax and so on. A vector field V is said to be isovector field if &I c I, where Z& is the Lie derivative of forms over the vector field V.

(3.7)

1051

Isogroups of the semilinear hyperbolic equation u,, =f(u)

Isovectors of semilinear hyperbolic equations Using the transport property of the exterior differential Lie derivative of cu, /3 are given as follows:

forms given by equation

L,(cu) = LY,

(3.7), the (3.8)

E,(B)=5rB+w~a-Eda;

(3.9)

where a; cl, 5 are arbitrary functions of t, x, u, y, z, W is a l-form, and I&(cu) and I&(/3) are = VJ da + d(V_i (u),

(3.10)

E&3) = VJ dp + d(VJ /3).

(3.11)

&(a)

In equations differentiation

(3.10) and (3.11), _I denotes of differential forms. Assume

inner

multiplication

and d denotes

VJa=G,

exterior (3.12)

and W=Adt+B&+Cdz+Ddy+Edu,

(3.13)

where A, B, C, D and E are arbitrary functions of t, x, u, y, z. On making use of equations (3.1), (3.10) and (3.12) in equation (3.8), expanding by using necessary operations given in Edelen [ll], and collecting the coefficients of similar l-forms, we get a system of first order partial differential equations which after eliminating yields V” = -GY,

(3.14)

V’ = -G,,

(3.15)

V”=G-yG,--zG,,

(3.16)

Vy = G, + yG,,

(3.17)

V’ = G, + .zG,,.

(3.18)

Further, on combining equations (3.2), (3.3), (3.9), (3.11) and (3.13) and performing similar expansions and collecting the coefficients of similar 2-forms, we arrive at the following system of partial differential equations v: + v: - f (u)VG = L!j1- cz + 5, V:-(V;+V:)-f’(u)V”=-&f(u)+Ay-Bz, -V:+f(u)Vx,=A -f (u)V:

f (u)V:

= B + Ey,

(3.21) (3.22)

= -cy,

= --or

-f (u)V;

(3.20) + Ez,

v: + f (u)Vi

(3.19)

(3.23)

+ 5,

(3.24)

+ V; = -Dz,

(3.25)

v:=c,

(3.26)

v;=o,

(3.27)

D = 0,

(3.28)

On eliminating the arbitrary functions 5, E1, A, B, C, D and E we get the following system of equations: V: -f ‘(u)V” -f (u)V.: +f (u)K

-f (u)v;}

- y(f (u)VZ - Vi} = 0,

(3.29)

yv: + f (u)Vi + v: = 0,

(3.30)

v; -f (u)VG = 0,

(3.31)

v;=o.

(3.32)

#O

ll-u

k, sin u

n

_____.

1

-

(

+4

+ %)

+ a*)

+ PJ

-(-alx

-(a,x

+ (13)

-(-02x

na,;+~1~x+a~)

-(w

-(*tx + *A

-(*lx + a,)

-(w

- n,(k - I))! + CI1

-(-a,r

-(qt

-(a$

+ 03)

+ 02)

+4

-(nb,;+b,r+b3)

-(a,

+ u3)

u3)

-(-o,l+

-(-o,t

+ IIJ

-(-o,r

- 41)

V’

*zu

(I

,+u

a,x+b,t+

0

0

0

a,+b, n

T(LI + bu)

112

( 1

PI +

0

P,(X, 4 + OlU

V”

@I+ QdY

+ d

%Y

-0IY

-%Y

a,(l+ny)+a,y

(a I + ao)y

(*1+ Q,)Y

P’rx +

“IY

Plz + YM

VY V’

(a*--*lb

~___

and g(l) are

--(I,z

QIL

%L

b,(l+nz)+b,z

(aok - 0

(**-*lb

PI, +

-a,1

P,r+4P;+4

Where a,p4, b,, b,, c,-c4 and &, are arbitrary constants. and y, = -a,/k,t - l/k, j g(t) dr -us/k,, where f(x) In row 1 P1=xIg(r)dt+tIf(x)dx+a,*r+n,, p,=- n,/kgc - l/k, If(x) dx -q/k, respectively arbitrary functions of x and f and as-a8 are arbitrary constants. In row 3, p, has to satisfy the condition pIn - bp, + aa = 0.

c 1ezm + c*em + c,e-@& + c4e-20@E

____~__

7,

(a+bu)“,k#O,

o+bu

Arbitrary

-P,(x)

a constant

-k,

V”

f(u)

Table 1

Isogroups of the semilinear hyperbolic equation u,, =f(u)

in equations (3.29)-(3.32),

On making use of equations (3.14)-(3.18) Ga + zG,

+f(u)Gx,

-f’(u)(G

-yG,

-tW

+f(u)(Gz

we get

+ G + G,z

+Y(--G, G,, +f(u)Gzz

1053

+fWGyr)

- Gm -.fWCJ

= 0,

(3.34)

+ YG,, = 0,

G, + zG,, +f(u)Gyy

(3.35)

= 0,

(3.36)

Gzy = 0, Solving equations (3.34)-(3.36),

(3.33)

we get (3.37)

G=h+~2U+~3y+iU4%

where p,, pz are functions of x, t and ,u3 is a function of x and p4 is a function of t alone. Further, by using equation (3.37) in equation (3.33), we obtain Pltr

+

w2tz

+

zJ?L2% +

d4fw

-f’(Ukh

+

wd

+f(uM

+

Pd

-Y/&f = 0.

(3.38)

Even though it is possible to solve equation (3.38) for arbitrary f(u), we confine our attention only to the possible forms of f(u), which are physically realizable. For such different forms of f(u), we solve equation (3.38) for pI, c(~, p3 and p4 and substitute the resulting values in equations (3.37), (3.14)-(3.18). Thus, we arrive at different forms of V’, V”, Vu, V”, Vy, which are listed in Table 1. From the table, one can easily observe that for each of the form of f(u) that is listed in it, the isovectors V”, V’, V” constitute Lie algebras whose dimensions depend upon the number of arbitrary constants occurring in their expressions.

4. SIMILARITY

TRANSFORMATIONS, AND SIMILARITY

SIMILARITY SOLUTIONS

REDUCTIONS

Herein, we have confined our attention only to five physically interesting and different choices of f(u) corresponding to which the isovectors are given in Table 1. As it is known, using &vector, the orbital equations are constructed and their solutions result in similarity tr~sfo~ations. These simiiarity transformations are used to reduce equation (2.1) to an ordinary ~~erenti~ equation that has been solved for ~o~es~n~ng choices of f(u). Case I: f(u) = (a + bu)&, k # 0, 1 where a and b are constants Corresponding equations

to this case, the isovector dx* -= ds

-(w*

given in Table

+ a2),

x*(o) =x,

dt* = -(al - a,(k - l))t* + a3, Q

-

%=;(a Solving equations (4.1)-(4.3),

1, row 5 result in the orbital

+bu*),

t*(O) = t,

(4.1) (4.2)

u*(o) = 24.

we get x*+A3 -=ew x+A, t* -

+A,

t +

=

A4

u* + afb u+afb

(4.4)

’

e-a&

,

= &@ ’

(4.5) (4.6)

1054

0.

From equations (4.4)-(4.6),

we obtain similarity transformations

P. BHUTAN1

and K. VIJAYAKUMAR

given as follows

x + A3

E=

(t +

jlpk

u = - ; + (t +

(4.7)

’

A,ykq(g),

where & = az/ao and L, = aJa,k. On making use of equations (4.7)-(4.8) in equation (2.1) for f(u) = (a + bu)“, we obtain the ordinary differential equation satisfied by q(E): &j” - bkrjk = 0.

(4.9)

71= &E1’l-k,

(4.10)

A solution to equation (4.9) is given by

where A0 is an arbitrary constant whose value is A0 = (b(1 - k)2)1’1--k. On making use of equation (4.10) in equation (4.7) and (4.8), we get u = - ; + A,(@ + A3)(t + &))“‘-k,

(4.11)

a new solution to equation (2.1) for f(u) = (a + bu)k, k # 1. Choosing a, = 0 in equations (4.1)-(4.3), and proceeding as mentioned above, we arrive at the following similarity transformations (4.12)

‘z = (x + &)(f + W

(4.13)

Lf= V(5)> where & = u2/u1, 12.,= u3/u1. On making use of equations (4.12) and (4.13) in equation for the case f(u) = (a + bu)k, we get the following ordinary differential equation in ~(5) &” + q’ = (a -t br/)k,

(2.1)

(4.14)

Under the transformations R = (a + b~)~-l’l-k, Q = b(u + bq)-kq’, to first order ordinary differential equation and is given as follows

dQ

-kRk-‘Q* - Q + b dR= 1 --RRR’Q

equation (2.14) is reduced

(4.15)

l-k

For the exceptional case k = 1 which corresponds to the case f(u) = a + bu, following the same procedure as for k # 1 we arrive at the following ODE for equation (2.1): &7” + 2~7’- bq = 0,

(4.16)

where g = (x + &)(t + A,),

u = V(5)

(4.17)

and A3=

@*la,),

Ll=

(%/a

Equation (4.16) is a particular case of the Bessel-Clifford b = -1, can be expressed as 7 = 5-“*{W,(2I&

equation,

+ GK(2VQ).

the solution to which, for

(4.18)

Case II: Liouville equation: f(u) = en”, n # 0 Corresponding

to this case, isovectors,

as given in Table 1, row 6, are used to obtain the

Isogroupsof

the semilinearhyperbolicequation u,, =f(u)

1055

orbital equations dt* - = -(b2t* + b3), ds dx* -= ds

+2x*

+ a3),

du* az+bz -=--_ ds n’ On solving equations (4.19)-(4.21), variable q(E) in the form

t*(O) = t,

(4.19)

X*(o) =x,

(4.20)

u*(o) = U.

we obtain the similarity variable x + A1

5=(t + A,): 24 =;

{log(t +

A*}-(l+r)+ q(E)},

(4.21) 5 and the dependent

(4.22)

(4.23)

where A, = a3/a2, A2= b3/b2 and r = azlbz. On making use of equations (4.22) and (4.23) in equation (2.1) for f(u) = en@,we get the following ordinary differential equation for q(E): r(EqW+ q’) + ne” = 0. An interesting feature of the analysis equation (4.24) via Noether’s theorem. Bhutani and Mital [12] and Bhutani and hence formulated the functional F in the

(4.24)

corresponding to this case is the exact solution of To this effect, using the mathematical tools given in Vijayakumar [13], we have proved the existence and following form:

F[q] = I(-;

r&f2 + mfJ) d&

(4.25)

Thus, the Lagrangian L, leading to equation (4.24) is given as L= -ir&‘2+ne?

(4.26)

For obtaining the first integral of equation (4.24) we have to prove the invariance of fundamental integral (f L de). Thus, we look for a one-parameter infinitesimal group of trnasformations of the form: E = E + =(E, 11)+ O(E2),

(4.27)

4 = rl + Ef(E, rl) + O(E2).

(4.28)

Necessary and sufficient conditions for the fundamental integral (1 L dE) to be invariant under the transformations (4.27) and (4.28) as given by Rund [3], is (4.29) Substituting for L and its derivatives in equation (3.29) and collecting in descending order t@e coefficients of various powers of q’, and setting these coefficients equal to zero, we obtain the following system of partial differential equations for t and 5: t; = 0,

CE= 0,

g + ts = 0,

(4.30) (4.31)

1056

0. P. BHUTAN1 and K. VIJAYAKUMAR

Solving equations (4.30)-(4.32),

we get c= -k,,

(4.33)

r=k,L

(4.34)

where k, is an arbitrary constant. Further, by using Noether’s theorem, written as

the first integral can be

1t+$T=O.

(

L-q'$

(4.35)

On making use of equations (4.36), (4.33) and (4.34) in equation (4.35), we obtain

Equation (4.36) represents a first integral to equation (4.24). In order to solve equation (4.36), we transform it to the form which is devoid of the middle term in the sense that the coefficient of the derivative of the new dependent variable is zero. Consequently, using the substitution w = ceV

(4.37)

in equation (4.36), we get

Combining equations (4.22), (4.23)

(4.37) and the solution to equation (4.38), we get

UC; log r n

[

1

(

2n (x + A,)(t + A2) sech* 1 1% (+

(t + Ap + A,))‘/2

111

’

(4.39)

where c is a constant of integration. Equation (4.39) represents a solution to equation (2.1) for the case f(u) = enu, n # 0, which does not seem to have been reported earlier?. Case ZZZ. Sine-Gordon

equation: f(u) = k, sin u

‘For the case f(u) = k0 sin u, where k,, is an arbitrary constant such that k0 = f 1, the isovectors given in Table 1, row 7 result in the following orbital equations: dt* -_= ds

-

(a2t*+ cd,

dx* = -(-a2x*

+ a3),

iis

du* -= ds Solving equations (4.40)-(4.42), given by

0,

t*(O) = t,

n*(o) =x

(4.41)

u*(o) = u.

the similarity variable g and the new dependent

variable n are

5 = (x + %)(t + 6,)

(4.43)

u = n(5),

(4.44)

where 8r = ad/a1 and e2 = -a3/a2. On making use of equations (4.43) and (4.44) in equation (2.1) for f (u) = k 0 sin u, we arrive at the following differential equation for q(g): ~~“+~‘-kkosin~=O.

(4.45)

tin order to show that the isovector approach is quite exhaustive we have in the Appendix obtained two more solutions of Liouville equation which do not seem to have been reported earlier.

1057

Isogroups of the semilinear hyperbolic equation u,, = f(u)

(4.45) is not solvable exactly by using any of the available techniques as mentioned in [12,13], we, therefore, transform it to a well known form by making the substitution w = eiV. Thus, equation (4.45) is reduced to As equation

I2

I

w”=!L/;+$(w2-1),

(4.46)

which is an equation of PainlevC type of the third kind (Ince [12]). Case IV: Phi four equation: f(u) = u 3 - u

Corresponding orbital equations

to this case, the isovectors given in Table 1, row 8, result in the following dt* -_= ds

-@It* + a2),

$ =-(-a+* du* -= ds

Solving equations (4.47)-(4.49), variable 7 which are given as

+ a3),

o 9

(4.47)

t*(O) = t,

x*(o) =x,

(4.4)

u*(o) = u.

we get the similarity variable

(4.49)

5 and the new dependent

5 = (x + &)(t + &),

(4.50)

u = V(f),

(4.51)

where f& = a3/a1 and e4 = -a2/al. Substituting equation (4.50) and (4.51) in equation (2.1) for f(u) = u3 - u, we obtain the following ordinary differential equation for a(E): zj#‘++$+~=o.

(4.52)

It may be remarked that equation (4.52) is neither solvable by using any of the available techniques nor reducible to any PainlevC transcendents. However, for the choice al = 0, isovectors, given in Table 1, row 8, lead to the following similarity transformations: L$=X-ct,

(4.53)

u = V(5),

(4.54)

where c = a2/a3. On making further use of equations (4.53) and (4.54) in equation (2.1) for the case f(u) = u3 - u, we obtain the following ordinary differential equation for q(E): crj” + r/3 - ?j = 0,

(4.55)

whose solution can be expressed in the following form

where c2 is another constant of integration. obtain

Combining equations (4.53), (4.54) and (4.56), we

u=tisech(-$=(x-ct)+q).

Equation (4.57) represents

a soliton type of solution and has an importance

(4.57) of its own.

Case V For the choice of the function f(u) = c,e2@@‘+ c2eaW + c3emBW+ c4e-2Bou, where cl, c2, c3, cq and j&, are arbitrary constants, the isovectors, given in Table 1, row 9, lead to the following ES30:8-G

10.58

0. P. BHUTAN1 and K. VIJAYAKUMAR

similarity transformation: 5 = (x + W(r + W,

(4.58)

u = 77(&

(4.59)

where A1 = a2/a1 and A2= a3/a1. On making use of equations (4.58) and (4.59) in equation (2.1) for this choice of f(u), we find that r,?(E) satisfies the following ordinary differential equation: &I” 4 11’= cie280V+ c2eBolf+ c3e-BoV+ cqe-zzBOri. On using the substitution

log V = &q we arrive at another ordinary differential

(VV”_

P)l$+

(4.60) equation in V:

Vv’-po(cJ4+C2V3+CgV+C~)=0.

(4.61)

Equation (4.61) is neither solvable exactly nor reducible to any of the Painleve transcedents. It may be remarked here that the other alternative of solving the above ODE via invariant variation principle as for equation (4.61) also has failed to provide the first integral for the choices of cl, c2, c3 and c4. However, for a choice a, = 0, we find a travelling wave type of transformation from the isovectors that is given by

g = X - ct,

(4.62)

u = r(E),

where c = a3/a2. On using equation (4.62) in equation (2.1) for the case f(u) = clezBaV+ czeaotl+ c3e+01) + cqe-2B0t), we arrive at the following ordinary differential equation: 01” + cle280r)+ czeBoq+ c3e+Oq f c4e-280*)= 0. On integrating once and making the substitution integrating again we find that f

f

(4.63)

/Jo7 = log v(E) in the resulting equation

(dv)/Vh,v4+a,v3+a3v2+a4v+a5=~+c1,

and

(4.44)

where al = -cl&&

a2 = -cz/%/c,

a3 = -2c&Qc,

a4 = c&%/c

and a 5 = c4&Jc and co and cl are constants of integration. It can be easily seen that quadrature given in equation (4.64) is an elliptic integral, whose solution can be written in terms of Jacobi’s elliptic functions. Further, for various choices of the values of the constants a,, a*, a3, a4 and a5, one may lead to many sub cases of equation (4.64). For an exhaustive study of the integral (4.64) we refer to our forthcoming paper [15], in which we have obtained quite a few interesting and new exact solutions to equations such as Liouville, Sine-Gordon, Double Sine-Gordon, Mikhaliov and Dodd-Bullough equations.

REFERENCES [l] 0. P. BHUTAN1 and K. VIJAYAKUMAR, Inr. 1. Engng Sci. 28(S), 375 (1990). [2] G. W. BLUMAN and J. D. COLE, Similarity Methods for Differential Equations. Springer, Berlin (1974). J. Math. Mech. 18, 1025 (1969). [3] H. RUND, Hamilton-Jacobi Theory in the Calculus of Vuriations. Princeton, New Jersey (1966). [4] A. BARONE, F. ESPOSITO, C. J. MAGEE and A. C. SCOTT, RiuCtu Nuouo Cimenro l(2), 227 (1971). [5] L. I. SHIFF, Phys. Rev. 84, 1 (1951). [6] J. K. PERRING and T. H. R. SKYRME, Nucl. Phys. 31,550 (1962). [7] F. CALOGERO, Stud. Appl. Math. 70, 189 (1984). [8] L. P. EISHENHART, A Treatise on the Diflerentiai Geometry of Curves and Surfaces. Mass. Gin., Boston (1909). {9] W. F. SHADWICK, J. Math. Phys. 19(11), 2312 (1978). [lo] P. A. CLARKSON, P. J. OLVER, 3. B. McLEOD and A. RAMANI, Integrab~y of Klein-Gordon Eq~rio~. University of Minne~ta Math. Rep., 83-159 (1986). [ll] D. G. B. EDELEN, Applied Erlerior Cafcuius. Wiley, New York (1985).

1059

Isogroups of the semilinear hyperbolic equation u,, =f(u)

0. P. BHUTAN1 and P. MITAL, Znt. J. Engng Sci. 23(3), 353 (1985). 0. P. BHUTAN1 and K. VIJAYAKUMAR, .I. Aust. Math. Sot. Ser. B32, 457 (1991). E. L. INCE, Ordinary Differential Equations. Dover, New York (1956). K. VIJAYAKUMAR, 0. P. BHUTAN1 and M. H. M. MOUSSA, New similarity solutions of Klein-Gorden type equations. Ph.D. thesis, III-Delhi (1991). [16] H. D. WAHLQUIST and F. B. ESTABROOK, J. Math. Phys. 16(l), 1 (1975).

[12] [13] [14] [15]

(Received and accepted 21 November

1991)

APPENDIX For the case of Liouville equation (case II), on making the arbitrary constants a, # 0, a2 = 0 = a3 = b, = b, that occur in Table 1, row 6, we find that the isovectors lead to the following orbital equations: t*2

dt ’ -=-nb,T, ds

x*2

dx* -=-Ids

2 ’

$$=a,x* On solving equations (Al)-(A3),

+ b,P,

t*(0) = t,

(Al)

x*(o) =x,

642)

u*(o) = u.

(A3)

we obtain the similarity variable 5 and the dependent variable ~(5) in the form:

g=- a,x - b,t a,b,xt

(A4)

’

On making use of equations (A4) and (A5) in equation (2.1) for f(u) = enu, we arrive at the following ODE: q’ + k,e” = 0,

(A6)

where k, = nbT/a,. On solving equation (A6), we obtain two different solutions, which are given as (A7)

?=log(&)~ and r~= log{c, sech’(c,g + c~)},

(A8)

where k, = 2c, is equation (A8) and c and cs are constants of integrations. On combining equations (A7) and (A8) with ;re;ti;; (A4) and (A5) respectively, we arrive at two different new solutions to equation (2.1) for f(u) = e”” that are

u

Aog n

-2% nb,(a,x

649)

- b,t + a,b,cxt)2

and 1 n

u =-log

b2a

&sech’ 1

a2

‘.

(AlO)