Exact solution of variable coefficient mixed hyperbolic partial differential problems

(~

Applied Mathematics Letters

Applied Mathematics Letters 16 (2003) 309-312

PERGAMON

www.elsevier.com/locate/aml

E x a c t S o l u t i o n of Variable Coefficient M i x e d H y p e r b o l i c Partial Differential P r o b l e m s M.

J. RODRIGUEZ-ALVAREZ, G. RUBIO AND Instituto de Matem~tica Multidisciplinar Universidad Polit~cnica de Valencia, Spain

L. JODAR

~mat, upv. es

A. E. P o s s o Departamento de Matem~ticas Universidad Tecnol6gica de Pereira, Colombia possoa©andromeda, utp. edu. co

(Received April 2002; accepted May 2002) A b s t r a c t - - T h i s paper is concerned with the construction of exact series solution of mixed variable coefficient hyperbolic problems. (~) 2003 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - H y p e r b o l i c mixed problem, Exact solution.

1. I N T R O D U C T I O N In recent papers [1,2], exact series solutions of certain hyperbolic mixed problems have been given using separation of variables technique. Here we consider a variable coefficient mixed hyperbolic system of the form 1

Uxx(x,t)-q(x)U(z,t)=-~Vtt(x,t), alV(O,t)+blUx(O,t)=O, a2V(p,t) + b2Ux(p,t) = O, U(x,O)=f(x), Ut(x,O)=g(x), where

q(x)

is real,

r(t)

00,

(1.2)

t > O,

(1.3)

00,

[a21+lb21>0.

(1.5)

Conditions on the coefficients and d a t a functions are given in Section 2 in order to guarantee an exact series solution of problem (1.1)-(1.5). This paper has been supported by the Spanish A.E.C.I., the C.I.C.Y.T. Grant DPI2001-2703-C02-02, and D.G.I.C. Y.T. Grant BFM 2000-C04-04. 0893-9659/03/$ - see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(02)00197-0

Typeset by . 4 ~ ¢ S - ~

310

M.J.R.ODRIGUEZ-ALVAREZ et al.

2. E X A C T

SERIES

SOLUTION

Following the ideas developed in [2], we propose a candidate series solution of problem (1.1)(1.4) of the form U(x, t) = E(anzn(t) + bnwn(t))~on(x), (2.1) n>l

where {~,~(x)} is the eigenfunction system of the Sturm-Liouville problem ~ " ( x ) + (A - q ( x ) ) ~ ( ~ ) = 0,

0 < • < p,

al~v(0) + bl~'(0) = 0,

(2.2)

a2~(p) q- b2~flt(P) : 0, and if An is the n th eigenvalue of (2.2), then the pair {z,~,wn} are the solutions of

Y"(t) + A.r(t)Vn(t) = 0,

(2.3)

t > 0,

satisfying zn(0) = 1,

z~(0) = 1;

wn(0) = 0,

w~(0) = 1;

(2.4)

and

f~ f(x)~n(x) dx an = f ~ ( ~ n ( x ) ) 2 d x '

f~ g(x)~n(x) dx b,~ =

fop(~n(x)) 2dx '

n > 1.

(2.5)

q2 = max{q(x); 0 < x < p}.

(2.6)

Let us denote ql = min{q(x); 0 < x < p},

By [1; 3, p. 264], the n th eigenvalue of problem (2.2) satisfies n27r 2 (rt q- 1)27r2 p2 A-ql _< An ~_ p2 + q2,

n > 1.

(2.7)

If r(t) is continuously differentiable in 0 < t < T and

M(T) = max{r(t); 0 < t < T}, M'(T) = max {r'(t); 0 < t < T } , m(T) = min{r(t); 0 < t < T},

(2.s)

taking no large enough so that An > 0 for n > no, then by Theorem 4 of [4] it follows that

Iz'(t)l _< L ~ ,

Iz,~(t)l < L,

I~o,~(t)l _<

0 < t < T, (2.9)

L

Iw~(t)l < L

,

n >_ no,

where V m----~ exp ~, ~

(2.1o)

]"

Note that by (2.3), (2.7), and (2.9) for n > no one gets

Iz~(t),