AbstractThis paper is concerned with the construction of exact series solution of mixed variable coefficient hyperbolic problems

Jódar, L.

Posso, A.E.

Rodriguez-Alvarez, M.J.

Rubio, G.

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(~  Applied 
Mathematics 
Letters 
PERGAMON Applied Mathematics Letters 16 (2003) 309-312 
www.elsevier.com/locate/aml 
Exact  Solut ion of Variable Coefficient Mixed 
Hyperbol ic  Partial Differential Problems 
M.  J. RODRIGUEZ-ALVAREZ,  G .  RUBIO AND L. JODAR 
Instituto de Matem~tica Multidisciplinar 
Universidad Polit~cnica de Valencia, Spain 
<mjrodri><grubio><lj odar>~mat, upv. es 
A. E. Posso  
Departamento deMatem~ticas 
Universidad Tecnol6gica de Pereira, Colombia 
possoa©andromeda, utp. edu. co 
(Received April 2002; accepted May 2002) 
Abst rac t - -Th is  paper is concerned with the construction ofexact series olution of mixed variable 
coefficient hyperbolic problems. (~) 2003 Elsevier Science Ltd. All rights reserved. 
Keywords--Hyperbol ic mixed problem, Exact solution. 
1. INTRODUCTION 
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),  0<x<p,  t>0,  (1.1) 
alV(O,t)+blUx(O,t)=O, t>0,  (1.2) 
a2V(p,t) + b2Ux(p,t) = O, t > O, (1.3) 
U(x,O)=f(x), Ut(x,O)=g(x), 0<x<p,  (1.4) 
where q(x) is real, r(t) >0 and 
lall+]bll >0,  [a21+lb21>0. (1.5) 
Conditions on the coefficients and data 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. Typeset by .4~¢S-~ 
PII: S0893-9659 (02)00197-0 
310 M.J.R.ODRIGUEZ-ALVAREZ et al. 
2. EXACT 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, t > 0, (2.3) 
satisfying 
and 
Let us denote 
zn(0) = 1, z~(0) = 1; wn(0) = 0, w~(0) = 1; (2.4) 
f~ f(x)~n(x) dx f~ g(x)~n(x) dx 
an = f~(~n(x))2dx ' b,~ = fop(~n(x)) 2dx ' n > 1. (2.5) 
ql = min{q(x); 0 < x < p}, q2 = max{q(x); 0 < x < p}. (2.6) 
By [1; 3, p. 264], the n th eigenvalue of problem (2.2) satisfies 
n27r 2 (rt q- 1)27r 2
p2 A-ql _< An ~_ p2 + q2, 
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}, 
n > 1. (2.7) 
(2.s) 
taking no large enough so that An > 0 for n > no, 
Iz,~(t)l < L, 
where 
L 
I~o,~ (t)l _< 
Iz'(t) l  _< 
Iw~(t)l < 
then by Theorem 4 of [4] it follows that 
L ~ ,  0 < t < T, 
(2.9) 
L , n >_ no, 
V m----~ exp ~, ~ ] "  
Note that by (2.3), (2.7), and (2.9) for n > no one gets 
Iz~(t),<LM(t) (n ~ l )T r )  2 ] -- -kq2 , 
\ 
LM(T) [((n~1)~)2+ I I/2 
l~"(t)l < vr ~ 
0<t<T,  
0<t<T.  
(2.1o) 
(2.11) 
Part ia l  Differential Prob lems 311 
By (2.2) and (2.7) we have 
- I~n(x) ] ,  0 < x < p. (2.12)  
Now we are concerned in bounding the eigenfunctions ~,~(x) as well as the Sturm-Liouville 
coefficients a~, bn, defined by (2.5). In order to prove that (2.1) gives a rigorous solution of 
problem (1.1)-(1.4), it is sufficient to show that series (2.1) as well as 
E {a~z,(t) + bnwn(t) }~(x),  (2.13) 
n>_l 
E {anz~(t) + b,~wn(t)} ~on(x), (2.14) 
n>l  
are locally uniformly convergent in a rectangle [0, p] x [to - 5, to + 5] = R(to, 5) for 0 < 5 < to < 
to + 5 _< T. All these series will be uniformly convergent in R(to, 5) if the series ~>l{n2lanl + 
nlb~l}n21~vn(x)l is uniformly convergent for 0 _< x _< p. Thus, we are interested in conditions 
which guarantee the uniform convergence of series 
~_n4lanl@n(x)h ~-~n31b=ll~o~(x)l, 0<z<p.  (2.15) 
n_>l n>l  
Let us suppose that 
q(x) is a function of bounded variation in 0 < x < p, (2.16) 
let us denote by Vq(O,p) the total variation of q(x) in [0,p] and take nl large enough so that 
An > q2, for all n > nl. (2.17) 
By applying Prufer transformation [3,5] to problem (2.2) and using Theorem 4.1 of [4], it follows 
that 
I~(x ) l  _< CqMq, I~'n(x)l < CqMa~/An - q(p), 0 < x < p, 
i 1 ) Cq = 1 + An - q(0) M a = exp ( Vq(0_p) ' ~k2 [(nTr//p) 2 -- Vq(O,p)]  ' n ~_ n 1. ¢12.18) 
Note that by (2.18) the series appearing in (2.15) is uniformly convergent in 0 _< x < p if 
coefficients an, bn satisfy 
lan[ = 0 (n-S) ,  Ibn] = 0 (n -5) l, n --~ ~.  (2.19) 
By Theorem 5 of [5, p. 273], and the proof of Theorem 4 of [2], condition (2.19) holds true if 
function f(x) and g(x) satisfy the following conditions: 
elf(O) + blf ' (0)  = 0, a2f(p) + b2f'(p) = 0, 
(2.20) 
alg(O) + blg'(O) = O, a2g(p) +b2g'(p) = 0, 
f(x) is six times differentiable, f(6)(x) is piecewise continuous and has (2.21) 
a bounded variation in [0,p] with f(2~)(0) = f(20(p) = 0, for i = 1, 2, 3. 
g(x) is five times differentiable, g(5)(x) is piecewise continuous and has (2.22) 
a bounded variation in [0,p] with g(2~)(0) = g(2i)(p) = 0, for i = 1, 2. 
Summarizing, the following result has been established. 
THEOREM 1. Consider real valued functions f(x), g(x), q(x), and r(t) where r(t) is continuous 
and positive, q(z) satisfies (2.16) and f(x), g(x) satisfy conditions (2.20)-(2.22). If {z~, w~} are 
solutions of (2.3) satisfying (2.4) and an, bn are defined by (2.5), then U(x, t) given by (2.1) is a 
solution of problem (1.1)-(1.5). 
312 M. J .  RODRIGUEZ-ALVAREZ et al. 
REFERENCES 
1. L. J6dar and M.D.B.osell6, Mixed problems for separate variable coefficient wave equation. The non-Dirichlet 
case. Continuous numerical solutions with a priori error bounds, Mathl. Comput. Modelling 30 (9/10), 1-22, 
(1999). 
2. P. Almenar and L. Jddar, Mixed problems for separated variable coefficient wave equations. Analytic- 
numerical solutions with a prior/error bounds, J. Comput. Applied Math. 134, 301-323, (2001). 
3. H. Sagan, Boundary and Eigenvalues Problems in Mathematical Physics, Dover, New York, (1989). 
4. L. J6dar and A.E. Posso, Analytic numerical approximation with a pr/or/error bound for the wave equation 
with time dependent coefficient, Mathl. Comput. Modelling 29 (6), 1-14, (1999). 
5. G. Birkhoff and G.C. P~ta, Ordinary Differential Equations, Third Edition, John-Wiley, New York, (1978). 


Exact solution of variable coefficient mixed hyperbolic partial differential problems 

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Exact solution of variable coefficient mixed hyperbolic partial differential problems

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