Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

This paper is concerned with the construction of accurate continuous numeri cal solutions for partial self-adjoint differential systems of the type (P( t) u(t))(t) = Q(t)u(xx), u(0, t) = u(d, t) = 0, u(x,0) = f(x), u(t)(x, 0) = g(x), 0 less than or equal to x less than or equal to d, t greater than or equal to 0, where P(t), Q(t) are positive definite R-r x r-valued function s such that P'(t) and Q'(t) are simultaneously semidefinite (positive or ne gative) for all t greater than or equal to 0. First, an exact theoretical s eries solution of the problem is obtained using a separation of variables t echnique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following questio n is addressed. Given T > 0 and an admissible error epsilon > 0 how to cons truct a continuous numerical solution whose error with respect to the exact series solution is smaller than epsilon, uniformly in D(T) = {(x, t); 0 le ss than or equal to x less than or equal to d, 0 less than or equal to t le ss than or equal to T}. Uniqueness of solutions is also studied. (C) 1999 E lsevier Science Ltd. All rights reserved.