Exact and approximate solutions with A priori error bounds for systems of time-dependent wave equations

In this paper, mixed problems for systems of partial differential equations of the type u(tt) = C(t)u(xx), 0 < x < d, t > 0, where C(t) is a continuously differentiable R-rxr valued symmetric positive definite matrix function and u(x,t) is a R-r-valued vector are considered. First, uniqueness of solutions and the existence of an exact series solution is proved using a matrix separation of variables technique. Given an admissible error epsilon and a bounded subdomain D(b) = {(x,t); 0 less than or equal to x less than or equal to d; 0 less than or equal to t less than or equal to b}, a continuous numerical solution is constructed so that the approximation error is less than epsilon uniformly for (x,t) in D(b).