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 utt = C(t)uxx, 0 < x < d, t < 0, where C(t) is a continuously differentiable Rr × r valued symmetric positive definite matrix function and u(x, t) is a Rr-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 ϵ and a bounded subdomain D(b) = {(x, t); 0 ≤ x ≤ d; 0 ≤ t ≤ b}, a continuous numerical solution is constructed so that the approximation error is less than ϵ uniformly for (x, t) in D(b).