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 numerical solutions for partial self-adjoint differential systems of the type (P(t) u"t)"t = Q(t)u"x"x, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), u"t(x, 0) = g(x), 0 @? x @? d, t >- 0, where P(t), Q(t) are positive definite oR^r^x^r-valued functions such that P'(t) and Q'(t) are simultaneously semidefinite (positive or negative) for all t >= 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error @e > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than @e, uniformly in D(T) = {(x, t); 0 @? x @? d, 0 @? t @? T}. Uniqueness of solutions is also studied.