Analytic-numerical solutions with a priori bounds for matrix-vector parabolic partial differential equations

We consider a parabolic matrix-vector system in which the diffusion matrix may be time dependent. For the time-independent case we construct approximate solutions with guaranteed error bounds using spectral information from certain matrix-vector Sturm-Liouville problems. For the time-dependent case we employ an approximation procedure which reduces the problem, on each time-step, to the time-independent case. We give an algorithm which may be used a priori at each time-step in the time-dependent case to guarantee accuracy to a specified tolerance.