Continuous numerical solutions and error bounds for time dependent systems of partial differential equations: Mixed problems

The aim of this paper is to construct continuous numerical solutions with a prefixed accuracy in a bounded domain Ω(t0, t1) = [0, p] × [t0, t1], for mixed problems of the type ut(x, t) − D(t)uxx(x, t) = 0, 0 < x < p, t> 0, subject to u(0, t) = u(p, t) = 0 and u(x, 0) = F(x). Here, u(x, t) and F(x) are r-component vectors and D(t) is a Cr × r valued two-times continuously differentiable function, so that D(t1)D(t2) = D(t2)D(t1) for t2 ≥ t1 > 0 and there exists a positive number δ such that every eigenvalue z of (D(t) + DH(t))/2 with t> 0 is bigger than δ.