Analytical-numerical solution of coupled singular mixed diffusion type problems

The aim of this paper is to construct analytical approximate solutions for systems of partial differentiable equations of the form Eut(x, t) − = Auxx(x, t) = 0 for x ϵ [0, 1] and t > 0, subject to u(0, t) = 0and Bu(1, t) + Cux(1, t) = 0 for t > 0 and u(x, 0) = f(x), x ϵ [0, 1]. Here, u(x, t), f(x) are m-component vectors and E, A, B, C are mth-order complex matrices where B and C are not simultaneously singular and E is singular. Scalar regular Sturm-Liouville problems are used to find exact series solution; the series is truncated, then its matrix exponential terms replaced by Taylor series approximation. With appropriate selections, the resulting approximate solution will be arbitrarily close (uniformly) on a domain [0, 1] × [to, t1] where to > 0.