Constructive approximations of mixed variable coefficient diffusion problems

This paper deals with the construction of analytical approximations of the problem uxx = (a(x)/b(t))ut, 0 < x < L, t > 0, u(x,0) = ƒ(x), 0 ≤ x ≤ L. First, the uniqueness of solution is studied and an exact series solution is constructed. Given ϵ > 0, 0 < t0 < t1 and D(t0, t1) = {(x,t); 0 ≤ x ≤ L, t0 ≤ t ≤ t1} the truncation index n0 of the series solution is determined in terms of the data so that the error is less than e uniformly in D(t0,t1). Since the truncated series approximation is expressed in terms of exact eigenvalues λ1,…, λn0 and eigenfunctions w1(x),…,wn0(x), the admissible errors when one approximates λn by and wn(x) by, 1 ≤ n ≤ n0, are determined so that the global error of an analytical approximation of the problem, constructed in terms of , , 1 ≤ n ≤ n0, is less than ϵ uniformly in D(t0, t1).