Six point implicit methods for the pure second derivatives of the solution of first type boundary value problem for one dimensional heat equation

Abstract

We construct six point implicit difference boundary value problem for the first derivative of the solution u (x, t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Furthermore, for the second order pure derivatives of u (x, t) special six point implicit difference boundary value problems are proposed. A uniform approximation of the order O (h2 + τ2) (second order accurate in the spatial varıable x and second order accurate in time t) where h is the step size in spatial variable x and τ is the step size in time is achieved. It is assumed that the initial function belongs to the Hölder space C10+α, 0 < α < 1, the heat source function is from the Hölder space Cx,t8+α,4+α2, the boundary functions are from C5+α2, and between the initial and the boundary functions the conjugation conditions of orders q = 0, 1, 2, 3, 4, 5 are satisfied. Theoretical results are justified by numerical examples.