Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

Abstract

We construct four-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. Also, for the second derivatives of u(x, t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Holder space C8+alpha 0 < alpha < 1, the heat source function given in the heat equation is from the Holder space C-x,t(6+alpha, 3+alpha/2), the boundary functions are from C4+alpha/2, and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h(2) + tau) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and tau is the step size in time. Theoretical results are justified by numerical examples.