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

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 C 8+α ,0 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 +τ) (second order accurate in the spatial variable x and first order accurate in time t ) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.