Six Point Implicit Methods for the Approximation of the Derivatives of the Solution of First Type Boundary Value Problem for Heat Equation

Abstract

We construct six point symmetric implicit difference boundary value problems for the first derivative of the solution $u\left( x,t\right)$ of the first type boundary value problem for one dimensional heat equation with respect to the time variable t and for the pure second derivative of the solution with respect to the spatial variable x. Furthermore, for the approximation of the pure second derivative of the solution with respect to time t,  and for the approximation of the mixed derivative of the solution special six point implicit difference methods are proposed. It is proved that the constructed implicit schemes converge with the order $O\left( h^{2}+\tau ^{2}\right)$ where, h is the step size in x and $\tau$ is the step size in time to the corresponding exact derivatives under the assumption that the given boundary value problem of one dimensional heat equation has the initial function belonging to the Hölder space $C^{10+\alpha },$ $0<\alpha <1,$ the heat source function from the Hölder space $C_{x,t}^{8+\alpha ,4+\frac{\alpha }{2}},$ the boundary functions from $C^{5+\frac{\alpha }{2}},$ 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.