O(h3) order of accurate difference method for solving the first derivative of the Laplace equation with mixed boundary value problem

Abstract

An approach has been obtained for the solution and derivative of the solution of the mixed boundary value problem on a rectangular domain. It is assumed that the boundary values on the sides of the rectangle have fourth derivatives that satisfy the Hölder condition. It is also assumed that besides the continuity condition at the corners of the rectangle, the compatibility condi- tions obtained from the Laplace equation for the second and fourth derivatives of the boundary values are also fulfilled. When all these conditions considered, the approximate solution of the first derivative with mixed boundary value problem on a square grid O(h3) result was obtained. In order to support the theoretical results given in the study, some numerical experiments are presented.