A new finite difference algorithm for boundary value problems involving transmission conditions

Abstract

Abstract The finite difference method (FDM) is used to find an approximate solution to ordinary and partial differential equations of various type using finite difference equations to approximate derivatives. The idea is to replace ordinary or partial derivatives appearing in the boundary-value problem by finite differences that approximate them. There is an extensive literature on this topic. But, as a rule, ordinary differential equations or partial differential equations were studied without an internal singular point and without corresponding transmission conditions. It is our main goal here to develop finite difference method to deal with an boundary value problem involving additional transmission conditions at the interior singular point. In this study, we have proposed a new modification of classical FDM for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transmission conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To demonstrate the efficiency and reliability of the proposed modification of FDM an illustrative example is solved b y this method. The obtained results are compared with those obtained by the standard FDM and by the analytical method. Corresponding graphical illustrations are also presented.