Development of the Finite Difference Method to solve a new type Sturm-Liouville problems

Abstract

The purpose of this study is to present a new modification of finite difference method (FDM) for approximating the solution of the two-interval boundary value problems for second order differential equations, whose main feature is the nature of the imposed conditions. Namely, the investigated problems contains not only boundary conditions at the points of the considered interval, but also an additional conditions at one interior point of interaction, so-called transmission conditions. Naturally, the analysis of two-interval boundary-value problems is more complicated and it is not clear how to extend the classical FDM to such type problems. The proposed modification of FDM tested on two model problems with known exact solutions. The obtained result are illustrate the applicability and efficiency of our own algoritm, which can be readily extended to all many-interval problems.