Numerical Treatment of Sturm-Liouville Problems with Two Interior Transition Points

Abstract

Boundary value problems (BVPs) for second-order differential equations, particularly Sturm-Liouville equations, are fundamental in modeling numerous phenomena across the natural sciences. A significant challenge arises when these problems involve additional transmission conditions at internal points. This paper presents a numerical framework for solving Sturm-Liouville boundary value problems characterized by two internal transmission points. We extend the Finite Difference Method (FDM) to effectively address such problems, with a particular emphasis on ensuring accuracy at the critical transmission points x = ±1/3. The methodology, which generalizes the approach for a single discontinuity, is implemented and tested on a mathematical model. Numerical results demonstrate that the proposed scheme successfully handles the derivative discontinuities, and maintains a second-order convergence rate O(h^2), and shows excellent agreement with the exact solution, proving the robustness and reliability of the developed framework for multi-point interaction models.