Finite Difference Approach for Fourth-Order Impulsive Sturm-Liouville Boundary Value Problems

This paper presents a finite difference method to solve a novel type fourth-order boundary value problem with impulsive conditions. These differential equations, which model deflections in beams, provide insights into various applications in fields such as civil, mechanical, and aeronautical engineering. Analytical solutions to boundary value problems are often challenging to derive, highlighting the need for robust numerical methods. In this study, a formula for finite difference approximation is derived by using Taylor series expansions at selected grid points. By transforming differential equations into algebraic systems, the unknown solutions are determined based on the grid points. The proposed method is validated through a numerical example involving a fourth-order impulsive linear boundary value problem, and the results demonstrate its effectiveness.