Stability and Convergence Analysis of an Exact Finite Difference Scheme for Fredholm Integro-Differential Equations

Abstract: This report addresses the boundary value problem for a second-order linear singularly perturbed FIDE. Traditional methods for solving these equations often face stability issues when dealing with small perturbation parameters. We propose an exact finite difference method to solve these equations and provide a detailed stability and $\varepsilon$-uniform convergence analysis. Our approach is validated with an example, demonstrating its uniform convergence and applicability, with a convergence order of 1. The results illustrate the method's robustness in handling perturbation effects efficiently.