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

TL;DR

This paper introduces an exact finite difference scheme for solving second-order linear singularly perturbed Fredholm integro-differential equations, providing stability and convergence analysis with demonstrated robustness and uniform convergence.

Contribution

The paper presents a novel exact finite difference method with stability and ε-uniform convergence analysis for singularly perturbed FIDEs, addressing stability issues in traditional approaches.

Findings

Method achieves uniform convergence with order 1.

Validated with an example demonstrating robustness.

Handles perturbation effects efficiently.

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.