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

Mehebub Alam, Rajni Kant Pandey

TL;DR

The paper addresses a boundary-value problem for a second-order singularly perturbed Fredholm integro-differential equation and proposes an exact finite difference method to achieve $\varepsilon$-uniform convergence with order 1. It constructs a nonstandard discretization using a denominator function $\psi_i^2$ and applies the composite trapezoidal rule for the integral term, resulting in a linear system $(M+S)u=F$. The authors prove a discrete minimum principle, establish stability, and derive $\|U-u\|_{\infty}\le C h$ under suitable smoothness and coupling conditions, ensuring $\varepsilon$-uniform convergence. A numerical example demonstrates uniform convergence and robustness of the method across small perturbation parameters, highlighting its practical applicability in handling boundary-layer behavior.

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.

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

TL;DR

The paper addresses a boundary-value problem for a second-order singularly perturbed Fredholm integro-differential equation and proposes an exact finite difference method to achieve -uniform convergence with order 1. It constructs a nonstandard discretization using a denominator function and applies the composite trapezoidal rule for the integral term, resulting in a linear system . The authors prove a discrete minimum principle, establish stability, and derive under suitable smoothness and coupling conditions, ensuring -uniform convergence. A numerical example demonstrates uniform convergence and robustness of the method across small perturbation parameters, highlighting its practical applicability in handling boundary-layer behavior.

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 -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.
Paper Structure (5 sections, 4 theorems, 49 equations, 1 table)

This paper contains 5 sections, 4 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Literature review
  3. Numerical discretization
  4. Stability and convergence analysis
  5. Conclusion

Key Result

Lemma 4.1

(Discrete Minimum Principle). Let $u_{i}$ be any mesh function that satisfies $u_{0} \geq 0, u_{N} \geq 0$, and $L_{\varepsilon}^{N} u_{i} \leq 0, i=1,2, \ldots, N-1$. Then $u_{i} \geq$$0, i=1,2, \ldots, N$.

Theorems & Definitions (9)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • Example 1