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Exponentially Fitted Finite Difference Approximation for Singularly Perturbed Fredholm Integro-Differential Equation

Mehebub Alam, Rajni Kant Pandey

TL;DR

This work tackles second-order singularly perturbed Fredholm integro-differential equations by introducing an exponentially fitted finite-difference scheme on a piecewise uniform mesh. The method uses integral identities with exponential basis functions, the composite trapezoidal rule, and interpolating quadrature rules to capture boundary and interior layers, yielding an ε-uniform convergence rate of $O(\mathcal{N}^{-2} \ln \mathcal{N})$ in the discrete max norm. A rigorous error analysis is conducted via remainder estimates and a discrete Green's function framework, establishing stability under a bound on $|\lambda|$ and providing explicit error bounds. Numerical results across multiple $\epsilon$ values confirm the theoretical predictions, showing near-second-order uniform convergence and robustness of the approach for SPFIDEs. The findings offer a practical and reliable tool for accurately solving SPFIDEs where traditional schemes fail near layers.

Abstract

In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of boundary and interior layers within the domain. To overcome these challenges, we develop a fitted second-order difference scheme that can capture the layer behavior of the solution accurately and efficiently, which is again, based on the integral identities with exponential basis functions, the composite trapezoidal rule, and an appropriate interpolating quadrature rules with the remainder terms in the integral form on a piecewise uniform mesh. Hence, our numerical method acts as a superior alternative to the existing methods in the literature. Further, using appropriate techniques in error analysis the scheme's convergence and stability have been studied in the discrete max norm. We have provided necessary experimental evidence that corroborates the theoretical results with a high degree of accuracy.

Exponentially Fitted Finite Difference Approximation for Singularly Perturbed Fredholm Integro-Differential Equation

TL;DR

This work tackles second-order singularly perturbed Fredholm integro-differential equations by introducing an exponentially fitted finite-difference scheme on a piecewise uniform mesh. The method uses integral identities with exponential basis functions, the composite trapezoidal rule, and interpolating quadrature rules to capture boundary and interior layers, yielding an ε-uniform convergence rate of in the discrete max norm. A rigorous error analysis is conducted via remainder estimates and a discrete Green's function framework, establishing stability under a bound on and providing explicit error bounds. Numerical results across multiple values confirm the theoretical predictions, showing near-second-order uniform convergence and robustness of the approach for SPFIDEs. The findings offer a practical and reliable tool for accurately solving SPFIDEs where traditional schemes fail near layers.

Abstract

In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of boundary and interior layers within the domain. To overcome these challenges, we develop a fitted second-order difference scheme that can capture the layer behavior of the solution accurately and efficiently, which is again, based on the integral identities with exponential basis functions, the composite trapezoidal rule, and an appropriate interpolating quadrature rules with the remainder terms in the integral form on a piecewise uniform mesh. Hence, our numerical method acts as a superior alternative to the existing methods in the literature. Further, using appropriate techniques in error analysis the scheme's convergence and stability have been studied in the discrete max norm. We have provided necessary experimental evidence that corroborates the theoretical results with a high degree of accuracy.
Paper Structure (6 sections, 4 theorems, 83 equations, 2 figures, 1 table)

This paper contains 6 sections, 4 theorems, 83 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed difference scheme
  4. Error analysis
  5. Numerical results
  6. Conclusion

Key Result

Lemma 1

Consider that $f,\ a \in C^{2}(\bar{\Gamma}),\ K \in C^{2}(\bar{\Gamma} \times \bar{\Gamma})$ with $a(\xi) \geq \bar{a}>0$, and where $\bar{K}=\underset{\xi \in \bar{\Gamma} }{\max} \int_0^T |K(\xi,\zeta)| \mathrm{d} \zeta,$ then, the solution $\mathit{v}$ of (equation:1)-(equation:2) follows:

Figures (2)

  • Figure 1: Solution plot of Example \ref{['Example1']} with $\epsilon = 2^{-24}$ and $\mathcal{N} = 128$.
  • Figure 2: Loglog plots of maximum pointwise error with various values of $\epsilon$ for Example \ref{['Example1']}

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Example 1