A numerical approach for singularly perturbed reaction diffusion type Volterra-Fredholm integro-differential equations

The goal of this study is to introduce a second-order computational method to solve Volterra-Fredholm integro-differential equations involving boundary layers. To solve numerically, we establish a finite difference scheme on the Shishkin mesh using a composite trapezoidal formulae for the integral part and interpolating quadrature rules and the linear exponential basis functions for the differential part. We establish that the numerical scheme and rate of convergence are both second-order and converge uniformly with reference to the small ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-parameter. The efficacy of the approach is supported by testing the numerical scheme’s performance.