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

• Published in: Journal of applied mathematics & computing Vol. 69; no. 5; pp. 3601 - 3624
• Main Author: Durmaz, Muhammet Enes
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-10-2023; Springer Nature B.V
• Subjects: Basis functions; Boundary layers; Computational Mathematics and Numerical Analysis; Convergence; Differential equations; Finite difference method; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Quadratures; Theory of Computation; Finite difference scheme; Uniform convergence; Shishkin mesh; 65R20; Singular perturbation; 65L20; Integro-differential equation; 65L12; 65L11
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-023-01895-3

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
-parameter. The efficacy of the approach is supported by testing the numerical scheme’s performance.