A Second-Order Numerical Approximation for Volterra-Fredholm Integro-Differential Equations With Boundary Layer and an Integral Boundary Condition

Abstract

This study introduces a novel second-order computational technique to effectively tackleVolterra-Fredholm integro-differential equations, which are characterized by integral conditions andboundary layers. Initially, some analytical properties of the solution are given. Then, the approachinvolves implementing a finite difference scheme on the piece-wise uniform mesh (Shishkin type mesh).It integrates a composite trapezoidal formula for the integral component and utilizes interpolatingquadrature rules and linear exponential basis functions for the differential part. The analysis ofthe method demonstrates that both the numerical scheme and its convergence rate exhibit second-order accuracy, ensuring uniform convergence with respect to the small parameter in the discretemaximum norm. Finally, two test examples are given.