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

This study introduces a novel second-order computational technique to effectively tackle Volterra-Fredholm integro-differential equations, which are characterized by integral conditions and boundary layers. Initially, some analytical properties of the solution are given. Then, the approach involves 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 interpolating quadrature rules and linear exponential basis functions for the differential part. The analysis of the 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 discrete maximum norm. Finally, two test examples are given.