Numerical scheme for singularly perturbed Fredholm integro-differential equations with non-local boundary conditions

This paper solves a class of Fredholm integro-differential equations involving a small parameter with integral boundary conditions numerically. The solution to the problem possesses boundary layers at both end boundaries. A central difference scheme is used for approximating the derivatives. In contrast, the trapezoidal rule is used for the integral term, provided an appropriately adapted mesh is considered, namely Shishkin and Bakhvalov–Shishkin meshes. The proposed numerical method presents a uniform second-order convergence rate regardless of the perturbation parameter. Furthermore, using a post-processing technique, we have significantly improved the convergence from second to fourth order. The effectiveness of the proposed approach is validated through some numerical examples.