A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term

This article introduces a numerical method to solve a singularly perturbed Fredholm integro-differential equation of second order with a discontinuous source term. To effectively handle the problem, we utilized the finite difference method on an adaptive mesh. This adaptive mesh, generated via the grid equidistribution method, significantly improves solution accuracy and resolves issues inherent in traditional numerical methods. Our numerical results provide evidence of \(\varepsilon -\) $\epsilon -$ uniform convergence rates and prove the effectiveness of our approach. Overall, our scheme offers an efficient solution for singularly perturbed Fredholm integro-differential equations that include discontinuous source terms.