Spline-Based Computational Technique for Singularly Perturbed Fredholm Integro-Differential Problems

In this work, using a spline-based discretization, we develop a computational approach for singularly perturbed Fredholm integro-differential equations. The scheme addresses the challenges of the singular perturbation parameter ϵ through a tension and compression spline technique, coupled with Simpson’s rule for quadrature approximations. We analyze the stability and convergence properties of the proposed algorithm. Through the computation of maximum absolute errors on varying mesh sizes, we demonstrate the method’s effectiveness. Numerical results indicate that the scheme yields accurate solutions and exhibits a consistent rate of convergence for arbitrarily small values of ϵ.