A Numerical Technique for Solving High-Order Fredholm Integro-Differential Equations Via Laguerre Polynomial Approximation

Abstract

This work presents an efficient numerical technique for solving Fredholm integro-differential equations (FIDEs) using Laguerre polynomial basis functions. The proposed method transforms the given integro-differential equation into an equivalent integral form, which is subsequently approximated through the collocation technique at standard collocation points. The resulting system of algebraic equations is solved using matrix inversion procedures to obtain the approximate numerical solution. The existence, uniqueness, continuity, and convergence of the proposed method are established using the Banach fixed-point theorem. Three numerical examples covering first, second, and third-order FIDEs are solved to demonstrate the applicability, accuracy, and computational efficiency of the method. Numerical results show that the proposed method achieves absolute errors ranging from 0.00 to 7.05 × 10⁻⁵ at truncation order N = 5, which are substantially smaller than those reported for existing methods. Specifically, for the first-order problem, the method produces exact solutions at all evaluation points, while for the second- and third-order problems, the maximum absolute errors are 7.05 × 10⁻⁵ and 0.00, respectively. These results confirm the superior accuracy and computational efficiency of the proposed approach.

DOI: https://doi.org/10.5281/zenodo.22860161