Fredholm Integro-Differential Equations: Numerical Solution Using Collocation Approximation

This paper presents an efficient numerical technique for the solution of high-order Fredholm integro-differential equations using Laguerre polynomial approximation. The given equation is first transformed into an equivalent integral formulation and subsequently discretized into a system of algebraic equations using standard collocation points. The resulting algebraic system is solved via matrix inversion to obtain the unknown coefficients. These coefficients are then substituted into the Laguerre polynomial expansion to generate the approximate solution. Numerical examples are provided to validate the accuracy and effectiveness of the proposed method. The results indicate that the method compares favorably with existing techniques and, in several cases, yields highly accurate approximations to the exact solution.