Application of the Laguerre Perturbed Galerkin Analysis Method for Solving Higher-Order Integro-Differential Equations

Abstract

This study presents the development and implementation of a novel numerical method, the Laguerre Perturbed Galerkin (LPG) method, for solving higher-order integro-differential equations. The method leverages the advantages of Laguerre polynomials as basis functions while incorporating Chebyshev polynomials as perturbation terms to enhance both accuracy and efficiency. In the LPG method, the solution is approximated using Laguerre polynomials of degree N, with the residual error minimized via the Galerkin approach. Chebyshev polynomials are introduced as perturbation terms to further refine the solution. The residual is systematically reduced to a system of (N + 1) equations, which is then solved to determine the unknown coefficients of the approximating Laguerre polynomials. Comparative analyses demonstrate that the LPG method achieves superior accuracy and faster convergence rates compared to existing techniques, particularly for higher-order integro-differential equations. The findings contribute to the advancement of numerical methods in this domain, providing a powerful computational tool for scientists and engineers.