A LAGUERRE-PERTURBED GALERKIN METHOD FOR NUMERICAL SOLUTION OF HIGHER-ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

This study presents a novel Laguerre-Perturbed Galerkin (LPG) method for the numerical solution of higher-order nonlinear integro-differential equations. The method integrates Laguerre polynomials as primary basis functions with shifted Chebyshev polynomial perturbations to improve approximation precision. Nonlinear terms are handled via quasilinearization, converting the problem into a sequence of linear systems solvable within a Galerkin projection framework. The LPG approach is tested on benchmark nonlinear Volterra and Fredholm integro-differential equations, exhibiting superior convergence rates and accuracy compared to existing techniques such as decomposition methods and wavelet collocation. Testing on classic Volterra and Fredholm examples shows LPG pulling ahead, errors drop from about   at N=5 to a tiny   at N=10, with faster exponential convergence than methods like Sharif et al.'s (2020) decomposition or Amin et al.'s (2023) wavelets,which confirm the method's robustness across different orders and nonlinearities. The LPG method's adaptability positions it as a valuable tool for modeling complex phenomena in physics, engineering, and applied mathematics, with opportunities for further extensions to fractional and partial integro-differential systems.