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