Gauss-QuadratureCollocationAlgorithmforSolvingFractionalIntegro-DifferentialEquationsviaLucasPolynomials

Abstract

This study presents an innovative spectral technique based on Lucas polynomials to numerically approximate solutions for linear and nonlinear fractional-order integro-differential equations. The proposed methodology integrates a Gauss quadrature rule with a collocation framework, effectively converting the governing fractional equations and their boundary constraints into a system of algebraic equations. By determining the unknown expansion coefficients, highly accurate numerical solutions are obtained. Several numerical experiments are conducted to demonstrate the robustness, precision, and computational efficiency of the developed scheme. Comparisons with existing analytical and numerical results, including previous Fibonacci-Fejér [1] approaches, confirm the superiority and reliability of the Lucas-Gauss methodology, making it a valuable asset in computational mathematics and applied sciences.