(untitled) (Contemporary Mathematics, 2024)
In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.