fractional integro-differential equation, caputo derivative operator, jacobi polynomials, operational matrices, convergence, Applied mathematics. Quantitative methods, T57-57.97

This study aims to address a specific class of mathematical problems known as fractional integro-differential equations. These equations are used to model various phenomena„ including heat conduction in materials with memory, damping laws, diffusion processes, earthquake models, fluid dy-namics, traffic flow, and continuum mechanics. This research focuses on problems where the fractional derivative operator is defined in the Caputo sense. Our proposed methodology employs an operational approach based on the use of shifted Jacobi polynomials. We derive operational matrices for fractional integration and product, which are then applied to approx-imate solutions for both linear and nonlinear problems. By using these matrices in conjunction with the collocation method, we transform the orig-inal problem into a system of algebraic equations. Notably, our approach is simpler and more cost-effective compared to established methods such as Adomian decomposition, Homotopy perturbation, Sinc-collocation, and Legendre wavelet techniques. We provide several illustrative examples to validate our method’s effectiveness and reliability. Additionally, we present theorems concerning the existence of a unique solution and the convergence of our proposed approach.