Stability and Convergence of Spectral Approach for Fractional Integral-Differential Equation based on Legendre Basis

Background: In this study, we use operational Tau method (OTM) for finding the answer for fractional integral-differential equations (FIDEs). Methods: We prove that the approximated solutions of the Legendre Tau method converge to the exact solution in the L² norm. Also, some numerical findings are presented to clearly show the better performance of the proposed approach. Results: Outcomes reveals that the spectral approach based on the shifted Legendre basis can be considered as a structurally simple method that is typically applied for numerical solve of FIDEs. Also, our concentration restricted to linear Volterra FIDEs, we propose the approach to be developed to more common FIDEs. Despite the relatively low degrees utilized the numerical findings demonstrate the better performance of the spectral approach, in real condition, by considering the Legendre basis. Conclusion: Although the spectral rate of convergence illustrates the error of the Legendre spectral method demonstrates a tendency to increase fast.