Shifted Legendre Fractional Pseudo-spectral Integration Matrices for Solving Fractional Volterra Integro-Differential Equations and Abel's Integral Equations

Shifted Legendre polynomials (SLPs) with the Riemann–Liouville fractional integral operator have been used to create a novel fractional integration tool. This tool will be called the fractional shifted Legendre integration matrix (FSL B-matrix). Two algorithms depending on this matrix are designed to solve two different types of integral equations. The first algorithm is to solve fractional Volterra integro-differential equations (VIDEs) with a non-singular kernel. The second algorithm is for Abel’s integral equations. In addition, error analysis for the spectral expansion has been proven to ensure the expansion’s convergence. Finally, several examples have been illustrated, including an application for the population model.