An efficient numerical method based on ultraspherical polynomials for linear weakly singular fractional Volterra integro-differential equations

The linear weakly singular fractional Volterra integro-differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy.