An $hp$-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $hp$-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented.