Superconvergent results for fractional Volterra integro-differential equations with non-smooth solutions

Abstract

This article focuses on finding the approximate solutions of fractional Volterra integro-differential equations with non-smooth solutions using the shifted Jacobi spectral Galerkin method (SJSGM) and its iterated version. To deal with the singularity present in the kernel of the transformed weakly singular Volterra integral equation, we convert it into an equivalent weakly singular Fredholm integral equation. We first directly apply our proposed methods to this equivalent transformed equation and obtain improved convergence results by incorporating the singularity of the kernel function into the shifted Jacobi weight function. Further, we introduce a smoothing transformation and discuss the regularity of the transformed solution, and achieve superconvergence results for all $\gamma \in (0,1)$. Additionally, we obtain super-convergence results for classical first-order Volterra integro-differential equations. Finally, numerical examples with a comparative study are provided to validate our theoretical results and verify the efficiency of the proposed methods. We show that the convergence rates can be obtained to the desired degree by increasing the value of the smoothing index $\varrho$ $(1<\varrho \in N)$, where $N$ stands for the set of natural numbers.