Convergence analysis of Jacobi spectral tau-collocation method in solving a system of weakly singular Volterra integral equations

Abstract

The main purpose of this paper is to solve a system of weakly singular Volterra integral equations using the Jacobi spectral tau-collocation method from two perspectives. Since the solutions of the main system exhibit discontinuity at the origin, classical Jacobi methods may yield less accuracy. Therefore, in the first approach, we transform the proposed system through a suitable transformation into an alternative type whose solutions are as smooth as desired. Subsequently, we derive a matrix formulation of the method and analyze its convergence properties in both L2 and L∞-norms.In the second approach, instead of employing a smoothing transformation, we select fractional Jacobi polynomials as basis functions for the approximation space. This choice is motivated by their similar behavior to the exact solutions. We then derive a matrix formulation of the method and perform an error analysis analogous to the first approach. Finally, we present several illustrative examples to demonstrate the accuracy of our method.