A product integration method for nonlinear second kind Volterra integral equations with a weakly singular kernel (with application to fractional differential equations)

Abstract

This paper presents a novel product integration method that provides an appropriate numerical solution for nonlinear weakly singular Volterra integral equations (WSVIEs). Extensive research in the literature has focused on studying the existence and uniqueness of solutions to these equations. However, when solving the WSVIEs, the solution may exhibit a singular behavior near the initial point of the integration interval, which can pose challenges for numerical computation. In these cases, we propose a smoothing change of variables that transforms the equation into one with a smooth solution, while still being weakly singular. We provide a convergence analysis and determine the order of convergence. The effectiveness of the proposed method is then demonstrated through the solution of various test problems.