Convergence Analysis of the Tau Method for Volterra–Fredholm Integrodifferential Equations With Weakly Singular Kernel

This work focuses on the analysis of the Volterra–Fredholm integral equation (VFIE) with a weakly singular kernel, which is a transformed form of a partial integrodifferential equation (PIDE) with a weakly singular kernel. We present an approach based on the spectral Tau method for determining the unknown function in the VFIE formulation with respect to the spatial variable. In this method, the approximate solution is expanded using Legendre polynomials, reducing the problem to a system of algebraic equations obtained through the Tau method applied to the VFIE. Furthermore, convergence analysis and error estimation of the Tau method are provided for the VFIE formulation. Numerical results, obtained by applying the Tau method to the integral VFIE formulation—our proposed approach—as well as directly to the PIDE, demonstrate the superior numerical stability of the integral-based method.