In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.

First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $\nu$-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in ^[1]. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.