A Tau Method for Solving Second-Order Partial Integro-Differential Equations with Weakly Singular Kernels

This paper introduces a numerical method to approximate the solutions of initial-boundary value problems for a specific class of partial integro-differential equations. The approach utilizes Volterra Fredholm Integro Differential with Weakly singular kernel for spatial derivatives and the backward Tau method for temporal derivatives. The study delves into detailed Converting BVP to Volterra Fredholm Integral Differential Equation of the Second Kind with tau methode and using the Legendre or Chebyshev polynomials, proving their convergence and stability. The proposed method is then applied to several test cases, with the numerical outcomes compared with The Finite Element Method . The results demonstrate the computational efficiency of the method, leading to the conclusion that it is effective for solving initial-boundary value problems.