On the convergence analysis of the Tau method applied to fourth-order partial differential equation based on Volterra-Fredholm integral equations

Abstract

This paper presents a study of the performance of the Tau method using orthogonal polynomials as basis functions for solving fourth-order partial differential equations with boundary conditions. Because of the good numerical stability properties of integral operators in compare to differential operator, we first convert this PDE problem to Volterra-Fredholm integral equation and then apply the numerical Tau method to solve the obtained integral equation. Applying the Tau method yields a system of the ordinary differential equation such that this system is solved by piecewise polynomial collocation methods. Convergence analysis and error estimation of the Tau method are discussed. The advantages of converting PDE to integral equation are shown by the numerical examples. For this aim, we consider two cases to solve the proposed examples. In case1, we apply the Tau method to solve the converted problem (Volterra-Fredholm integral equation) and in case 2, we solve PDE problem directly by Tau method. Comparing the numerical results, we observe that the obtained errors in case 1 are less than the errors in case 2.