Generalized Auto-Convolution Volterra Integral Equations: Numerical Treatments

Abstract

In this paper, we use the operational Tau method based on orthogonal polynomials to achieve a numerical solution of generalized autoconvolution Volterra integral equations. Displaying a lower triangular matrix for basis functions, the corresponding solution is represented in matrix form, and an infinite upper triangular Toeplitz matrix is used to show the matrix representation of the integral part of the autoconvolution integral equation. We also investigate solvability of the obtained nonlinear system with infinite dimensional space and examine the convergence analysis of this method under the $L^2-$ norm. Finally, the efficiency of the operational Tau method is studied by numerical examples.