Tau-Collocation Method for Weakly Singular Volterra Integral Equations and Related Special Cases

Abstract

The present study examines the implementation of the tau-collocation method for solving a class of Volterra integral equations and related cases which their kernels contain (special) weak singularity of type (x2 − s2)−1/2. These types of equations can be written in the form of the so-called cordial Volterra integral equations and so inherit their properties. We will recall some conditions on the kernel and forcing function for which the existence and uniqueness of a solution has been proven. Then we will discuss regularity conditions for the solution of same types equations which indicate that unlike the standard Volterra integral equations with singularity of the form (x − s)−α, 0 < α < 1, these types of equations have regular solutions if the kernel and forcing functions are sufficiently smooth. This property allows us to use the classical Jacobi polynomials as a basis functions for collocation method. For this method, we will first derive a matrix formulation that makes it easy to implement. We will prove convergence of the method by providing an error bound. © 2026 University of Guilan.