University of Guilan Journal of Mathematical Modeling 2345-394X 14 1 2026 03 01 Tau-collocation method for weakly singular Volterra integral equations and related special cases 161 175 9094 10.22124/jmm.2025.30587.2742 EN Sedaghat Shahmorad Department of Applied Mathematics, University of Tabriz, Tabriz-Iran Mahdi Mostafazadeh Department of Applied Mathematics, University of Tbariz, Tabriz Fevzi Erdogan Department of Mathematics, Van Uzuncu Yil University, Van, Turkey Journal Article 2025 05 05 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 $(x^2-s^2)^{-1/2}$. These types of equations can be written in the form of the so-called \textit{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 $(x-s)^{-\alpha}$, $0<\alpha<1$, 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. https://jmm.guilan.ac.ir/article_9094_4fc9d29d0d798536c0e26a59ed6413dc.pdf