A new Tau-collocation method with fractional basis for solving weakly singular delay Volterra integro-differential equations

The main purpose of this paper is to introduce a new formulation of the Tau-collocation method for solving a class of nonlinear weakly singular delay Volterra integro-differential equations based on fractional Müntz Jacobi polynomials. So the paper consists of two main parts: studying the regularity of solution and introducing a simple structure and efficient numerical method. The method generates approximate solution with fractional power terms having the same behavior of the exact solution of the given problem. The convergence analysis of the method is also investigated in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-norm. Some illustrative examples are given to confirm the theoretical results and the accuracy of the approximate solution. The paper is closed by providing real application of the method to some physical problems.