An error estimation of a Nyström type method for integral-algebraic equations of index-1

This paper presents a numerical method based on the first kind of Chebyshev polynomials for solving a coupled system of Volterra integral equations of the second and first kind. For sake using the theory of orthogonal Chebyshev polynomials, we use some variable transformations to change the mentioned system into a new system on the interval [-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1, 1]$$\end{document}. The integral-algebraic equations belong to the class of moderately ill-posed problems. The main idea in the numerical method is that we will approximate the product of the kernels and solutions which using this idea, we achieve an accurate algorithm. Due to the presence of the first kind Volterra integral equation, convergence analysis can be challenging. We analyze the convergence of this method by computation of over estimate for errors. Finally, the numerical examples confirm the validity of the convergence analysis.