Convergence analysis of spectral methods for high-order nonlinear Volterra integro-differential equations

In this paper, we propose and analyze Chebyshev spectral collocation approximation for high-order nonlinear Volterra integro-differential equations. Under reasonable assumptions on the nonlinearity, it is shown that this numerical method converges exponentially in both L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}-norm and 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. Numerical results of several test examples are presented and comparisons are made with some existing numerical methods to prove the superiority and the effectiveness of the proposed method.