Solvability of Volterra integral equations of third kind by modified Bernstein operators

Abstract

In this paper, we apply modified Bernstein operators Bm & lowast;(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B<^>{*}_{m}(f)$\end{document} to obtain the solution of Volterra integral equations of third type. We construct the numerical scheme, study the convergence analysis and the corresponding rate of convergence of third type Volterra integral equations. We concentrate on analysing the convergence of the suggested solution approach in order to verify it theoretically. We also aim to evaluate an upper limit for supxj is an element of[0,1]|f(xj)-Bm & lowast;(fm(xj))|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup \limits _{{x_{j}}\in [0,1]} \big | f (x_{j})-B<^>{*}_{m}( f_{m}(x_{j})) \big |$\end{document} where it must finally go to zero in the limiting case. Finally, some numerical examples have been presented to demonstrate the usefulness of this new method for solving third-type Volterra integral equations.