Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

This paper presents a novel numerical approach for solving linear Fredholm integro-differential equations. We integrate the backward finite difference method with the Nystr & ouml;m method to reduce the system size by half compared to the method proposed by Tair et al. (J. Appl. Math. Comput. Mech. 20(3):53-64, 2021). This reduction enhances computational efficiency and is particularly advantageous for large integration intervals. Our method ensures convergence by constructing a new norm for RN+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>{N+1}$$\end{document}. Numerical tests demonstrate the superiority of our method in terms of execution time and accuracy. The results contribute to the ongoing efforts in solving integro-differential equations more effectively, offering a robust tool for applications in various scientific and engineering domains.