Convergence of a refined iterative method and its application to fractional Volterra-Fredholm integro-differential equations

Our research introduces an innovative iterative method for approximating fixed points of contraction mappings in uniformly convex Banach spaces. To validate the stability of this iterative process, we provide a comprehensive theorem. Through detailed examples and graphical analysis, we demonstrate that our method outperforms previous approaches for contraction mappings, including those developed by Agarwal, Gursoy, Thakur, Ali, and D & lowast;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D<^>{**}$$\end{document}, using MATLAB software for implementation and comparison. Moreover, we investigate the effect of various parameters on the convergence behavior of our proposed method. By comparing it with existing iterative schemes through a specific example, we highlight the efficiency and robustness of our approach. Additionally, we establish a significant result concerning data dependence for an approximate operator, utilizing our iterative process to show how small changes in data can affect the outcome. Finally, we apply our fundamental findings to a practical problem by estimating solutions for a fractional Volterra-Fredholm integro-differential equation. This application not only illustrates the practical utility of our method but also underscores its potential for solving complex problems in applied mathematics.