A novel fixed point approach based on Green's function for solution of fourth order BVPs

In recent years of the literature, fixed point iteration schemes have been studied theoretically for many nonlinear problems, including differential and integral equations; however, numerically, there has been a little work on these iterative schemes for finding numerical solutions of differential equations. The main purpose of this paper is to propose a new numerically efficient iterative approach based on fixed point theory and Green's function. First, we compute a Green's function for a wider class fourth order BVPs and express the sought solution as a fixed point of a certain integral operator. We show by Banach Contraction Principle that this new integral operator admits a unique fixed point which is the solution for a given BVP. We embed the integral operator into our iterative scheme and obtain a generalized iterative scheme. We prove the convergence of our new scheme under possible mild assumptions. Instead of stability, we prove a weak w2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w<^>{2}$$\end{document}-stability result for our scheme which is a natural and weak notion of the stability for any given iterative scheme. To support numerically the convergence of our scheme, we provide two different examples. Using these examples, we compare the high accuracy of our new scheme with the classical approaches of the literature. As an application of our scheme, we show that the scheme can be used for finding solutions for the Bratu's problem in a Banach space setting. At the end of the paper, we include some interesting discussion about the main results of the paper.