On the solutions of three-point boundary value problems using variational-fixed point iteration method

Abstract

Given a three-point fourth-order boundary value problems y ðivÞ þ pðxÞy 000 þ qðxÞy 00 þ rðxÞy 0 þ sðxÞy ¼ f ðxÞ; a x b such that yðaÞ ¼ yðbÞ ¼ y 00 ðbÞ ¼ y 00 ðaÞ ¼ 0; a a b; where p; q; r; s; f 2 C½a; b , we combine the application of variational iteration method and fixed point iteration process to construct an iterative scheme called variationalfixed point iteration method that approximates the solution of three-point boundary value problems. The success of the variational or weighted residual method of approximation from a practical point of view depends on the suitable selection of the basis function. The method is self correcting one and leads to fast convergence. Problems were experimented to show the effectiveness and accuracy of the proposed method.Keywords Fixed point iteration Á Variational iteration method Á Lagrange's multiplier and boundary value problems.

Comparison of Basic Iterative Methods Used to Solve of Heat and Fluid Flow Problems

“…There has been a great deal of interest in finding a solution to the problem of second-order boundary values in differential equations in recent years. Particular emphasis is placed on determining the existence and approximation of the solution of twoor three-point boundary value problems [8,9]. Mann and his colleagues also generalised the variational iteration method for proper treatment of the boundary value problem in [10].…”

Approximation of Iterative Methods for Altering Points Problem With Applications

“…So far, numerous problems in nonlinear analysis involving contraction and nonexpansive mappings have been solved using this fundamental result. In recent past, the fixed point iterative schemes have been examined considerably to study monotone variational inequalities, problems from nonlinear analysis and applied mathematics such as initial and boundary value problems, image recovery problems, image restoration problems, image processing problems, for more details, see [12,13,25,26,32]. Note that, Mann-like iterative algorithms and its variant forms are efficiently applied for solving several nonlinear problems.…”

A Variational-Fixed Point Iterative technique for the Solution of Second Order Differential Equations

“…Many methods have been developed and others are still being developed. Some of these developed methods have their own limitations which have to undergo changes either to shed more light on their behavior or improve on their accuracy (Araghi F. M. A., 2011;He J. H., 2012;Kilicman A.,Wadai M., 2016). Few, among many methods developed, are the variational iteration method, the fixed point iteration method and so on.…”

“…Researchers have shown that the variational iteration method is flexible with good convergence property (He J. H.,2000). However, the success of the method depends on the careful choice of the starting function which is usually done by guessing (Daniel T., et al, 2012;Wu G. C., 2013;Kilicman A.,Wadai M., 2016;Lu J., 2007. In this study, we propose a Variational-Fixed Point Iterative technique where the starting function is determined by the use of the Galerkin approximation. The presentation of this article is as follows: In Section 2 we present the methodology, Section 3 deals with the numerical results and finally discussion and conclusion are given in the last section.…”