New Family of Iterative Methods for Solving Nonlinear Models

Abstract

We introduce a new family of iterative methods for solving mathematical models whose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.

Fourth-order iterative method without calculating the higher derivatives for nonlinear equation

“…Using equation (20) in equation (19), one can get the following iterative method free from second derivatives.…”

“…where y k ¼ x k À fðx k Þ f 0 ðx k Þ for all k P 1. It is obvious to see that the iterative method equation (19) needs to calculate the second derivative of the function f(x). In order to avoid computing the second derivative, we introduce an approximant of the second derivative by using Taylor series.…”

“…The test equations are presented as below, most of them could be found in the literatures 17–19 or some references mentioned previously. …”

“…11 Starting with an given initial approximation x 0 , all numerical computations are carried out by using Mathematica software. For e ¼ 10 À14 , we choose the following stopping criteria so that the computer programs for iterative computations are terminated when the criteria are satisfied simultaneously: The test equations are presented as below, most of them could be found in the literatures [17][18][19] or some references mentioned previously. Table 1 are the different test equations f i ¼ 0; i ¼ 1; 2; .…”

By using a new iterative method to the generalized system Zakharov-Kuznetsov and estimate the best parameters via applied the pso algorithm

“…One of the most prominent problems we face in mathematical sciences, especially in numerical analysis, is solving non-linear equations. The new iterative method is one of the best modern methods for solving non-linear equations in addition to the Homotopy perturbation method, and decomposition method and others [19][20][21].…”