New Family of Iterative Methods for Solving Nonlinear Models

Ali, Faisal; Aslam, Waqas; Ali, Kashif; Anwar, Muhammad; Nadeem, Akbar

doi:10.1155/2018/9619680

Cited by 15 publications

(15 citation statements)

References 21 publications

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

Section: The Iterative Methodsmentioning

confidence: 99%

“…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.…”

Section: The Iterative Methodsmentioning

confidence: 99%

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

Section: Numerical Examplesmentioning

confidence: 99%

“…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; .…”

Section: Numerical Examplesmentioning

confidence: 99%

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

2019

Journal of Algorithms & Computational Technology

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In this paper, we present firstly several one-step iterative methods including classical Newton's method and Halley's method for root-finding problem based on Thiele's continued fraction. Secondly, by using approximants of the second derivative and the third derivative, we obtain an improved iterative method, which is not only two-step iterative method but also avoids calculating the higher derivatives of the function. Analysis of its convergence shows that the order of convergence of the modified iterative method is four for a simple root of the equation. Finally, to illustrate the efficiency and performance of the proposed method, we give some numerical experiments and comparison.

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

Section: The Iterative Methodsmentioning

confidence: 99%

Section: The Iterative Methodsmentioning

confidence: 99%

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

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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

2019

Journal of Algorithms & Computational Technology

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Aldabagh

2020

IJEECS

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In this paper, a new iterative method was applied to the Zakharov-Kuznetsov system to obtain the approximate solution and the results were close to the exact solution, A new technique has been proposed to reach the lowest possible error, and the closest accurate solution to the numerical method is to link the numerical method with the pso algorithm which is denoted by the symbol (NIM-PSO). The results of the proposed Technique showed that they are highly efficient and very close to the exact solution, and they are also of excellent effectiveness for treating partial differential equation systems.

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Iteration Methods with an Auxiliary Function for Nonlinear Equations

Ali

Aslam

Khalid

et al. 2020

Journal of Mathematics

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Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

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New Family of Iterative Methods for Solving Nonlinear Models

Cited by 15 publications

References 21 publications

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

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

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

Iteration Methods with an Auxiliary Function for Nonlinear Equations

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