A single parameter fourth-order Jarrat-type iterative method for solving nonlinear systems

Jia Yu; Xiaofeng Wang; Jia Yu; Xiaofeng Wang

doi:10.3934/math.2025360

AIMS Mathematics

2025, Volume 10, Issue 4: 7847-7863. doi: 10.3934/math.2025360

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Research article

A single parameter fourth-order Jarrat-type iterative method for solving nonlinear systems

Jia Yu ,
Xiaofeng Wang ^,

School of Mathematical Sciences, Bohai University, Jinzhou 121000, Liaoning, China

Received: 29 December 2024 Revised: 14 March 2025 Accepted: 28 March 2025 Published: 03 April 2025
MSC : 65B99, 65H05

In this paper, a Jarrat-type iterative method for solving nonlinear systems is proposed through the weight function method. It is proved that the convergence order of the iterative method reaches fourth order in Banach space. The stability of this fourth-order iterative method was analyzed using real dynamics theory. The number of fixed points and critical points was solved using real dynamics tools, and the dynamic plane was drawn. Through the dynamic plane, it can be observed that the stability of the iterative method is best when the parameters are $ 0\le\beta < \frac{423}{8} $. Next, we compare the efficiency of the proposed method with two fourth-order iterative methods. The results show that when $ \beta = \frac{9}{8} $, our method shows better efficiency in terms of calculation time, calculation error, and calculation efficiency. In addition, we also use this method to successfully solve the Hammerstein-type integral equation, boundary value problem, heat conduction problem in partial differential equation, and other nonlinear equation systems. The experimental results are consistent with the theoretical analysis, which further confirms the accuracy of the method.
- Jarrat-type iterative method,
- nonlinear systems,
- stability analysis
Citation: Jia Yu, Xiaofeng Wang. A single parameter fourth-order Jarrat-type iterative method for solving nonlinear systems[J]. AIMS Mathematics, 2025, 10(4): 7847-7863. doi: 10.3934/math.2025360

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Abstract

In this paper, a Jarrat-type iterative method for solving nonlinear systems is proposed through the weight function method. It is proved that the convergence order of the iterative method reaches fourth order in Banach space. The stability of this fourth-order iterative method was analyzed using real dynamics theory. The number of fixed points and critical points was solved using real dynamics tools, and the dynamic plane was drawn. Through the dynamic plane, it can be observed that the stability of the iterative method is best when the parameters are $ 0\le\beta < \frac{423}{8} $. Next, we compare the efficiency of the proposed method with two fourth-order iterative methods. The results show that when $ \beta = \frac{9}{8} $, our method shows better efficiency in terms of calculation time, calculation error, and calculation efficiency. In addition, we also use this method to successfully solve the Hammerstein-type integral equation, boundary value problem, heat conduction problem in partial differential equation, and other nonlinear equation systems. The experimental results are consistent with the theoretical analysis, which further confirms the accuracy of the method.

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