A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications

Dongdong Ruan; Xiaofeng Wang; Dongdong Ruan; Xiaofeng Wang

doi:10.3934/era.2025065

Electronic Research Archive

2025, Volume 33, Issue 3: 1398-1413. doi: 10.3934/era.2025065

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A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications

Dongdong Ruan ,
Xiaofeng Wang ^,

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

Received: 08 January 2025 Revised: 27 February 2025 Accepted: 28 February 2025 Published: 11 March 2025

In this paper, the local convergence of a high-order Chebyshev-type method without the second derivative is studied. We study the convergence under $ \omega $-continuity conditions based on the first derivative. The uniqueness of the solution and the radii of convergence domains are obtained. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. In addition, the attractive basins of the family with different parameters are studied, which can show the different stability of the family. Finally, in numerical experiments, the iterative method is used to solve different nonlinear models, including vertical stresses, civil engineering problem, blood rheology model, and so on. Theoretical results of convergence criteria are verified.
- high-order method,
- local convergence,
- nonlinear equation,
- Chebyshev-type method,
- radius of convergence
Citation: Dongdong Ruan, Xiaofeng Wang. A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications[J]. Electronic Research Archive, 2025, 33(3): 1398-1413. doi: 10.3934/era.2025065

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Abstract

In this paper, the local convergence of a high-order Chebyshev-type method without the second derivative is studied. We study the convergence under $ \omega $-continuity conditions based on the first derivative. The uniqueness of the solution and the radii of convergence domains are obtained. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. In addition, the attractive basins of the family with different parameters are studied, which can show the different stability of the family. Finally, in numerical experiments, the iterative method is used to solve different nonlinear models, including vertical stresses, civil engineering problem, blood rheology model, and so on. Theoretical results of convergence criteria are verified.

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