Predictor-corrector higher derivative-free algorithms for nonlinear equations with basin of attraction analysis

Ali Hasan Ali; Iman A. Abdul Samad; Huda J. Saeed; Firas Ghanim; Alina Alb Lupaș; Ali Hasan Ali; Iman A. Abdul Samad; Huda J. Saeed; Firas Ghanim; Alina Alb Lupaș

doi:10.3934/math.2026679

AIMS Mathematics

2026, Volume 11, Issue 6: 16550-16569. doi: 10.3934/math.2026679

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Research article Topical Sections

Predictor-corrector higher derivative-free algorithms for nonlinear equations with basin of attraction analysis

1.
Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq
2.
Technical Engineering College, Al-Ayen University, Dhi Qar, 64001, Iraq
3.
Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates
4.
Department of Mathematics and Computer Science, University of Oradea, Romania

Received: 07 January 2026 Revised: 06 March 2026 Accepted: 17 March 2026 Published: 09 June 2026
MSC : 65B99, 65H05

This work presents two new numerical methods for finding the zeros of nonlinear equations, based on the Adomian decomposition approach and the quadrature rule. Using a predictor–corrector scheme, the proposed iterative methods provide a high order of convergence and do not require second derivatives, thereby reducing computational time. The methods are tested on several functions, and their efficiency is demonstrated through comparisons with various established techniques. To support the numerical findings, graphical analyses are provided. Furthermore, a study of the basins of attraction is conducted to validate the stability of the proposed methods.
- quadrature rule,
- Adomian decomposition approach,
- iterative methods,
- basins of attraction,
- numerical analysis
Citation: Ali Hasan Ali, Iman A. Abdul Samad, Huda J. Saeed, Firas Ghanim, Alina Alb Lupaș. Predictor-corrector higher derivative-free algorithms for nonlinear equations with basin of attraction analysis[J]. AIMS Mathematics, 2026, 11(6): 16550-16569. doi: 10.3934/math.2026679

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Abstract

This work presents two new numerical methods for finding the zeros of nonlinear equations, based on the Adomian decomposition approach and the quadrature rule. Using a predictor–corrector scheme, the proposed iterative methods provide a high order of convergence and do not require second derivatives, thereby reducing computational time. The methods are tested on several functions, and their efficiency is demonstrated through comparisons with various established techniques. To support the numerical findings, graphical analyses are provided. Furthermore, a study of the basins of attraction is conducted to validate the stability of the proposed methods.

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