Fractional-order multi-parametric methods for nonlinear problems

Ahmad Alalyani; Ahmad Alalyani

doi:10.3934/math.2026244

AIMS Mathematics

2026, Volume 11, Issue 3: 5911-5935. doi: 10.3934/math.2026244

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Fractional-order multi-parametric methods for nonlinear problems

Ahmad Alalyani ^,

Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65526, Saudi Arabia

Received: 13 November 2025 Revised: 03 February 2026 Accepted: 02 March 2026 Published: 09 March 2026
MSC : 65H04, 65H05, 65B99, 65Y15, 65Y04

In this research, a two-step bi- and tri-parametric iterative method is developed for solving nonlinear equations employing the Caputo fractional derivative. The theoretical convergence analysis demonstrates that the proposed schemes achieve an order of convergence of $ 2\tau+1 $, where $ \tau $ denotes the order of the Caputo fractional operator. In addition, fractal analysis is employed to identify effective initial values that enhance numerical performance. To compare the suggested scheme's efficacy and stability to existing approaches, several nonlinear engineering problems are examined. The numerical results demonstrate that, compared with existing methods, the proposed schemes achieve lower residual errors, higher convergence rates, reduced memory consumption, improved error profiles, and superior computational orders of convergence, thereby making them a more efficient and reliable alternative for solving problems in science and engineering.
- fractional order scheme,
- convergence analysis,
- stability analysis,
- dynamical planes,
- biomedical engineering applications
Citation: Ahmad Alalyani. Fractional-order multi-parametric methods for nonlinear problems[J]. AIMS Mathematics, 2026, 11(3): 5911-5935. doi: 10.3934/math.2026244

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Abstract

In this research, a two-step bi- and tri-parametric iterative method is developed for solving nonlinear equations employing the Caputo fractional derivative. The theoretical convergence analysis demonstrates that the proposed schemes achieve an order of convergence of $ 2\tau+1 $, where $ \tau $ denotes the order of the Caputo fractional operator. In addition, fractal analysis is employed to identify effective initial values that enhance numerical performance. To compare the suggested scheme's efficacy and stability to existing approaches, several nonlinear engineering problems are examined. The numerical results demonstrate that, compared with existing methods, the proposed schemes achieve lower residual errors, higher convergence rates, reduced memory consumption, improved error profiles, and superior computational orders of convergence, thereby making them a more efficient and reliable alternative for solving problems in science and engineering.

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