A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type

Godwin Amechi Okeke; Akanimo Victor Udo; Rubayyi T. Alqahtani; Nadiyah Hussain Alharthi; Godwin Amechi Okeke; Akanimo Victor Udo; Rubayyi T. Alqahtani; Nadiyah Hussain Alharthi

doi:10.3934/math.20231458

AIMS Mathematics

2023, Volume 8, Issue 12: 28488-28516. doi: 10.3934/math.20231458

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Research article

A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type

1.
Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology Owerri, P.M.B. 1526 Owerri, Imo State, Nigeria
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

Received: 07 August 2023 Revised: 05 September 2023 Accepted: 07 October 2023 Published: 19 October 2023
MSC : 34A08, 47J25, 47J26

In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.
- fixed point,
- rate of convergence,
- AG iterative scheme,
- Caputo fractional differential equation,
- weak convergence,
- $ \mathcal{J} $-stability
Citation: Godwin Amechi Okeke, Akanimo Victor Udo, Rubayyi T. Alqahtani, Nadiyah Hussain Alharthi. A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type[J]. AIMS Mathematics, 2023, 8(12): 28488-28516. doi: 10.3934/math.20231458

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Abstract

In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.

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