Analysis of nonlinear ordinary differential equations with the generalized Mittag-Leffler kernel

Abdon Atangana; Jyoti Mishra; Abdon Atangana; Jyoti Mishra

doi:10.3934/mbe.2023875

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 19763-19780. doi: 10.3934/mbe.2023875

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Analysis of nonlinear ordinary differential equations with the generalized Mittag-Leffler kernel

Abdon Atangana ^{1,2
,
,},
Jyoti Mishra ³

1.
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
3.
Gyan Ganga Institute of Technology and Sciences, Department of Mathematics, Jabalpur, Madhya Pradesh, India

Academic Editor: Carlo Cattani

Received: 30 June 2023 Revised: 30 August 2023 Accepted: 05 September 2023 Published: 30 October 2023

The Picard iterative approach used in the paper to derive conditions under which nonlinear ordinary differential equations based on the derivative with the Mittag-Leffler kernel admit a unique solution. Using a simple Euler approximation and Heun's approach, we solved this nonlinear equation numerically. Some examples of a nonlinear linear differential equation were considered to present the existence and uniqueness of their solutions as well as their numerical solutions. A chaotic model was also considered to show the extension of this in the case of nonlinear systems.
- nonlinear equation,
- Picard iteration,
- Atangana-Baleanu derivative,
- Euler and Heun's approaches
Citation: Abdon Atangana, Jyoti Mishra. Analysis of nonlinear ordinary differential equations with the generalized Mittag-Leffler kernel[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19763-19780. doi: 10.3934/mbe.2023875

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Abstract

The Picard iterative approach used in the paper to derive conditions under which nonlinear ordinary differential equations based on the derivative with the Mittag-Leffler kernel admit a unique solution. Using a simple Euler approximation and Heun's approach, we solved this nonlinear equation numerically. Some examples of a nonlinear linear differential equation were considered to present the existence and uniqueness of their solutions as well as their numerical solutions. A chaotic model was also considered to show the extension of this in the case of nonlinear systems.

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