Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches

Chatthai Thaiprayoon; Jutarat Kongson; Weerawat Sudsutad; Chatthai Thaiprayoon; Jutarat Kongson; Weerawat Sudsutad

doi:10.3934/math.2025405

AIMS Mathematics

2025, Volume 10, Issue 4: 8827-8872. doi: 10.3934/math.2025405

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Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches

1.
Research Group of Theoretical and Computational Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
2.
Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand

Received: 14 January 2025 Revised: 04 April 2025 Accepted: 08 April 2025 Published: 17 April 2025
MSC : 34A08, 34A34, 34D20, 65L05

This work studies a fractional model in the context of the fractal-fractional operator involving a power law kernel for waste plastic in the ocean consisting of waste plastic material, marine debris, and reprocessing. The novelty of this study lies in utilizing the fractal-fractional framework to analyze the behavior of the proposed model and derive qualitative theoretical results. We investigated the equilibrium points and analyzed their stability using the basic reproduction number. We examined the global stability of possible equilibrium points, as well as their unstable conditions. The sensitivity analysis can also help to better understand the model's dynamics better. The existence and uniqueness of results were studied by utilizing Banach's contraction mapping principle, while Ulam's stability for the proposed model was also investigated. Furthermore, we derived the numerical algorithms by utilizing four different techniques, including the decomposition, Adams-Bashforth, Newton polynomial, and predictor-corrector methods. By providing the input factor values, we illustrated the graphic numerical simulations to enhance our understanding of the waste plastic process and optimize the control strategies. Moreover, we compare ordinary, fractional, and fractal-fractional derivatives in the sense of the Caputo type with the reported real data of $ 70 $ years.
- waste plastic management,
- Volterra-type Lyapunov function,
- fractal-fractional derivative,
- Ulam-Hyers stability,
- numerical scheme
Citation: Chatthai Thaiprayoon, Jutarat Kongson, Weerawat Sudsutad. Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches[J]. AIMS Mathematics, 2025, 10(4): 8827-8872. doi: 10.3934/math.2025405

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Abstract

This work studies a fractional model in the context of the fractal-fractional operator involving a power law kernel for waste plastic in the ocean consisting of waste plastic material, marine debris, and reprocessing. The novelty of this study lies in utilizing the fractal-fractional framework to analyze the behavior of the proposed model and derive qualitative theoretical results. We investigated the equilibrium points and analyzed their stability using the basic reproduction number. We examined the global stability of possible equilibrium points, as well as their unstable conditions. The sensitivity analysis can also help to better understand the model's dynamics better. The existence and uniqueness of results were studied by utilizing Banach's contraction mapping principle, while Ulam's stability for the proposed model was also investigated. Furthermore, we derived the numerical algorithms by utilizing four different techniques, including the decomposition, Adams-Bashforth, Newton polynomial, and predictor-corrector methods. By providing the input factor values, we illustrated the graphic numerical simulations to enhance our understanding of the waste plastic process and optimize the control strategies. Moreover, we compare ordinary, fractional, and fractal-fractional derivatives in the sense of the Caputo type with the reported real data of $ 70 $ years.

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