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

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.