EXPLORING THE IMPACT OF FRACTAL–FRACTIONAL DERIVATIVES ON CHAOTIC SYSTEMS AND HEAT TRANSFER

Abstract

This study presents a novel application of fractal–fractional differential equations (FFDEs) to model heat and mass transfer phenomena in porous media, particularly in the context of chaotic systems. We introduce a new formulation of the fractal–fractional heat and advection–diffusion equations, incorporating Caputo–Fabrizio operators to capture the complexities of anomalous diffusion in heterogeneous porous structures. The existence and uniqueness of solutions are rigorously established using the Banach contraction principle, while stability analyses ensure the robustness of the proposed model. Furthermore, numerical simulations illustrate the model’s capability to describe intricate transport dynamics, including the emergence of chaotic behavior arising from fluid instabilities in porous channels. By extending classical chaotic models such as the Lorenz–Lü–Chen system to the fractal–fractional framework, this research provides deeper insights into nonlinear transport mechanisms in porous media. The application of the Adomian decomposition method enables highly accurate solutions, demonstrating its effectiveness in solving nonlinear FFDEs. These findings have significant implications across various scientific and engineering disciplines, including energy systems, environmental science, and biomedical engineering, by offering a refined approach to modeling complex dynamical processes with memory and hereditary effects.