THE CAPUTO EQUATION AS A MODEL OF PARTICLE SPREADING IN A NON-CLASSICAL DIFFUSION PROCESS: A SPECIAL CASE

Abstract

The main objective of this article is to present the application of the Mittag-Leffler function in anomalous diffusion modeling using the Caputo equation. Anomalous diffusion refers to non-classical diffusion processes in which traditional models fail to accurately capture the dynamics. The Caputo equation, which involves a fractional derivative of order in the Caputo sense, provides a powerful tool for describing these types of phenomena. The model we consider in this research presents particle diffusion , which represents the concentration or density of particles at position and time . The Laplacian operator captures the spatial diffusion process, and the diffusion coefficient governs the diffusion rate. An important aspect of this study is the integration of the Mittag-Leffler function, which arises in the solution of the Caputo equation. By solving the Caputo equation with the appropriate initial and boundary conditions, the concentration profile is obtained. The Mittag-Leffler function plays a key role in this solution because it accurately captures the memory-dependent behavior and the nonlocal nature of anomalous diffusion. A distinctive feature of this model is the presence of the fractional derivative of order in the Caputo sense, which captures the memory-dependent behavior and non-local nature of the diffusion process, allowing the representation of anomalous diffusion phenomena. In this paper, an important contribution is evidenced in the use of the Mittag-Leffler function to explore the behavior of anomalous diffusion processes and obtain information about the complex dynamics of particle propagation in various physical systems. Keywords: The Mittag-Leffler Function, Fractional Derivative, Anomalous Diffusion Processes, The Caputo Equation DOI： https://doi.org/10.35741/issn.0258-2724.58.4.91