Fractional Calculus-Based Modeling and Numerical Simulation of Anomalous Diffusion Processes

Abstract

Anomalous diffusion processes are nonlocal processes with memory behavior that are commonly observed in complex systems, like porous media, biological tissue, and financial dynamics. The classical integer-order diffusion models are usually incapable of capturing these effects because they are not able to model any long-range temporal dependencies. To overcome this shortcoming, this paper develops a time-fractional partial diffusion model of anomalous diffusion by using a fractional calculus model. An approximate time scheme and scheme based on the space operator are discretised to come up with a numerical scheme that has better efficiency and accuracy. The stability and convergence of the proposed method are thoroughly studied, and the method has been proved to be reliable in different discretization parameters. Various experiments are conducted to perform numerical simulations on various values of the order of the fractional derivatives, which underscores the role of the memory effects on diffusion processes. The findings show the improved accuracy and consistency with the classical diffusion models and existing fractional methods. The main contributions made by this work are; formulation of effective fractional diffusion model, a stable and convergent numerical method and systematic analysis of abnormal diffusion behavior under different conditions of the system.