Authors: Jag Pratap Singh Yadav
Abstract: Classical diffusion models based on Fick’s law assume Brownian motion and local transport, leading to a mean squared displacement that grows linearly with time. However, many physical, biological, and engineering systems exhibit anomalous diffusion, where the mean squared displacement follows a power law in time rather than a linear relationship. Such behavior commonly arises in heterogeneous porous materials, crowded biological environments, polymeric systems, and disordered media, where long trapping times and memory effects invalidate standard integer-order diffusion equations. Despite significant progress in fractional modeling, there remains a need for mathematically consistent and computationally efficient formulations that clearly link the physical origin of anomalous transport to robust numerical implementation. In this paper, we develop a time-fractional diffusion model using the Caputo fractional derivative to represent memory-dependent transport induced by heavy-tailed waiting times. Starting from the conservation of mass and a constitutive relation with temporal memory, we derive a physically meaningful fractional diffusion equation. An analytical solution for a benchmark initial-boundary value problem is presented using Laplace and Fourier transforms, and a numerical approximation based on the L1 finite difference scheme is constructed. The stability and convergence properties of the numerical method are discussed. Numerical experiments demonstrate that the fractional order controls the transition from normal to subdiffusive transport and accurately reproduces power-law mean squared displacement behavior. The model captures anomalous transport with significantly fewer parameters than multi-scale classical alternatives. These results show that fractional differential equations provide an effective and parsimonious framework for describing memory-driven diffusion processes, with direct relevance to transport in porous media, biological tissues, and complex soft matter systems.
DOI: http://doi.org/10.5281/zenodo.70
