Generalized diffusion equation with nonlocality of space-time: analytical and numerical analysis

TL;DR

This paper derives and analyzes a generalized diffusion equation incorporating space-time nonlocality and fractality using fractional calculus, providing analytical and numerical insights into its spectral properties and velocity behaviors.

Contribution

It introduces a new generalized diffusion equation with fractional derivatives accounting for space-time nonlocality and fractality, along with analytical and numerical analysis of its spectral characteristics.

Findings

Spectral analysis of the generalized diffusion equation

Numerical results on phase and group velocities

Impact of fractality indexes on diffusion dynamics

Abstract

Based on the non-Markov diffusion equation taking into account the spatial fractality and modeling for the generalized coefficient of particle diffusion $D^{\alpha\alpha'}(\mathbf{r},\mathbf{r}';t,t')=W(t,t')\bar{D}^{\alpha\alpha'}(\mathbf{r},\mathbf{r}')$ using fractional calculus the generalized Cattaneo--Maxwell--type diffusion equation in fractional time and space derivatives has been obtained. In the case of a constant diffusion coefficient, analytical and numerical studies of the frequency spectrum for the Cattaneo--Maxwell diffusion equation in fractional time and space derivatives have been performed. Numerical calculations of the phase and group velocities with change of values of characteristic relaxation time, diffusion coefficient and indexes of temporal $\xi$ and spatial $\alpha$ fractality have been carried out.