Generalized fractional diffusion equation with arbitrary time varying diffusivity

Anomalous diffusion processes in many complex systems are frequently described by the diffusion equation with a time-dependent diffusion coefficient. This paper introduces an exact solution to the broad classes of the fractional diffusion equation with the arbitrary time-dependent diffusion coefficient by using the Laplace-Fourier technique. The Riesz fractional derivative serves to replace the Laplacian operator, while the new regularized Caputo-type fractional derivative is employed instead of the time derivative. We examine our results by introducing and analyzing the most three common cases that represent dif-fusivity that varying with time. Our calculation shows exact matching with the probability distribution function and mean square displacement illustrated in the literature. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper presents an exact solution for a wide range of fractional diffusion equations with arbitrary time-dependent diffusion coefficients using the Laplace-Fourier technique. The results are validated by examining three common cases of varying diffusivity with time and show exact matching with probability distribution functions and mean square displacements as illustrated in existing literature.