Fractional Fokker-Planck equation from non-singular kernel operators

TL;DR

This paper introduces a generalized fractional Fokker-Planck equation using non-singular memory kernels, providing analytical solutions that reveal non-Gaussian, unimodal or bimodal distributions linked to anomalous diffusion.

Contribution

It develops a new fractional Fokker-Planck framework with non-singular kernels and variable diffusion coefficients, expanding the understanding of memory effects in anomalous diffusion.

Findings

Analytical solutions for Caputo-Fabrizio and Atangana-Baleanu kernels.

Emergence of non-Gaussian distributions with long and short tails.

Distribution modality depends on the diffusion index .

Abstract

Fractional diffusion equations imply non-Gaussian distributions that generalise the standard diffusive process. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels, differently from the fractional operator defined in the literature. In this work we propose a generalisation of the Fokker-Planck equation in terms of a non-singular fractional temporal operator and considering a non-constant diffusion coefficient. We obtain analytical solutions for the Caputo-Fabrizio and the Atangana-Baleanu fractional kernel operators, from which non-Gaussian distributions emerge having a long and short tails. In addition, we show that these non-Gaussian distributions are unimodal or bimodal according if the diffusion index $\nu$ is positive or negative respectively, where a diffusion coefficient of the power law type $\mathcal{D}(x)=\mathcal{D}_0|x|^{\nu}$ is considered. Thereby, a class of anomalous diffusion phenomena connected with fractional derivatives and with a diffusion coefficient of the power law type is presented. The techniques employed in this work open new possibilities for studying memory effects in diffusive contexts.