Fractional Fokker-Planck equation from non-singular kernel operators

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.