A fractional Fokker–Planck equation for non-singular kernel operators

Abstract

Fractional diffusion equations imply non-Gaussian distributions that allow the characterisation of a wide variety of diffusive processes. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels. 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 unimodal or bimodal behavior according to whether the diffusion index ν is positive or negative respectively, where the diffusion coefficient of the power law type is considered. Thereby, we show that several types of diffusion can be characterised in the context of fractional kernel operators.