  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.
