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.

References in corpus (19)

• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• A new fractional operator of variable order: application in the description of anomalous diffusion
• The role of fractional time-derivative operators on anomalous diffusion
• Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions
• High order schemes for the tempered fractional diffusion equations
• A model of non-Gaussian diffusion in heterogeneous media
• Economic interpretation of fractional derivatives
• Infinite densities for Lévy walks
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• Discrete-time quantum walk with feed-forward quantum coin
• Heterogeneous diffusion in comb and fractal grid structures
• Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa
• New representations of pi and Dirac delta using the nonextensive-statistical-mechanics q-exponential function
• Anomalous diffusion on a fractal mesh
• Lévy processes on a generalized fractal comb
• Thermodynamic Framework for Compact q-Gaussian Distributions
• Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution
• Convergence of the probability of large deviations in a model of correlated random variables having compact-support $Q$-Gaussians as limiting distributions

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• Critical patch size reduction by heterogeneous diffusion
• Efficiency of random search with space-dependent diffusivity