Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

Abstract

The numerical solutions to a non-linear Fractional Fokker-Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of nonlocal transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.

Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations

“…Moreover, by Lemmas 1 and 5, (21) and (22), {(F 1 u)(t) : u ∈ B r } is an equicontinuous and uniformly bounded set. Then, F 1 is a completely continuous operator on B r .…”

“…The fractional Langevin equation was introduced by Mainardi et al in the early 1990s [14,15]. Much work since then has been devoted to the study of the fractional Langevin equations in the field physics (e.g., [16][17][18][19][20][21][22]). Moreover, the fractional Langevin equations have been applied to describe various anomalous diffusive process, such as single file diffusion and crossover dynamics between different diffusive regimes (see, e.g., [23][24][25][26]).…”

Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

“…Recently, fractional differential equations (FDEs) has been able to be used extensively as the generalized type of integer-order differential equations, including ordinary differential equations and partial differential equations. FDEs have attracted the researchers' attention for modeling real-world phenomena, such as modeling anomalous diffusion using a nonlinear fractional Fokker-Planck equation with fractional velocity derivatives and Langevin dynamics to elucidate the effect of non-local transport in the plasma turbulence [27]. More examples of applications of FDEs for real-world problems can be found in [28][29][30] and the references therein.…”

Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

“…For examples, one can refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Besides, there are many results about fractional equations such as [16][17][18][19][20][21][22]. However, in the real world, at certain moments, many behaviors in neural networks may experience a sudden change.…”