An Analytical Study of Fractional Klein–Kramers Approximations for Describing Anomalous Diffusion of Energetic Particles

Abstract

In this article, anomalous transport models of energetic particles in space plasmas are developed by deriving the fractional force-less Fokker–Planck equation and the fractional diffusion-advection equation from the Klein–Kramers equation. Analytical solutions of the space-time fractional equations are obtained by use of the Caputo and the Riesz fractional derivatives with the Laplace–Fourier technique. The solutions are given in terms of Fox’s H-function and the profiles of the particles densities are discussed in each case for different values of fractional orders.

Appendix: Mittag–Leffler and Fox’s H-functions

In this appendix, we present the Mittag–Leffler and Fox’s H-functions, which have been used in our calculations. The Mittag–Leffler function [43] of the first kind is the generalization of the exponential function and defined as

$$\begin{aligned} E_{\alpha }(x)=\sum \limits _{n=0}^{\infty }\frac{x^{n}}{\varGamma (\alpha n+1)}. \end{aligned}$$

(A.1)

Also, the Mittag–Leffler of the second kind [43] is defined as

$$\begin{aligned} E_{\alpha ,\beta }(x)=\sum \limits _{n=0}^{\infty }\frac{x^{n}}{\varGamma (\alpha n+\beta )}. \end{aligned}$$

(A.2)

The Fox H-function [54] is the generalized Mellin-Barnes integral in the following manner

$$\begin{aligned} H_{p,q}^{m,n}\left( x\right)&=H_{p,q}^{m,n}\left[ x\left| \begin{array} [c]{l} \,(a_{1},A_{1}),....,(a_{p},A_{p})\\ \,(b_{1},B_{1}),....,(b_{q},B_{q}) \end{array} \right. \right] \nonumber \\&=\frac{1}{2\pi i}\int \nolimits _{L}\varTheta (s)\ x^{-s}ds, \end{aligned}$$

(A.3)

where

$$\begin{aligned} \varTheta (s)=\frac{\prod \limits _{j=1}^{m}\varGamma (b_{j}+B_{j}s)\ \prod \limits _{j=1}^{n}\varGamma (1-a_{j}-A_{j}s)\ }{\prod \limits _{j=m+1}^{q} \varGamma (1-b_{j}-B_{j}s)\ \prod \limits _{j=n+1}^{p}\varGamma (a_{j}+A_{j}s)}. \end{aligned}$$

(A.4)

m, n, p and q are integers satisfying $0\le n\le p,1\le m\le q$ ,$A_{j},B_{j}\in {\mathbb {R}} ^{+}$and $a_{j}.b_{j}\in {\mathbb {R}}$ or ${\mathbb {C}} .$

The Fourier Cosine Transform of the Fox’s H-function [55] is given by

$$\begin{aligned}&\int _{0}^{\infty }\left| y\right| ^{\rho -1}H_{p,q}^{m,n}\left[ \omega \left| y\right| ^{\mu }\left| \begin{array} [c]{l} \,(a_{p},A_{p})\,\\ \,(b_{q},B_{q}) \end{array} \right. \right] \cos (yx)dy\nonumber \\&\quad =\frac{\pi }{\left| x\right| ^{\rho }}H_{q+1,p+2}^{n+1,m}\;\left[ \frac{\left| x\right| ^{\mu }}{\omega }\left| \begin{array} [c]{l} \text { }(\,1-b_{q},\text { }B_{q}),\text { }(\frac{1+\rho }{2},\frac{\mu }{2})\\ \text { }(\rho ,\mu ),(1-a_{p},A_{p}),(\frac{1+\rho }{2},\frac{\mu }{2})\text { } \end{array} \right. \right] . \end{aligned}$$

(A.5)

The derivative of the Mittag–Leffler function is a special case of the Fox’s H-function

$$\begin{aligned} E_{\alpha ,\beta }^{n}(x)=H_{1.2}^{1,1}\left[ -x\left| \begin{array} [c]{l} (-n,1)\\ (0,1)\text { }(1-(\alpha n+\beta ),\alpha ) \end{array} \right. \right] . \end{aligned}$$

(A.6)

The Fox’s H-function can be expanded as a computable series [47] in the form

$$\begin{aligned} H_{p,q}^{m,n}\left( x\right)&=\sum _{h=1}^{m}\sum _{v=0}^{\infty } \frac{\prod \limits _{j=1,j\ne h}^{m}\varGamma (b_{j}+B_{j}(b_{h}+v)/B_{h} )\ }{\prod \limits _{j=m+1}^{q}\varGamma (1-b_{j}+B_{j}(b_{h}+v)/B_{h} )\ \ }\nonumber \\&\quad \times \frac{\prod \limits _{j=1}^{n}\varGamma (1-a_{j}+A_{j}(b_{h}+v)/B_{h} )}{\prod \limits _{j=n+1}^{p}\varGamma (a_{j}-A_{j}(b_{h}+v)/B_{h})}\frac{(-1)^{v}x^{(b_{h}+v)/B_{h}}}{v!B_{h}}. \end{aligned}$$

(A.7)

which is an alternating series and thus shows slow convergence. For large argument $\left| x \right| \rightarrow \infty$, the contour integral can be evaluated using the residue theorem and the Fox functions expanded as a series over the residues [45]

$$\begin{aligned} { H }_{ p,q }^{ m,n }(x)\sim \sum _{v=0 }^{\infty }{Res(\varTheta (s){x}^{s})}. \end{aligned}$$

(A.8)

to be taken at the points $s=({a}_{j}-1-v)/{A}_{j}$, for $j=1,..,n$.