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On fractional approximations of the Fokker–Planck equation for energetic particle transport

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Abstract

To determine the main effects of anomalous diffusion upon the propagation of energetic particles, the fractional modified telegraph equation is solved using the Laplace–Fourier technique and given in terms of the Fox H and M-Wright functions. Examples are used to illustrate the qualitative differences between the fractional telegraph, fractional advection–diffusion and fractional diffusion equations. The profiles of the particles densities are discussed in each case for different values of fractional orders.

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Acknowledgements

The author is very grateful to anonymous referees whose questions, remarks, and suggestions, which allowed to improve the paper and significantly clarify its meaning.

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Correspondence to Ashraf M. Tawfik.

Appendices

Appendix A: Caputo and Riesz fractional derivatives

In this appendix, we introduce two fractional derivatives, one in the Caputo sense, used to consider time variables, and the other is a Riesz fractional derivative, which will be associated with the space variable. Furthermore, we introduce the Laplace transform of the Caputo derivative and Fourier transform of the Riesz derivative.

The Caputo fractional derivative [53] is a proper definition of a fractional derivative that can provide initial conditions with a physical interpretation for a differential equation of fractional order. Caputo’s definition of the derivative of any continuous function \(\phi (t)\) is given as follows:

$$\begin{aligned} ^{c}D_{t}^{\alpha }\phi (t)= {\frac{1}{\varGamma (n-\alpha )}\int _{0}^{t}(t-\tau )^{n-\alpha -1\,}\frac{\hbox {d}^{n}}{\hbox {d}\tau ^{n}}\phi (\tau )~\hbox {d}\tau ,\, n-1<\alpha <n,n\ge 0}, \end{aligned}$$
(A.1)

with a Laplace transform that obeys [54]

$$\begin{aligned} L\left\{ ^{c}D_{t}^{\alpha }\phi (t),s\right\} =s^{\alpha }\phi (s)-\sum _{k=0}^{n-1}s^{\alpha -1-k}\;\phi ^{k}(0^{+}),n-1<\alpha \le n,n\in N. \end{aligned}$$
(A.2)

The proper generalization of the Laplacian is the so-called Riesz fractional derivative [55], which is given for any continuous function f(x) by

$$\begin{aligned} \frac{\partial ^{\beta }f(x)}{\partial \left| x\right| ^{\beta }}=\frac{1}{\pi }\sin \left( \frac{\pi }{2}\beta \right) ~\varGamma (1+\beta )~\int \limits _{0}^{\infty }\frac{f(x+\xi )-2f(x)+f(x-\xi )}{\xi ^{1+\beta }}\hbox {d}\xi . \end{aligned}$$
(A.3)

The regularized form above is valid for \(0<\beta <2\), with the Fourier transform property [55]

$$\begin{aligned} {\mathcal {F}} \left\{ \frac{\partial ^{\beta }f(x)}{\partial \left| x\right| ^{\beta } },k\right\} =-\left| k\right| ^{\beta }\phi (k),\beta >0. \end{aligned}$$
(A.4)

Appendix B: M-Wright, Mittag–Leffler and Fox’s H-functions

In this appendix, we present the M-Wright, Mittag–Leffler and Fox’s H-functions, that have been used in our calculations.

The general Wright function denoted by \(W_{\lambda ,\mu }(x)\) is defined by the series representation, as [56]

$$\begin{aligned} W_{\nu ,\mu }(x)= {\displaystyle \sum \limits _{n=0}^{\infty }} \frac{x^{n}\ }{n!\ \varGamma (n\nu +\mu )},\quad \nu >-1,\mu \in C. \end{aligned}$$
(B.1)

In their analysis of time-fractional diffusion, Mainardi et al. introduced the M-Wright function defined as [38]

$$\begin{aligned} M_{\nu }(x)=W_{-\nu ,1-\nu }(-x)= {\displaystyle \sum \limits _{n=0}^{\infty }} \frac{\left( -x\right) ^{n}\ }{n!\ \varGamma (-\nu n+1-\nu )}. \end{aligned}$$
(B.2)

The M-Wright/Mainardi function has some explicit representations that should be mentioned, namely,

$$\begin{aligned} M_{0}(x)&=\exp (-x),\ \ \ \ M_{1/3}(x)=3^{2/3}Ai(x/\root 3 \of {3})\nonumber \\ M_{1/2}(x)&=\frac{1}{\sqrt{\pi }}\exp (-x^{2}/4)\ \ \ \ \ M_{1} (x)=\delta (x-1), \end{aligned}$$
(B.3)

where Ai(x) is the Airy function [57].

The Mittag–Leffler function [58] of the first kind is a generalization of the exponential function and defined as

$$\begin{aligned} E_{\alpha }(x)=\sum \limits _{n=0}^{\infty }\frac{x^{n}}{\varGamma (\alpha n+1)}. \end{aligned}$$
(B.4)

The Mittag–Leffler function of the second kind [58] is defined as

$$\begin{aligned} E_{\alpha ,\beta }(x)=\sum \limits _{n=0}^{\infty }\frac{x^{n}}{\varGamma (\alpha n+\beta )}, \end{aligned}$$
(B.5)

with the property

$$\begin{aligned} E_{a,b}\left( x+y\right) = {\displaystyle \sum \limits _{n=0}^{\infty }} \frac{x^{n}}{n!}E_{a,b}^{n}\left( y\right) . \end{aligned}$$
(B.6)

The Fox H-function [34] generalizes the Mellin–Barnes function as

$$\begin{aligned} H_{p,q}^{m,n}\left( x\right)&=H_{p,q}^{m,n}\left[ x\left| \begin{array}[c]{l} \,(a_{1},A_{1}),\ldots ,(a_{p},A_{p})\\ \,(b_{1},B_{1}),\ldots ,(b_{q},B_{q}) \end{array} \right. \right] \nonumber \\&=\frac{1}{2\pi i}\int \nolimits _{L}\varTheta (s)\ x^{-s}\hbox {d}s, \end{aligned}$$
(B.7)

where

$$\begin{aligned} \varTheta (s)=\frac{\prod \nolimits _{j=1}^{m}\varGamma (b_{j}+B_{j}s)\ \prod \nolimits _{j=1}^{n}\varGamma (1-a_{j}-A_{j}s)\ }{\prod \nolimits _{j=m+1}^{q} \varGamma (1-b_{j}-B_{j}s)\ \prod \nolimits _{j=n+1}^{p}\varGamma (a_{j}+A_{j} s)}. \end{aligned}$$
(B.8)

Here mnp 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 derivative of the Mittag–Leffler function is a special case of the Fox H-function

$$\begin{aligned} E_{\alpha ,\beta }^{n}(y)=H_{1.2}^{1,1}\left[ -y\left| \begin{array}[c]{l} (-n,1)\\ (0,1)(1-(\alpha n+\beta ),\alpha ) \end{array} \right. \right] . \end{aligned}$$
(B.9)

The Fourier cosine transform of the Fox H-function [19] is given by

$$\begin{aligned}&\int _{0}^{\infty }\left| y\right| ^{\rho -1}H_{p,q}^{m,n}\left[ \omega \left| y\right| ^{\mu }\left| \begin{array}[c]{l} \,(a_{p},A_{p})\, \\ \,(b_{q},B_{q}) \end{array} \right. \right] \cos (yx)\hbox {d}y\nonumber \\&\quad =\frac{\pi }{\left| x\right| ^{\rho }}H_{q+1,p+2}^{n+1,m}\; \left[ \frac{\left| x\right| ^{\mu }}{\omega }\left| \begin{array}[c]{l} (\,1-b_{q},B_{p}),(\frac{1+\rho }{2},\frac{\mu }{2})\\ (\rho ,\mu ),(1-a_{p},A_{p}),\left( \frac{1+\rho }{2},\frac{\mu }{2}\right) \end{array} \right. \right] . \end{aligned}$$
(B.10)

The Laplace transform identity of the Fox H-function [35] is:

$$\begin{aligned} L^{-1}\left\{ s^{-\rho }\exp (-as^{\sigma };x\right\} =x^{\rho -1}H_{1,1} ^{1,0}\left[ ax^{-\sigma }\left| \begin{array}[c]{l} \,(\rho ,\sigma )\\ \,(0,1) \end{array} \right. \right] . \end{aligned}$$
(B.11)

The Fox H-function can be expanded as a computable series [34] of the form:

$$\begin{aligned} H_{p,q}^{m,n}\left( x\right) =&\sum _{h=1}^{m}\sum _{v=0}^{\infty } \frac{\prod \nolimits _{j=1,j\ne h}^{m}\varGamma (b_{j}+B_{j}(b_{h}+v)/B_{h} )}{\prod \nolimits _{j=m+1}^{q}\varGamma (1-b_{j}+B_{j}(b_{h}+v)/B_{h})}\nonumber \\&\times \frac{\prod \nolimits _{j=1}^{n}\varGamma (1-a_{j}+A_{j}(b_{h}+v)/B_{h} )}{\prod \nolimits _{j=n+1}^{p}\varGamma (a_{j}-A_{j}(b_{h}+v)/B_{h})}\frac{(-1)^{v}x^{(b_{h}+v)/B_{h}}}{v!B_{h}}. \end{aligned}$$
(B.12)

which is an alternating series and thus exhibits slow convergence.

For a large argument \(\left| x\right| \rightarrow \infty \), the contour integral can be evaluated using the residue theorem and the Fox H-function can be expanded as a series over the residues [59]

$$\begin{aligned} {H}_{p,q}^{m,n}(x)\sim \sum _{v=0}^{\infty }{Res(\varTheta (s){x}^{s})}. \end{aligned}$$
(B.13)

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

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Tawfik, A.M. On fractional approximations of the Fokker–Planck equation for energetic particle transport. Eur. Phys. J. Plus 135, 820 (2020). https://doi.org/10.1140/epjp/s13360-020-00848-0

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