Lévy stable distribution and space-fractional Fokker-Planck type equation

1 Introduction

In recent decades, the theory and applications of the fractional calculus have developed rapidly. The applied fields include viscoelasticity, anomalous diffusion, heat transfer, signal processing, dynamics and control, and so on (Podlubny, 1999; Kilbas et al., 2006). Different definitions and methods have also been proposed (Yang et al., 2013; Yang and Baleanu, 2013; Yang, 2012; Li et al., 2011a,b). Scientists and engineers have found the description of some phenomena is more accurate when the fractional derivative is used. In particular, anomalous diffusion can be characterized by fractional differential equations.

Let random variable $X(t)$ denote the location of diffusing particle with $X(0)=0$ and $p(x\text{,}t)$ be the probability density function for $X(t)$ . The time-fractional Fokker-Planck type equations are derived and solved in Hilfer (1995) and Rangarajan and Ding (2000); Space-fractional Fokker-Planck type equation is obtained in Compte (1996) and Yanovsky et al. (2000) using statistical methods. One-dimensional case with convective term without asymmetric term reads (Yanovsky et al., 2000)

Space-fractional Fokker-Planck type equations similar to the Eq. (1) are also considered in Chaves (1998) and Chechkin et al. (2002), but therein the exact analytic solutions for the problem (1) and (2) are given only for the cases of $?=1$ and $?=2$ . In the general case of $0<??2$ , the solution for the problem (1) and (2) is studied in the sequel. We obtain the exact analytic solution in terms of Fox H functions (Mathai and Saxena, 1978; Srivastava et al., 1982), and its series representation and asymptotic behavior are investigated.

2 Solution to the problem

Taking the Fourier transform for the problem (1) and (2) with respect to x we get

Lévy stable distribution $?(x)$ with index $?$ $(0<??2)$ is defined through the Fourier transform as (Feller, 1971; Fogedby et al., 1992; Zanette, 1997)

In order to obtain the inverse in (7) we rewrite it as

As $?=1$ , the first expression in Eq. (15) permits $|\mathit{Dt}/x|<1$ , while the second permits $|\mathit{Dt}/x|>1$ . Making use of the series expression of Fox functions and the formula $$\mathrm{?}(z)\mathrm{?}(1-z)=\frac{?}{\sin ?z}\text{,}$$ we get the series representations

3 Discussions and conclusions

As $?=1$ , the Cauchy distribution is obtained from Eqs. (16) and (17)

As $?=2$ , with the help of the identity $$\mathrm{?}\left(n+\frac{1}{2}\right)=\frac{\sqrt{?}(2n)!}{4^{n}n!}$$ and the expression in Eq. (17), the Gauss distribution is obtained

From Eqs. (16) and (17) we have the following asymptotic expressions

In Figs. 1 and 2, we plot the curves of $U(x\text{,}t)$ versus x for fixed values of t and different values of $?$ . Since for fixed t, the series in (16) converges faster for large $|x|$ while the second converges faster for small $|x|$ , the different ranges of x are chosen in Figs. 1 and 2.

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Figure 1

Curves of $U(x\text{,}0.5)$ versus x on the interval $1.5?x?25$ for $D=1$ and $?=1$ (thick dot line), 0.75 (solid line), 0.5 (dot line) and 0.25 (dash line).

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Figure 2

Curves of $U(x\text{,}5)$ versus x on the interval $-9?x?9$ for $D=1$ and $?=1$ (thick dot line), 1.25 (solid line), 1.5 (dot line), 1.75 (dash line) and 2 (dot-dash line).

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Inserting Eqs. (15)-(17), (20)-(23) into Eq. (8), we can obtain the Fox function representations, series representations and asymptotic behavior for the probability density $p(x\text{,}t)$ .

As $?=2$ , the Gauss distribution (19) describes the standard diffusion process. The mean square displacement is given by $?x^2?=2\mathit{Dt}$ . But as $0<?<2$ , from the asymptotic representation the mean square displacement $?x^2?$ is infinite. The Lévy stable distribution $U(x\text{,}t)$ describes anomalous diffusion (Shlesinger et al., 1993). In order to determine the diffusion velocity we adopt the following method.

Taking a fixed number d $(0<d<1)$ , we investigate $x_d$ such that