Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion
- PDF / 421,660 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 92 Downloads / 191 Views
Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion Longjin Lv · Weiyuan Qiu · Fuyao Ren
Received: 24 December 2011 / Accepted: 10 October 2012 / Published online: 23 October 2012 © Springer Science+Business Media New York 2012
Abstract In this paper, we aim to answer the question proposed by Magdziarz (Stoch. Proc. Appl. 119:3238–3252, 2009), i.e. we investigate the solution of an anomalous diffusion equation with time and space dependent force and diffusion coefficient. First, we try to find the stochastic representation of this equation, which means the PDF of this stochastic process is rightly the solution of the equation we aim to solve. Then, we also simulate the sample paths of the stochastic process. At last, taking advantage of the stochastic representation method, we employed Monte Carlo method to approximate the solution of the mentioned equation. Keywords Anomalous diffusion equation · Subordinated process · Stochastic representation · Numerical approximation 1 Introduction Recently, the diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion [3, 13, 14]. For instance, surface growth, transport of fluid in porous media [23], two-dimensional rotating flow [22], subrecoil laser cooling [1], diffusion on fractals [24], or even in multidisciplinary areas such as in analyzing the behavior of CTAM micelles dissolved in salted water [4, 18] or econophysics [20]. Note that the physical systems mentioned above essentially concern anomalous diffusion of the correlated type (both sub and super-diffusion [2] or of the Lévy type, see [21]). In general, anomalous diffusion may be classified by employing the second moment x 2 . When x 2 ∝ t γ , the value γ > 1 characterizes a super-diffusive process, γ < 1 a sub-diffusive one, and γ = 1 a normal diffusion. L. Lv () Department of Information Science and Engineering, Ningbo Institute of Technology, Zhejiang University, Ningbo 315000, China e-mail: [email protected] W. Qiu · F. Ren Department of Mathematics, Fudan University, Shanghai 200433, China
620
L. Lv et al.
In order to describe this phenomena clearly, one needs to introduce the fractional FokkerPlanck equation (FFPE) [13]. Several methods were introduced to get the solution and properties of the FFPE in [13, 15, 16]. However, the limitation of such an approach is that it does not allow one to construct and analyze sample paths of the underlying stochastic process. In [11], the authors introduced a simple and efficient method for computer simulation of sample paths of anomalous diffusion process described by the FFPE. It reveals that subdiffusion is actually a combination of two independent mechanisms: the first mechanism is the standard diffusion represented by a Itô process X(τ ), the second mechanism introduces the trapping events and is represented by the so-called inverse α-stable subordinator Sα (t). The subordinated process X(Sα (t)) combine
Data Loading...
