Numerical solution of the space fractional Fokker-Planck equation
Numerical solution of the space fractional Fokker-Planck equation
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DOI：
10.1016/j.cam.2003.09.028
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发表时间：
2004-04-01
影响因子：
2.4
通讯作者：
Turner, I
中科院分区：
文献类型：
--
作者：
Liu, F;Anh, V;Turner, I
传统的二阶福克-普朗克方程可能无法充分描述含水层中溶质的运动，因为它与布朗运动的动力学存在较大偏差。 α稳定类型的密度已被用来描述这些运动的概率分布。由此产生的这些运动的控制方程与传统的福克-普朗克方程类似，只是最高导数的阶数α是分数阶。本文考虑具有瞬时源的空间分数阶福克-普朗克方程（SFFPE）。提出了求解 SFFPE 的数值方案。使用分数导数的 Riemann-Liouville 和 Grunwald-Letnikov 定义，SFFPE 被转换为常微分方程 (ODE) 系统。然后用直线法求解ODE系统。评估具有恒定扩散系数的 SFFPE 的数值结果，以便与已知的解析解进行比较。具有随时间变化的扩散系数的 SFFPE 数值近似也可用于模拟具有 α 稳定密度的 Levy 运动。我们将证明 SFFPE 的数值方法能够更准确地模拟这些重尾运动。 (C) 2003 Elsevier B.V. 保留所有权利。
The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of alpha-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order alpha of the highest derivative is fractional.In this paper, a space fractional Fokker-Planck equation (SFFPE) with instantaneous source is considered. A numerical scheme for solving SFFPE is presented. Using the Riemann-Liouville and Grunwald-Letnikov definitions of fractional derivatives, the SFFPE is transformed into a system of ordinary differential equations (ODE). Then the ODE system is solved by a method of lines. Numerical results for SFFPE with a constant diffusion coefficient are evaluated for comparison with the known analytical solution. The numerical approximation of SFFPE with a time-dependent diffusion coefficient is also used to simulate Levy motion with alpha-stable densities. We will show that the numerical method of SFFPE is able to more accurately model these heavy-tailed motions. (C) 2003 Elsevier B.V. All rights reserved.