Numerical Algorithms for the Forward and Backward Fractional Feynman–Kac Equations
Numerical Algorithms for the Forward and Backward Fractional Feynman–Kac Equations
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DOI：
10.1007/s10915-014-9873-6
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发表时间：
2014-01
影响因子：
2.5
通讯作者：
W. Deng;Minghua Chen;E. Barkai
中科院分区：
文献类型：
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作者：
W. Deng;Minghua Chen;E. Barkai
Feynman-Kac方程是一类描述扩散运动泛函分布的偏微分方程。布朗泛函的概率密度函数（PDF）满足费曼-卡茨公式，是虚时间中的薛定谔方程。非布朗运动或异常扩散的泛函遵循分数阶Feynman-Kac方程（Carmi等人，J Stat Phys 141：1071-1092，2010），其中涉及分数阶实质导数。基于最近发展的分数阶实质导数离散格式（Chen和Deng arXiv：1310.3086 ），给出了数值求解正、倒向分数阶Feynman-Kac方程的算法;由于分数阶实质导数是非局部时空耦合算子，与普通分数阶导数相比，提出了新的挑战。两种方法（有限差分和有限元）离散的空间导数被认为是。对于分数阶后向Feynman-Kac方程，理论上讨论了一阶精度算法的数值稳定性和收敛性，并得到了最优估计.对所提出的前向和后向Feynman-Kac方程的一阶和高阶格式进行了大量的数值实验，以验证其有效性.
The Feynman–Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman–Kac formula, being a Schrödinger equation in imaginary time. The functionals of non-Brownian motion, or anomalous diffusion, follow the fractional Feynman–Kac equation (Carmi et al. in J Stat Phys 141:1071–1092, 2010), where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives (Chen and Deng arXiv:1310.3086 ), this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman–Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced compared with the ordinary fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman–Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman–Kac equations, extensive numerical experiments are performed to show their effectiveness.