分数物质导数：逐点表示与有限体积数值格式
Fractional material derivative: pointwise representation and a finite volume numerical scheme
AI总结：
本文给出分数物质导数的时空逐点局部表示，分析相关方程解的存在性，并提出稳定、收敛且保持概率守恒的有限体积格式，计算效率优于 Monte Carlo 方法。
摘要：
分数物质导数是作为控制 Lévy 游走（Lévy walks）标度极限动力学的分数算子出现的，Lévy 游走是一种起源于著名连续时间随机游走的随机过程。它通常被定义为 Fourier-Laplace 乘子，因此可被视为一种伪微分算子。本文证明，分数物质导数在时间和空间上存在局部的逐点表示。这使我们能够在一个局部可积函数空间上定义该算子，该空间大于原先要求 Fourier 变换和 Laplace 变换作为函数存在的空间。我们考虑了若干涉及分数物质导数的典型微分方程，并给出其解存在的条件。在某些情形下，可以求得解析解。对于一般的初值问题，我们设计了一种有限体积方法，并证明了其稳定性、收敛性以及概率守恒。数值算例验证了我们的解析结论。此外，数值实验表明，在概率密度函数推导问题上，所提出的数值格式在计算时间方面优于 Monte Carlo 方法。
英文摘要：
The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions. We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function's derivation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
