The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of L\'evy 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.

翻译：分数阶物质导数作为控制莱维行走（一种源于经典连续时间随机游走的随机过程）标度极限动力学的分数阶算子出现。该算子通常定义为傅里叶-拉普拉斯乘子，因此可视为伪微分算子。本文证明了分数阶物质导数存在时间与空间上的逐点局部表示，从而可将其定义在比原始空间（傅里叶与拉普拉斯变换存在作为函数）更广的局部可积函数空间上。我们研究了若干包含分数阶物质导数的典型微分方程，并给出其解存在的条件。在某些情形下可求得解析解。针对一般初值问题，我们设计了一种有限体积方法，并证明了其稳定性、收敛性与概率守恒性。数值实验验证了理论分析结果。此外，我们的数值实验表明，在概率密度函数推导问题中，所提数值方案的计算时间优于蒙特卡洛方法。