  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.
