Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution

In this paper, a multi-dimensional α-fractional diffusion-wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α/2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α-fractional diffusion-wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α-fractional diffusion-wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.