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

Abstract

In this paper, a multi-dimensional fractional wave equation that describes propagation of the damped waves is introduced and analyzed. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order $\alpha,\ 1\le \alpha \le 2$ both in space and in time. This feature is a decisive factor for inheriting some crucial characteristics of the wave equation like e.g. a constant phase velocity of the damped waves that are described by the fractional wave equation. Some new integral representations of the fundamental solution of the multi-dimensional wave equation are presented. In the one- and three-dimensional cases, the fundamental solution is obtained in explicit form in terms of elementary functions. In the one-dimensional case, the fundamental solution is shown to be a spatial probability density function evolving in time. However, for the dimensions grater than one, the fundamental solution can be negative and therefore does not allow a probabilistic interpretation anymore. To illustrate analytical findings, results of numerical calculations and numerous plots are presented.