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

TL;DR

This paper introduces and analyzes a multi-dimensional fractional wave equation with equal fractional derivatives in space and time, revealing unique properties of its fundamental solution, including explicit forms and probabilistic interpretations in specific dimensions.

Contribution

It presents a novel multi-dimensional fractional wave equation with explicit fundamental solutions and explores their properties, including probabilistic aspects in one dimension.

Findings

Fundamental solution explicitly derived in 1D and 3D.

In 1D, the solution is a probability density function.

In higher dimensions, the solution can be negative, losing probabilistic interpretation.

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.