doi: 10.1685/journal.caim.485

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

Yuri Luchko

Abstract


In this paper, a multi-dimensional fractional wave equation that describes propaga- tion of 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 α, 1 ≤ α ≤ 2 both in space and in time. This feature is a decisive factor for inher- iting some crucial characteristics of the wave equation, such as, a constant phase velocity of the damped waves which is now described by the fractional wave equation. Some new integral representations of the fundamental solution of the multi-dimensional wave equa- tion 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 greater than one, the fundamental solution can be negative, and therefore, does not allow a probabilistic interpretation. To illustrate the analytical findings, results of numerical calculations and numerous plots are presented.

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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909