Fractional wave equation and damped waves

TL;DR

This paper introduces a fractional wave equation with derivatives of order α in both space and time, revealing constant propagation velocities and probability density solutions with finite moments, advancing understanding of damped wave propagation.

Contribution

It presents a novel fractional wave equation with equal order derivatives in space and time, demonstrating key wave properties and providing analytical and numerical insights.

Findings

Propagation velocity remains constant and depends on α.

Fundamental solution is a spatial probability density with finite moments.

Numerical results support analytical findings.

Abstract

In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. 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. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and mass centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order $\alpha$. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time that possesses finite moments up to the order $\alpha$. To illustrate analytical findings, results of numerical calculations and numerous plots are presented.