1205.1199 (Yuri Luchko)

Fractional wave equation and damped waves    [PDF]

Yuri Luchko

In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the well studied fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order $\alpha,\ 1< \alpha \le 2$ both in space and in time. We show that this feature of the fractional wave equation is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum and the gravity center of its Green function. The Green function of the fractional wave equation can be interpreted as a spatial probability density function evolving in time that possesses finite moments up to the order $\alpha$. Because the maximum value of the Green function (wave amplitude) decreases with time whereas its location moves with a constant velocity, solutions to the fractional wave equation can be interpreted as damped waves. Remarkably, the product of the maximum location and the maximum value of the Green function is time-independent and just a function of $\alpha$. To illustrate analytical findings, results of numerical calculations and numerous plots are presented.

View original: http://arxiv.org/abs/1205.1199