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 α, 1 ⩽ α ⩽ 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 α. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time all whose moments of order less than α are finite. To illustrate analytical findings, results of numerical calculations and plots are presented.
Received 20 July 2012Accepted 14 February 2013Published online 13 March 2013
The author is thankful to Professor Francesco Mainardi for useful and stimulating discussions regarding the subject of the paper during author's visit to the University of Bologna in December 2011 and appreciates the constructive remarks and suggestions of the anonymous referees that helped to improve the paper.
Article outline: I. INTRODUCTION II. ANALYSIS OF THE FRACTIONAL WAVE EQUATION A. Problem formulation B. Fundamental solution of the fractional wave equation C. Fundamental solution as a pdf D. Extrema points, gravity and “mass” centers of Gα, and location of its energy E. The velocities of the damped waves III. DISCUSSION OF THE OBTAINED RESULTS AND PLOTS IV. CONCLUSIONS AND OPEN QUESTIONS
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