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Fractional wave equation and damped waves. (English) Zbl 1302.35408

In this paper, the author deals with a fractional wave equation describing the propagation of damped waves. Namely, for any differentiability parameter \(1\leq \alpha \leq 2\), the following equation, containing fractional derivatives of the same order \(\alpha\) both in space and in time, is investigated: \[ D^{\alpha}_t u(x,t) = - (-\Delta)^{\frac{\alpha}{2}}u(x,t) \quad x \in \mathbb{R}^N, \;t\geq 0, \] where \(D^{\alpha}_t\) denotes the Caputo time-fractional derivative of order \(\alpha\), and \((-\Delta)^{\frac{\alpha}{2}}\) is the standard fractional Laplacian.
Under prescribed initial conditions, \[ u(x,0)=\varphi(x) \quad \text{and} \;\frac{\partial u}{\partial t}(x,0) = 0 \quad \text{for} \;x \in \mathbb{R}^N, \] the author presents some results for the solutions \(u=u(x,t)\) to the equation above. In particular, the attention is focused on the corresponding fundamental solution \(G_\alpha\), that is taking \(\varphi=\delta\) in the display above, where \(\delta\) is the Dirac delta function.
Passing through the Fourier transform and making use of the Mittag-Leffler function together with applications of the Mellin integral transform, the author recovers the explicit representation formula for the fundamental solution. Then it is shown that, in view of the special form of the considered equation, the fractional wave equation inherits some crucial characteristics of the wave equation such as, e. g., a constant propagation velocity of both the maximum and the gravity and mass centers of the fundamental solution \(G_\alpha\). Also, among other results, it is shown that the first, the second, and the Smith centro-velocities of the damped waves described by the fractional wave equation are constant and depend only on the equation order \(\alpha\).
Finally, some numerical computations and plots are given in order to illustrate the obtained analytical results.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
35L15 Initial value problems for second-order hyperbolic equations
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