Fractional wave equation and damped waves

By Yuri Luchko. 2012

140 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/27884ebdf5d4ba41ad90ad8d570764d1339cb25c

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In Applied Mathematics

In Modeling and Simulation

In Statistical and Nonlinear Physics

Citations in the corpus

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations (2015, background): “1) becomes a generalized time and space fractional advection–diffusion equation, which we refer to as a multi-term time fractional mixed wave–diffusion equation [63,64].”
On solutions of space–fractional diffusion equations by means of potential wells (2016, unknown): “(8) Space–time fractional wave equation 1 < α < 2, either 0 < β, γ ≤ 2, 0 < δ < 2, or 0 < β < 2, 0 < γ, δ ≤ 2, or 0 < β, δ ≤ 2, 0 < γ < 2, see for examples [5, 14].”