In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional diffusion-wave equation is nonrelativistic like the classical diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other.
Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 28, 2013; final manuscript received February 12, 2014; published online July 25, 2014. Assoc. Editor: Thomas J. Royston.
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Luchko, Y., and Mainardi, F. (July 25, 2014). "Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation." ASME. J. Vib. Acoust. October 2014; 136(5): 050904. https://doi.org/10.1115/1.4026892
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