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Research Papers

# Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation

[+] Author and Article Information
Yuri Luchko

Professor of Mathematics
Department of Mathematics,
Physics, and Chemistry,
Beuth Technical University of Applied Sciences,
Berlin 13353, Germany
e-mail: luchko@beuth-hochschule.de

Francesco Mainardi

Professor of Mathematical Physics,
Department of Physics and Astronomy,
Bologna University, and INFN,
Bologna 40126, Italy
e-mails: francesco.mainardi@unibo.it; francesco.mainardi@bo.infn.it

1Corresponding author.

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.

J. Vib. Acoust 136(5), 050904 (Jul 25, 2014) (7 pages) Paper No: VIB-13-1301; doi: 10.1115/1.4026892 History: Received August 28, 2013; Revised February 12, 2014

## Abstract

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 $β,1≤β≤2$ 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.

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## Figures

Fig. 1

The Green function Gc(x;ν):=Gc(x,1;ν). Plots for several different values of ν.

Fig. 2

The Green function Gs(x;ν):=Gs(x,1;ν). Plots for several different values of ν.

Fig. 3

Propagation velocity Vc(t,ν) of the maximum point of Gc in the log-lin scale

Fig. 4

Maximum locations and maximum values of Gc(x,t;ν)

Fig. 5

Maximum locations and maximum values of the Green function Gc(x,1;ν)

Fig. 6

Product of the maximum locations and maximum values of the Green function Gc(x,1;ν)

Fig. 7

Propagation velocity Vs(t,ν) of the maximum point of Gs

Fig. 8

Maximum locations and maximum values of the Green function Gs(x,1;ν)

Fig. 9

Location of the center of gravity of the Green function Gc(|x|,1;ν)

Fig. 10

Propagation velocity Vcg(t,ν) of the center of gravity of Gc

Fig. 12

Propagation velocity Vsg(t,ν) of the center of gravity of Gs

Fig. 11

Location of the center of gravity of the Green function Gs(x,1;ν)

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