Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

• Authors: Yuri Luchko , Francesco Mainardi
• Languages of publication: EN

Abstracts

EN

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed 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 and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

Keywords

EN

diffusion-wave equation; signalling problem; Mittag-Leffler function; Wright function; Mainardi function

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 6
• Pages: 666-675

Dates

published

1 - 6 - 2013

online

9 - 10 - 2013

Contributors

author

Yuri Luchko

• Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, D-13353, Berlin, Germany

author

Francesco Mainardi

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Identifiers

DOI

10.2478/s11534-013-0247-8