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

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order alpha, 1 a parts per thousand currency sign alpha a parts per thousand currency sign 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.