Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation    [PDF]

Yuri Luchko, Francesco Mainardi, Yuriy Povstenko
In this paper, the one-dimensional time-fractional diffusion-wave equation
with the fractional derivative of order $1 \le \alpha \le 2$ is revisited. This
equation interpolates between the diffusion and the wave equations that 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. For the time fractional diffusion-wave equation, the propagation
speed of a disturbance is infinite, but its fundamental solution possesses a
maximum that disperses with a finite speed. In this paper, the fundamental
solution of the Cauchy problem for the time-fractional diffusion-wave equation,
its maximum location, maximum value, and other important characteristics are
investigated in detail. To illustrate analytical formulas, results of numerical
calculations and plots are presented. Numerical algorithms and programs used to
produce plots are discussed.
View original: http://arxiv.org/abs/1201.5313