Cauchy and signaling problems for the time-fractional diffusion-wave equation

TL;DR

This paper investigates the properties of the time-fractional diffusion-wave equation, revealing how it interpolates between diffusion and wave behaviors, with finite propagation velocities but infinite spread of disturbances.

Contribution

It provides new analytical and numerical insights into the propagation characteristics of the time-fractional diffusion-wave equation for both Cauchy and signaling problems.

Findings

Propagation velocities of fundamental solutions are finite.

Disturbance spreads infinitely fast despite finite velocities.

Analytical and numerical descriptions of solution characteristics.

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 $\beta,\ 1 \le \beta \le 2$ 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 non-relativistic 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.