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

TL;DR

This paper investigates the propagation characteristics of the maximum point of the fundamental solution to the one-dimensional time-fractional diffusion-wave equation, revealing finite-speed dispersion despite infinite propagation speed.

Contribution

It provides detailed analysis and formulas for the maximum location and value of the fundamental solution, including numerical methods and visualizations.

Findings

Maximum point disperses with finite speed

Fundamental solution exhibits finite maximum propagation speed

Numerical algorithms for solution visualization

Abstract

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.