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

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.

22 pages 6 figures. This paper has been presented by F. Mainardi at the International Workshop: Fractional Differentiation and its Applications (FDA12) Hohai University, Nanjing, China, 14-17 May 2012

References in corpus (4)

• The fundamental solution of the space-time fractional diffusion equation
• Analytical properties and applications of the Wright function
• Fractional wave equation and damped waves
• Fractional Calculus in Wave Propagation Problems

Cited by in corpus (10)

• Fractional wave equation and damped waves
• Generalized diffusion-wave equation with memory kernel
• Subordination approach to multi-term time-fractional diffusion-wave equations
• Cauchy and signaling problems for the time-fractional diffusion-wave equation
• Data-driven Identification of 2D Partial Differential Equations using extracted physical features
• Fractional-Hyperbolic Systems
• Integral decomposition for the solutions of the generalized Cattaneo equation
• Solution of Linear Fractional Systems with Variable Coefficients
• Fractional Burgers wave equation
• Fractional Burgers wave equation on a finite domain