Physical interpretation of fractional diffusion-wave equation via lossy media obeying frequency power law

TL;DR

This paper provides a physical interpretation of the fractional diffusion-wave equation by linking it to frequency-dependent dissipation in lossy media, and establishes bounds on derivative orders that challenge some traditional assumptions.

Contribution

It introduces a new interpretation of FDWE through a frequency-dependent dissipative wave equation and derives bounds on derivative orders, clarifying its physical significance.

Findings

FDWE models frequency-dependent energy dissipation in complex media

A new bound on fractional derivative orders contradicts sub-diffusion assumptions

Standard derivation of diffusion from damped wave equations is physically inappropriate

Abstract

The fractional diffusion-wave equation (FDWE) is a recent generalization of diffusion and wave equations via time and space fractional derivatives. The equation underlies Levy random walk and fractional Brownian motion and is foremost important in mathematical physics for such multidisciplinary applications as in finance, computational biology, acoustics, just to mention a few. Although the FDWE has been found to reflect anomalous energy dissipations, the physical significance of the equation has not been clearly explained in this regard. Here the attempt is made to interpret the FDWE via a new time-space fractional derivative wave equation which models forequency-dependent dissipations observed in such complex phenomena as acoustic wave propagating through human tissues, sediments, and rock layers. Meanwhile, we find a new bound (inequality (6) further below) on the orders of time and space derivatives of the FDWE, which indicates the so-called sub-diffusion process contradicts the real world frequency power law dissipation. This study also shows that the standard approach, albeit mathematically plausible, is phyiscally inappropriate to derive the normal diffusion equation from the damped wave equation, also known as the Telegrapher's equation.