2003/03/17 by W. Chen, Sverre Holm, Chen, W. +2 · 3 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Adaptive Filtering Techniques #Fractional Differential Equations Solutions #Image and Signal Denoising Methods #math-ph #math.MP #math.PR #msc:26A33 #msc:26Axx #msc:60Gxx #msc:74Dxx
paper · pdf · doi:10.48550/arxiv.math-ph/0303040
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arxiv created 2003/03/17 · arxiv updated 2009/11/30
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.