频率幂律有耗介质分数阶扩散波动方程的物理解释
Physical interpretation of fractional diffusion-wave equation via lossy media obeying frequency power law
摘要：
分数阶扩散-波动方程（FDWE）是扩散和波动方程通过时间和空间分数阶导数的最新推广。该方程是利维随机游动和分数布朗运动的基础，在金融、计算生物学、声学等多学科应用的数学物理中非常重要。虽然FDWE已被发现反映异常的能量耗散，该方程的物理意义尚未在这方面得到清楚的解释。在这里，试图解释FDWE通过一个新的时间-空间分数阶导数波动方程模型forfrequency-dependent dissipation观察到的声波传播通过人体组织，沉积物和岩石层等复杂的现象。同时，我们发现了FDWE的时间和空间导数阶数的一个新的界（不等式（6）），这表明所谓的子扩散过程与现实世界的频率幂律耗散相矛盾。这项研究还表明，标准的方法，虽然在数学上是合理的，是phyiscally不适合从阻尼波方程，也被称为电报员的方程推导出正常的扩散方程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
