Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution
Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution
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DOI：
10.1016/j.camwa.2017.03.020
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发表时间：
2017-06
期刊：
影响因子：
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通讯作者：
L. Boyadjiev;Yuri Luchko
中科院分区：
文献类型：
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作者：
L. Boyadjiev;Yuri Luchko
本文引入了一个多维α-分数阶扩散波动方程，研究了其基本解的性质。这个方程可以从基本的连续时间随机游走方程中推导出来，它包含了α/2阶的Caputo时间分数阶导数和α阶的Riesz空间分数阶导数，使得导数的阶数之比等于传统扩散方程的二分之一。结果表明，α-分数阶扩散-波动方程同时继承了传统扩散方程和波动方程的一些性质。特别是在一维和二维情况下，α-分数阶扩散波动方程的基本解可以解释为一个概率密度函数，该方程所控制的随机过程的熵产率与常规扩散方程完全相同。另一方面，在三维情况下，该方程描述了一种传播相速度随时间变化的异常波传播。
In this paper, a multi-dimensional α-fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α/2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α-fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one-and two-dimensional cases, the fundamental solution to the α-fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.