Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824, USA.
Phys Rev E. 2019 Feb;99(2-1):022122. doi: 10.1103/PhysRevE.99.022122.
Superdiffusion, characterized by a spreading rate t^{1/α} of the probability density function p(x,t)=t^{-1/α}p(t^{-1/α}x,1), where t is time, may be modeled by space-fractional diffusion equations with order 1<α<2. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond this range (α>2), known as high-order diffusion or hyperdiffusion. Recently, space-time duality, motivated by Zolotarev's duality law for stable densities, established a link between time-fractional and space-fractional diffusion for 1<α≤2. This paper extends space-time duality to fractional exponents 1<α≤3, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order 2<α≤3 model subdiffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.
超级扩散的特征是概率密度函数 p(x,t)=t^{-1/α}p(t^{-1/α}x,1)的扩散速率 t^{1/α}，其中 t 是时间，可用分数阶扩散方程来建模，阶数 1<α<2。在生物物理学（钙火花扩散）、图像处理和计算流体动力学中有一些应用利用了此范围之外的整数阶和分数阶指数（α>2），称为高阶扩散或超扩散。最近，时空对偶性（受稳定密度的 Zolotarev 对偶律启发）为 1<α≤2 的分数阶和空间分数阶扩散建立了联系。本文将时空对偶性扩展到分数阶指数 1<α≤3，并给出了几个应用。特别是，将表明阶数为 2<α≤3 的空间分数阶扩散方程可以模拟亚扩散，并具有随机解释。还开发了用于模拟瞬态异常扩散的调和分数阶方程的时空对偶性。
