Super-diffusion, characterized by a spreading rate
of the probability density function $p(x,t) = t^{-1/\a...
...
... A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.
Super-diffusion, characterized by a spreading rate
of the probability density function
, where
is time, may be modeled by space-fractional diffusion equations with order
$1 < \alpha < 2$. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond than this range (
$\alpha > 2$), 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
$1 < \alpha \leq 2$. This paper extends space-time duality to fractional exponents
$1<\alpha \leq 3$, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order
$2<\alpha \leq 3$ model sub-diffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.