Space-Time Duality and High-Order Fractional Diffusion

Abstract

Super-diffusion, characterized by a spreading rate $t^{1/\alpha }$ of the probability density function $p(x,t)=t^{-1/\alpha }p\left(t^{-1/\alpha }x,1\right)$, where $t$ 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 $1<\alpha \le 2$. This paper extends space-time duality to fractional exponents $1<\alpha \le 3$, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order $2<\alpha \le 3$ model sub-diffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.

3 figures, Physical Review E

References in corpus (2)

• Hyperviscosity, Galerkin truncation and bottlenecks in turbulence
• Fractional Partial Differential Equations with Boundary Conditions

Cited by in corpus (2)

• Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion
• An inverse problem of determining fractional orders in a fractal solute transport model