STOCHASTIC SOLUTIONS FOR FRACTIONAL WAVE EQUATIONS

Space-time duality and high-order fractional diffusion

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.

Approximate analytical time-domain Green's functions for the Caputo fractional wave equation

The Caputo fractional wave equation [Geophys. J. R. Astron. Soc. 13, 529-539 (1967)] models power-law attenuation and dispersion for both viscoelastic and ultrasound wave propagation. The Caputo model can be derived from an underlying fractional constitutive equation and is causal. In this study, an approximate analytical time-domain Green's function is derived for the Caputo equation in three dimensions (3D) for power law exponents greater than one. The Green's function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1/(4πR). The approximate one dimensional (1D) and two dimensional (2D) Green's functions are also computed in terms of stable distributions. Finally, this Green's function is decomposed into a loss component and a diffraction component, revealing that the Caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation.