Space-fractional advection-diffusion and reflective boundary condition

Abstract

Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRWs) with transition probability densities showing space- and/or time-diverging moments were developed to account for anomalous behaviors. A broad class of CTRWs was shown to correspond, on the macroscopic scale, to advection-diffusion equations involving derivatives of noninteger order. In particular, CTRWs with Lévy distribution of jumps and finite mean waiting time lead to a space-fractional equation that accounts for superdiffusion and involves a nonlocal integral-differential operator. Within this framework, we analyze the evolution of particles performing symmetric Lévy flights with respect to a fluid moving at uniform speed . The particles are restricted to a semi-infinite domain limited by a reflective barrier. We show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator. Thus, the macroscopic space-fractional advection-diffusion equation obtained is different from that in an infinite medium.

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

“…Metzler, Klafter, 2000b;Metzler, Klafter, 2000a;Krepysheva et al 2006a;Krepysheva et al, 2006b) both for absorbing and reflective boundaries. Yet, to the authors' best knowledge, the analytical approach to subdiffusion through a two-layered medium and its connection with the underlying Monte Carlo microscopic dynamics are new and unexplored subjects.…”

Saturation dependence of dispersion in porous media

“…Note that the second form of Eq. (19), in the case that β = φ, generates the known form of non-universal scaling of the hydraulic conductivity derived by Balberg [85]. As can be seen in Figure 9, the time value at the peak of arrival time distribution (t p ) for h= 0, 10, and 15 cm is about 20, 200, and 3000 min, respectively.…”

“…Many alternative frameworks for understanding transport have been proposed, including the Fractional Advection-Dispersion Equation (FADE) [15][16][17][18][19][20][21][22], and the continuous time random walk (CTRW) [10,11,13,14,[23][24][25]. Since the FADE still underestimates solute arrivals at short times [11] and the dispersion coefficient of the FADE can still be scale-dependent [17], its relevance may be chiefly in the field of mathematics rather than to actual solute transport problems.…”

The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

“…Besides the unclear exact physical (or intuitive) meaning of a fractional derivative (see, e.g. [25]), there is the problem of the non-locality of fractional derivatives that implies a dependence of the explicit formulation of a fractional differential equation on the chosen boundary conditions, see, for example, Krepysheva et al [16] and Garcia-Garcia et al [3]. Another problem is how to treat sub-and superdiffusion simultaneously, as might be necessary in anisotropic media or in multi-process systems.…”