caa a22 4500 467891672 CHVBK 20180406152751.0 cr unu---uuuuu 170328e20060401xx s 000 0 eng 10.1007/s10955-006-9042-x doi (NATIONALLICENCE)springer-10.1007/s10955-006-9042-x On Using Random Walks to Solve the Space-Fractional Advection-Dispersion Equations [Elektronische Daten] [Zhang Yong, David Benson, Mark Meerschaert, Hans-Peter Scheffler] The solution of space-fractional advection-dispersion equations (fADE) by random walks depends on the analogy between the fADE and the forward equation for the associated Markov process. The forward equation, which provides a Lagrangian description of particles moving under specific Markov processes, is derived here by the adjoint method. The fADE, however, provides an Eulerian description of solute fluxes. There are two forms of the fADE, based on fractional-flux (FF-ADE) and fractional divergence (FD-ADE). The FF-ADE is derived by taking the integer-order mass conservation of non-local diffusive flux, while the FD-ADE is derived by taking the fractional-order mass conservation of local diffusive flux. The analogy between the fADE and the forward equation depends on which form of the fADE is used and on the spatial variability of the dispersion coefficient D in the fADE. If D does not vary in space, then the fADEs can be solved by tracking particles following a Markov process with a simple drift and an α-stable Lévy noise with index α that corresponds to the fractional order of the fADE. If D varies smoothly in space and the solute concentration at the upstream boundary remains zero, the FD-ADE can be solved by simulating a Markov process with a simple drift, an α-stable Lévy noise and an additional term with the dispersion gradient and an additional Lévy noise of order α−1. However, a non-Markov process might be needed to solve the FF-ADE with a space-dependent D, except for specific D such as a linear function of space. Springer Science+Business Media, Inc., 2006 Random walk nationallicence forward equation nationallicence fractional advection-dispersion equation nationallicence adjoint method nationallicence Yong Zhang Division of Hydrologic Sciences, Desert Research Institute, 89512, Reno, NV, USA aut Benson David Department of Geology and Geological Engineering, Colorado School of Mines, 80401, Golden, CO, USA aut Meerschaert Mark Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand aut Scheffler Hans-Peter Department of Mathematics, University of Nevada, 89557, Reno, NV, USA aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 123/1(2006-04-01), 89-110 0022-4715 123:1<89 2006 123 10955 https://doi.org/10.1007/s10955-006-9042-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-006-9042-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Yong Zhang Division of Hydrologic Sciences, Desert Research Institute, 89512, Reno, NV, USA aut NATIONALLICENCE 700 1- Benson David Department of Geology and Geological Engineering, Colorado School of Mines, 80401, Golden, CO, USA aut NATIONALLICENCE 700 1- Meerschaert Mark Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand aut NATIONALLICENCE 700 1- Scheffler Hans-Peter Department of Mathematics, University of Nevada, 89557, Reno, NV, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 123/1(2006-04-01), 89-110 0022-4715 123:1<89 2006 123 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer