On using random walks to solve the space-fractional advection-dispersion equations

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 Y. Zhang Division of Hydrologic Sciences, Desert Research Institute, Reno, NV 89512, USA E-mail: [email protected] D.A. Benson Department of Geology and Geological Engineering, Colorado School of Mines, Golden, CO 80401, USA E-mail: [email protected] M.M. Meerschaert Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand E-mail: [email protected] Hans-Peter Scheffler Department of Mathematics, University of Nevada, Reno, NV 89557, USA E-mail: [email protected]