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 can be solved by tracking particles following a Markov process with a simple drift and an alpha-stable Levy noise with index alpha 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 alpha-stable Levy noise and an additional term with the dispersion gradient and an additional Levy noise of order alpha-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.