MATHEMATICAL FOUNDATION FOR FADE PARTICLE TRACKING CODES
The space-fractional advection-dispersion equation (fADE), in which the dispersive flux is described by a fractional space derivative, has been applied to model the super-dispersion of solutes often observed in heterogeneous porous media. The fADE with variable coefficients can be solved by simulating a Markov process. We start with a nonlinear Langevin Stochastic Differential equation (SDE) driven by α-Stable Lévy noise, and then we derive the forward and backward equations associated with the Markov process solving this SDE. The SDE is chosen to make the forward equation equal the variable coefficient fADE. Solutions of the forward equation yield the transition densities of the Markov process. Hence, a particle tracking code that simulates the Langevin equation can be used to develop numerical solutions to the fADE with variable coefficients. We further prove, under certain assumptions on the drift and dispersion term of the Langevin SDE, that the Markov process exists and it is unique.