Variational solution of fractional advection dispersion equations on bounded domains in ℝd

In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd. Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation.