Space-time duality for the fractional advection-dispersion equation

Abstract

The fractional advection-dispersion equation replaces the second spatial derivative in the usual advection-dispersion equation with a fractional derivative in the spatial variable. It was first applied to tracer tests in underground aquifers, and later to tracer tests in rivers. An alternative model replaces the first time derivative with a fractional derivative in time. Previous work has shown that both models provide a reasonable fit to breakthrough curves in rivers, which has led to a controversy regarding the physically appropriate fractional model. This paper shows that the relevant space-fractional model is mathematically equivalent to the corresponding time-fractional model, thus resolving the controversy.