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Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order 0<β≤1, and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order 1<α≤2. We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.

Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation