Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 853127, 9 pages http://dx.doi.org/10.1155/2013/853127 Research Article Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation Abdon Atangana 1 and Adem Kilicman 2 1 Institute for Groundwater Studies, University of the Free State, P.O. Box 399, Bloemfontein, South Africa 2 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, P.O. Box 43400, Serdang, Selangor, Malaysia Correspondence should be addressed to Abdon Atangana; abdonatangana@yahoo.fr Received 24 January 2013; Accepted 1 March 2013 Academic Editor: Guo-Cheng Wu Copyright © 2013 A. Atangana and A. Kilicman. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fuid fow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the frst-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order 0<≤1, and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order 1<≤2. We derive the solution of the new equation in terms of Mittag-Lefer functions using Laplace transfrom. Some examples are given. Te results from comparison let no doubt that the FADE is better in prediction than ADE. 1. Introduction Te description of transport is closely related to the terms convection, difusion, dispersion, and retardation as well as decomposition. First, it is assumed that there are no inter- actions between the species dissolved in water and the sur- rounding solid phase [1]. Te primary mechanism for the transport of improperly discarded hazardous waste through the environment is by the movement of water through the subsurface and surface waterways. Studying this movement requires that one must be able to measure the quantity of waste present at a particular point in space time. Te mea- sure universally for chemical pollution is the concentration. Analytical methods that handle solute transport in porous media are relatively easy to use [1]. However, because of complexity of the equations involved, the analytical solu- tions are generally available restricted to either radial fow problems or to cases where velocity is uniform over the area of interest. Numerous analytical solutions are available for time-dependent solute transport within media having steady state and uniform fow. Tis work is devoted to the discus- sion underpinning the derivation of the analytical solution of space-time fractional derivative of advection-dispersion equation. 2. Governing Equations A relatively complete set of one-dimensional analytical solu- tions for convective-dispersive solute equations has been recently published by Van Genuchten and Alves in 1982 [2]. Here we shall review a case having a practical application. Let us consider a one-dimensional model consisting of infnitely ling homogenous isotropic porous media with steady state uniform fow with seepage velocity V. We inject a particular chemical from one end of the model for a period of time  0 such that the input concentration varies as an exponential function of time [3]. Te value of that chemical concentration at any time  and at a distance  from the injection boundary, allowing for the decay and adsorption, may be obtained from the solution of the following set of equations [3]:   2 (,)  2 − V (,)  −= (,)  , (1)