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Mathematical Problems in Engineering
Volume 2013, Article ID 853127, 9 pages
Research Article

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

1Institute for Groundwater Studies, University of the Free State, P.O. Box 399, Bloemfontein, South Africa
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, P.O. Box 43400, Serdang, Selangor, Malaysia

Received 24 January 2013; Accepted 1 March 2013

Academic Editor: Guo-Cheng Wu

Copyright © 2013 Abdon Atangana and Adem Kilicman. This 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 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 , and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order . 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.