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Mathematical Problems in Engineering
Volume 2013, Article ID 543026, 9 pages
Research Article

The Use of Fractional Order Derivative to Predict the Groundwater Flow

1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 9300, Bloemfontein, South Africa
2Department of Mathematics, Faculty of Art & Sciences, Celal Bayar University, Muradiye Campus, 45047 Manisa, Turkey

Received 2 July 2013; Revised 27 August 2013; Accepted 3 September 2013

Academic Editor: Tirivanhu Chinyoka

Copyright © 2013 Abdon Atangana and Necdet Bildik. 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.


The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis-time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes.