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Mathematical Problems in Engineering
Volume 2013, Article ID 543026, 9 pages
http://dx.doi.org/10.1155/2013/543026
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.

Linked References

  1. J. F. Botha and A. H. Cloot, “A generalised groundwater flow equation using the concept of non-integer order derivatives,” Water SA, vol. 32, no. 1, pp. 1–7, 2006. View at Google Scholar · View at Scopus
  2. A. Atangana, “Numerical solution of space-time fractional order derivative of groundwater flow equation,” in Proceedings of the International Conference of Algebra and Applied Analysis, p. 20, Istanbul, Turkey, June 2012.
  3. A. Atangana and J. F. Botha, “Generalized groundwater flow equation using the concept of variable order derivative,” Boundary Value Problems, vol. 2013, article 53, 2013. View at Publisher · View at Google Scholar
  4. J. Boonstra and R. A. L. Kselik, SATEM 2002: Software for Aquifer Test Evaluation, International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands, 2002.
  5. G. P. Kruseman and N. A. de Ridder, Analysis and Evaluation of Pumping Test Data, International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands, 2nd edition, 1990.
  6. H. Jafari and C. M. Khalique, “Homotopy perturbation and variational iteration methods for solving fuzzy differential equations,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 38–48, 2012. View at Google Scholar
  7. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, no. 2, pp. 73–99, 2012. View at Google Scholar
  8. A. Atangana and E. Alabaraoye, “Solving system of fractional partial differential equations arisen in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equation,” Advances in Difference Equations, vol. 2013, article 94, 2013. View at Publisher · View at Google Scholar
  9. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. Atangana and A. Secer, “A note on fractional order derivatives and table of fractional derivatives of some special functions,” Abstract and Applied Analysis, vol. 2013, Article ID 279681, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. C. Wu, “New trends in the variational iteration method,” Communications in Fractional Calculus, vol. 2, pp. 59–75, 2011. View at Google Scholar
  12. G. Jumarie, “On the representation of fractional Brownian motion as an integral with respect to (dt)a,” Applied Mathematics Letters, vol. 18, no. 7, pp. 739–748, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers and Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. A. Atangana, “New class of boundary value problems,” Information Sciences Letters, vol. 1, no. 2, pp. 67–76, 2012. View at Google Scholar
  16. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian Original, Gordon and Breach, Yverdon, Switzerland, 1993. View at MathSciNet
  17. C. V. Theis, “The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage,” Transactions, American Geophysical Union, vol. 16, no. 2, pp. 519–524, 1935. View at Google Scholar
  18. P. Flajolet, X. Gourdon, and P. Dumas, “Meilin transforms and asymptotics: harmonic sums,” Theoretical Computer Science, vol. 144, no. 1-2, pp. 3–58, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, pp. 35–43, 1993. View at Google Scholar
  20. A. Atangana and A. Kılıçman, “The use of Sumudu transform for solving certain nonlinear fractional heat-like equations,” Abstract and Applied Analysis, vol. 2013, Article ID 737481, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  21. M. G. M. Hussain and F. B. M. Belgacem, “Transient solutions of Maxwell's equations based on sumudu transform,” Progress in Electromagnetics Research, vol. 74, pp. 273–289, 2007. View at Google Scholar · View at Scopus
  22. S. Weerakoon, “The “Sumudu transform” and the Laplace transform—reply,” International Journal of Mathematical Education in Science and Technology, vol. 28, no. 1, pp. 159–160, 1997. View at Google Scholar
  23. F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer, Berlin, Germany, 1973. View at MathSciNet
  24. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet