About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2013 (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

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.

Linked References

  1. L. Javandel, C. Doughly, and F. C. Tsang, Groundwater Transport: Handbook of Mathematical Models, American Geophysical Union, 1984.
  2. M. T. van Genuchten and W. J. Alves, “Analytical solutions of the one dimensional convective solute transport equation,” US Department of Agriculture Technical Bulletin, vol. 1661, p. 149, 1982.
  3. G. Afken, Mathematical Methods for Physicists, Academic Press, London, UK, 1985.
  4. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  5. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press Inc., San Diego, Calif, USA, 1999. View at MathSciNet
  6. A. Y. Luchko and R. Groneflo, “The initial value problem for some fractional differential equations with the Caputo derivative,” Preprint series A08–98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
  7. R. L. Magin and M. Ovadia, “Modeling the cardiac tissue electrode interface using fractional calculus,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1431–1442, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent—part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
  9. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  10. A. Cloot and J. F. Botha, “A generalised groundwater flow equation using the concept of non-integer order derivatives,” Water SA, vol. 32, no. 1, pp. 55–78, 2006.
  11. A. Kilicman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  12. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, NY, USA, 1993. View at MathSciNet
  13. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at MathSciNet
  14. A. Anatoly, J. Juan, and M.S. Hari, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  15. S. Momani and Z. Odibat, “Numerical solutions of the space-time fractional advection-dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 6, pp. 1416–1429, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  16. V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. S. Duan, R. Rach, D. Bulean, 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.
  18. D. Q. Zeng and Y. M. Qin, “The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives,” Communications in Fractional Calculus, no. 3, pp. 26–29, 2012.
  19. A. Atangana and J. F. Botha, “Analytical solution of groundwater flow equation via Homotopy Decomposition Method,” Journal of Earth Science & Climatic Change, vol. 3, no. 115, p. 2157, 2012.
  20. N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517–529, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Singh, D. Kumar, and Sushila, “Homotopy perturbation Sumudu transform method for nonlinear equations,” Advances in Applied Mathematics and Mechanics, vol. 4, pp. 165–175, 2011.
  22. G. C. Wu and D. Baleanu, “Variational iteration method for the Burgers' flow with fractional derivatives—New Lagrange multipliers,” Applied Mathematical Modelling, vol. 37, no. 9, pp. 6183–6190, 2013. View at Publisher · View at Google Scholar
  23. Y. Chen and H.-L. An, “Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 87–95, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  24. A. Atangana and A. Secer, “Time-fractional Coupled—the Korteweg-de Vries Equations,” Abstract Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013.
  25. A. Abdon, “New class of boundary value problems,” Information Sciences Letters, vol. 1, no. 2, pp. 67–76, 2012.
  26. J. Hristov, “A short-distance integral-balance solution to a strong subdiffusion equation: a weak power-law profile,” International Review of Chemical Engineering-Rapid Communications, vol. 2, no. 5, pp. 555–563, 2010.
  27. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
  28. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Lévy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000.
  29. D. A. Benson, R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft, “Fractional dispersion, Lévy motion, and the MADE tracer tests,” Transport in Porous Media, vol. 42, no. 1-2, pp. 211–240, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  30. J. H. Cushman and T. R. Ginn, “Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux,” Water Resources Research, vol. 36, no. 12, pp. 3763–3766, 2000. View at Publisher · View at Google Scholar · View at Scopus
  31. B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, “Modeling Non-fickian transport in geological formations as a continuous time random walk,” Reviews of Geophysics, vol. 44, no. 2, Article ID RG2003, 2006. View at Publisher · View at Google Scholar · View at Scopus
  32. M. M. Meerschaert and H.-P. Scheffler, “Limit theorems for continuous-time random walks with infinite mean waiting times,” Journal of Applied Probability, vol. 41, no. 3, pp. 623–638, 2004. View at MathSciNet
  33. M. M. Meerschaert, J. Mortensen, and S. W. Wheatcraft, “Fractional vector calculus for fractional advection-dispersion,” Physica A, vol. 367, pp. 181–190, 2006.
  34. S. W. Wheatcraft and S. W. Tyler, “An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry,” Water Resources Research, vol. 24, no. 4, pp. 566–578, 1988. View at Scopus