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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 828764, 8 pages
http://dx.doi.org/10.1155/2013/828764
Research Article

Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
2Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
3Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, Balgat, 06530 Ankara, Turkey
4Institute of Space Sciences, P.O. Box MG-23, Magurele, 76900 Bucharest, Romania

Received 5 August 2013; Accepted 23 August 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 Abdon Atangana and Dumitru Baleanu. 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. A. Ashyralyev, F. Dal, and Z. Pinar, “On the numerical solution of fractional hyperbolic partial differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 730465, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. I. Podlubny and A. M. A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science, Institute of Experimental Physics, 1996.
  4. A. Yakar and M. E. Koksal, “Existence results for solutions of nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 267108, 12 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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
  6. A. Atangana and S. C. Oukouomi Noutchie, “Stability and convergence of a tme-fractional variable order Hantush equation for a deformable aquifer,” Abstract and Applied Analysis, vol. 2013, Article ID 691060, 8 pages, 2013. View at Publisher · View at Google Scholar
  7. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. B. Oldham and J. Spanier, The Fractional Calculus, vol. 111 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet
  10. B. Ross, Ed., Fractional Calculus and Its Applications, vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975. View at MathSciNet
  11. Y. Luchko and R. Gorenflo, “The initial value problem for some fractional differential equations with the Caputo derivative,” Preprint Series A08–98, Freic Universitat Berlin, 1998, Fachbreich Mathematik and Informatik.
  12. 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
  13. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  14. 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
  15. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  16. A. Atangana and A. Kilicman, “Analytical solutions of the space-time fractional derivative of advection dispersion equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 853127, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of navier-stokes equations on cantor sets,” Mathematical Problems in Engineering, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar
  18. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, Vienna, Austria, 1997. View at Zentralblatt MATH · View at MathSciNet
  19. M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524–529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. C. Tadjeran, M. M. Meerschaert, and H. P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation,” Journal of Computational Physics, vol. 213, no. 1, pp. 205–213, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. C. M. Chen, F. Liu, I. Turner, and V. Anh, “A Fourier method for the fractional diffusion equation describing sub-diffusion,” Journal of Computational Physics, vol. 227, no. 2, pp. 886–897, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara, “Matrix approach to discrete fractional calculus II: partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137–3153, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. E. Hanert, “On the numerical solution of space-time fractional diffusion models,” Computers and Fluids, vol. 46, no. 1, pp. 33–39, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 1760–1781, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. R. Lin, F. Liu, V. Anh, and I. Turner, “Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 435–445, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type,” Proceedings of the Cambridge Philosophical Society, vol. 43, no. 1, pp. 50–67, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. C. P. Li and C. X. Tao, “On the fractional Adams method,” Computers and Mathematics with Applications, vol. 58, no. 8, pp. 1573–1588, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus