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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 548292, 11 pages
http://dx.doi.org/10.1155/2012/548292
Research Article

A Characteristic Difference Scheme for Time-Fractional Heat Equations Based on the Crank-Nicholson Difference Schemes

Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

Received 25 April 2012; Revised 27 August 2012; Accepted 27 August 2012

Academic Editor: Dumitru Bǎleanu

Copyright © 2012 Ibrahim Karatay and Serife R. Bayramoglu. 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. X. Li, X. Han, and X. Wang, “Numerical modeling of viscoelastic flows using equal low-order finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 9–12, pp. 570–581, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. L. Bagley and P. J. Torvik, “Theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. M. Raberto, E. Scalas, and F. Mainardi, “Waiting-times and returns in high-frequency financial data: an empirical study,” Physica A, vol. 314, no. 1–4, pp. 749–755, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,” Physica A, vol. 284, no. 1–4, pp. 376–384, 2000. View at Publisher · View at Google Scholar
  5. 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. View at Publisher · View at Google Scholar · View at Scopus
  6. L. Galue, S. L. Kalla, and B. N. Al-Saqabi, “Fractional extensions of the temperature field problems in oil strata,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 35–44, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. I. Podlubny, Fractional Differential Equations: An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  8. X. Li, M. Xu, and X. Jiang, “Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 434–439, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. A. T. Machado, “Discrete-time fractional-order controllers,” Fractional Calculus & Applied Analysis, vol. 4, no. 1, pp. 47–66, 2001. View at Zentralblatt MATH
  10. O. P. Agrawal, O. Defterli, and D. Baleanu, “Fractional optimal control problems with several state and control variables,” Journal of Vibration and Control, vol. 16, no. 13, pp. 1967–1976, 2010. View at Publisher · View at Google Scholar
  11. D. Baleanu, O. Defterli, and O. P. Agrawal, “A central difference numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 15, no. 4, pp. 583–597, 2009. View at Publisher · View at Google Scholar
  12. Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006. View at Scopus
  13. Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28–39, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional wave equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 767–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. S. Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329–336, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. I. Podlubny, The Laplace Transform Method for Linear Differential Equations of Fractional Order, Slovac Academy of Science, Bratislava, Slovakia, 1994.
  18. B. Baeumer, D. A. Benson, and M. M. Meerschaert, “Advection and dispersion in time and space,” Physica A, vol. 350, no. 2–4, pp. 245–262, 2005. View at Publisher · View at Google Scholar · View at Scopus
  19. Z. Q. Deng, V. P. Singh, and L. Bengtsson, “Numerical solution of fractional advection-dispersion equation,” Journal of Hydraulic Engineering, vol. 130, no. 5, pp. 422–431, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, “Numerical methods for the solution of partial differential equations of fractional order,” Journal of Computational Physics, vol. 192, no. 2, pp. 406–421, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. 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
  23. I. Karatay, S. R. Bayramoğlu, and A. Şahin, “Implicit difference approximation for the time fractional heat equation with the nonlocal condition,” Applied Numerical Mathematics, vol. 61, no. 12, pp. 1281–1288, 2011. View at Publisher · View at Google Scholar
  24. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar
  25. M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation,” Journal of Computational Physics, vol. 211, no. 1, pp. 249–261, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. 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
  27. L. Su, W. Wang, and Z. Yang, “Finite difference approximations for the fractional advection-diffusion equation,” Physics Letters A, vol. 373, no. 48, pp. 4405–4408, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. A. M. Abu-Saman and A. M. Assaf, “Stability and convergence of Crank-Nicholson method for fractional advection dispersion equation,” Advances in Applied Mathematical Analysis, vol. 2, no. 2, pp. 117–125, 2007.
  29. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2087–2093, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience Publishers, New York, NY, USA, 1967.