Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 638026, 16 pages
http://dx.doi.org/10.1155/2012/638026
Research Article

The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations

1Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, 31521 Tanta, Egypt
2Department of Engineering Physics and Mathematics, Faculty of Engineering, Kafr El Sheikh University, Kafr El Sheikh, Egypt

Received 31 March 2012; Revised 8 May 2012; Accepted 23 May 2012

Academic Editor: Manfred Moller

Copyright © 2012 W. K. Zahra and S. M. Elkholy. 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. O. P. Agrawal and P. Kumar, “Comparison of five schemes for fractional differential equations,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds., pp. 43–60, 2007. View at Google Scholar
  2. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York, NY, USA, 1967. View at Zentralblatt MATH
  3. D. Baleanu and S. I. Muslih, “On Fractional Variational Principles,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds., pp. 115–126, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. M. Benghorbal, Power series solutions of fractional differential equations and symbolic derivatives and integrals [Ph.D. thesis], Faculty of Graduate Studies, The University of Western Ontario, Ontario, Canada, 2004.
  5. B. Bonilla, M. Rivero, and J. J. Trujillo, “Linear differential equations of fractional order,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds., pp. 77–91, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. X. Jiang, J. E. Carletta, and T. T. Hartley, “Implementation of fractional-order operators on field programmable gate arrays,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds., pp. 333–346, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. N. Kosmatov, “Integral equations and initial value problems for nonlinear differential,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2521–2529, 2009. View at Publisher · View at Google Scholar
  8. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  10. H. Nasuno, N. Shimizu, and M. Fukunaga, “Fractional derivative consideration on nonlinear viscoelastic statical and dynamical behavior under large pre-displacement,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds., pp. 363–376, 2007. View at Publisher · View at Google Scholar
  11. A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3877–3896, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  13. I. Podlubny, I. Petráš, B. M. Vinagre, P. O'Leary, and L'. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 281–296, 2002. View at Publisher · View at Google Scholar
  14. X. Su and S. Zhang, “Solutions to boundary-value problems for nonlinear differential equations of fractional order,” Electronic Journal of Differential Equations, vol. 2009, no. 26, pp. 1–15, 2009. View at Google Scholar · View at Scopus
  15. M. S. Tavazoei and M. Haeri, “A note on the stability of fractional order systems,” Mathematics and Computers in Simulation, vol. 79, no. 5, pp. 1566–1576, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, and W. A. Wakeham, “A fractional differential equation for a MEMS viscometer used in the oil industry,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 373–381, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Duan, J. An, and M. Xu, “Solution of system of fractional differential equations by Adomian decomposition method,” Applied Mathematics, A Journal of Chinese Universities Series B, vol. 22, no. 1, pp. 7–12, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. Garrappa, “On some explicit Adams multistep methods for fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 392–399, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. A. Ghorbani, “Toward a new analytical method for solving nonlinear fractional differential equations,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 49-50, pp. 4173–4179, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–11, 2008. View at Google Scholar · View at Zentralblatt MATH
  22. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Minsk, Belarus, 1st edition, 2006.
  23. S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 85–95, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1754–1758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. W. Chen, L. Ye, and H. Sun, “Fractional diffusion equations by the Kansa method,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1614–1620, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. K. Diethelm and G. Walz, “Numerical solution of fractional order differential equations by extrapolation,” Numerical Algorithms, vol. 16, no. 3-4, pp. 231–253, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017–1033, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. L. Galeone and R. Garrappa, “Fractional Adams-Moulton methods,” Mathematics and Computers in Simulation, vol. 79, no. 4, pp. 1358–1367, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. J. P. Roop, Variational solution of the fractional advection dispersion equation [Ph.D. thesis], Clemson University, Clemson, SC, USA, 2004.
  31. F. I. Taukenova and M. Kh. Shkhanukov-Lafishev, “Difference methods for solving boundary value problems for fractional-order differential equations,” Computational Mathematics and Mathematical Physics, vol. 46, no. 10, pp. 1871–1795, 2006. View at Publisher · View at Google Scholar
  32. W. K. Zahra and S. M. Elkholy, “Quadratic spline solution for boundary value problem of fractional order,” Numerical Algorithms, vol. 59, pp. 373–391, 2012. View at Publisher · View at Google Scholar
  33. P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1975.
  34. R. D. Russell and L. F. Shampine, “A collocation method for boundary value problems,” Numerische Mathematik, vol. 19, pp. 1–28, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH