About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2012 (2012), Article ID 346089, 17 pages
http://dx.doi.org/10.1155/2012/346089
Research Article

On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative

Department of Mathematics Engineering, Gümüşhane University, 29100 Gümüşhane, Turkey

Received 10 December 2011; Accepted 6 March 2012

Academic Editor: Ebrahim Momoniat

Copyright © 2012 Mehmet Merdan. 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. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  2. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  4. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  5. 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. View at Publisher · View at Google Scholar
  6. A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  7. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
  8. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  9. G. M. Zaslavsky, Hamiltonian Chaosand Fractional Dynamics, Oxford University Press, 2005.
  10. M. Merdan, A. Yıldırım, and A. Gökdoğan, “Numerical solution of time-fraction Modified Equal Width Wave Equation,” Engineering Computations. In press.
  11. G. A. Einicke, L. B. White, and R. R. Bitmead, “The use of fake algebraic Riccati equations for co-channel demodulation,” IEEE Transactions on Signal Processing, vol. 51, no. 9, pp. 2288–2293, 2003. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Gerber, B. Hasselblatt, and D. Keesing, “The riccati equation: pinching of forcing and solutions,” Experimental Mathematics, vol. 12, no. 2, pp. 129–134, 2003. View at Scopus
  13. R. E. Kalman, Y. C. Ho, and K. S. Narendra, “Controllability of linear dynamical systems,” Contributions to Differential Equations, vol. 1, pp. 189–213, 1963.
  14. S. Bittanti, P. Colaneri, and G. De Nicolao, “The periodic Riccati equation,” in The Riccati Equation, Communications and Control Engineering, pp. 127–162, Springer, Berlin, Germany, 1991.
  15. S. Bittanti, P. Colaneri, and G. O. Guardabassi, “Periodic solutions of periodic Riccati equations,” IEEE Transactions on Automatic Control, vol. 29, no. 7, pp. 665–667, 1984. View at Scopus
  16. B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979.
  17. W. T. Reid, Riccati Differential Equations: Mathematics in Science and Engineering, vol. 86, Academic Press, New York, NY, USA, 1972.
  18. H. Aminikhah and M. Hemmatnezhad, “An efficient method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 835–839, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. S. Abbasbandy, “Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 485–490, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. S. Abbasbandy, “Iterated He's homotopy perturbation method for quadratic Riccati differential equation,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 581–589, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. S. Abbasbandy, “A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59–63, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. N. A. Khan, A. Ara, and M. Jamil, “An efficient approach for solving the Riccati equation with fractional orders,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2683–2689, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 167–174, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. J. Cang, Y. Tan, H. Xu, and S. J. Liao, “Series solutions of non-linear Riccati differential equations with fractional order,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 1–9, 2009. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Jafari and H. Tajadodi, “He’s variational iteration method for solving fractional Riccati differential equation,” International Journal of Differential Equations, vol. 2010, Article ID 764738, 8 pages, 2010. View at Publisher · View at Google Scholar
  27. S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006. View at Publisher · View at Google Scholar · View at Scopus
  28. J. H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at Scopus
  29. J. H. He and X. H. Wu, “Variational iteration method: new development and applications,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007. View at Publisher · View at Google Scholar · View at Scopus
  30. J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999. View at Scopus
  31. G. Jumarie, “Stochastic differential equations with fractional Brownian motion input,” International Journal of Systems Science, vol. 24, no. 6, pp. 1113–1131, 1993. View at Publisher · View at Google Scholar
  32. G. Jumarie, “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,” Mathematical and Computer Modelling, vol. 44, no. 3-4, pp. 231–254, 2006. View at Publisher · View at Google Scholar · View at Scopus
  33. G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1659–1664, 2009. View at Publisher · View at Google Scholar · View at Scopus
  34. 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 Scopus
  35. G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” Applied Mathematics Letters, vol. 18, no. 7, pp. 817–826, 2005. View at Publisher · View at Google Scholar · View at Scopus
  36. M.-J. Jang, C.-L. Chen, and Y.-C. Liu, “Two-dimensional differential transform for partial differential equations,” Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261–270, 2001. View at Publisher · View at Google Scholar · View at Scopus
  37. N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,” Journal of King Saud University, vol. 23, no. 4, pp. 413–417, 2010. View at Publisher · View at Google Scholar · View at Scopus