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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 461970, 7 pages
http://dx.doi.org/10.1155/2013/461970
Research Article

The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations with the Riemann-Liouville Derivative

1Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, 06530 Ankara, Turkey
2Institute of Space Sciences, P.O. Box MG 23, Magurele, 077125 Bucharest, Romania
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Faculty of Basic Science, Babol University of Technology, P.O. Box 47148-71167, Babol, Iran
5Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

Received 6 April 2013; Accepted 22 May 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Dumitru Baleanu et al. 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. W. T. Reid, Riccati Differential Equations, vol. 86, Academic Press, New York, NY, USA, 1972, Mathematics in Science and Engineering. View at MathSciNet
  2. V. V. Kravchenko, Applied Pseudoanalytic Function Theory, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. Conte and M. Musette, “Link between solitary waves and projective Riccati equations,” Journal of Physics A, vol. 25, no. 21, pp. 5609–5623, 1992. View at MathSciNet
  4. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. 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 MathSciNet
  6. 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 MathSciNet
  7. 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 MathSciNet
  8. 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
  9. F. Geng, Y. Lin, and M. Cui, “A piecewise variational iteration method for Riccati differential equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2518–2522, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. F. Z. Geng and X. M. Li, “A new method for Riccati differential equations based on reproducing kernel and quasilinearization methods,” Abstract and Applied Analysis, vol. 2012, Article ID 603748, 8 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. N. A. Khan, A. Ara, and M. Jamil, “An efficient approach for solving the Riccati equation with fractional orders,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2683–2689, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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 MathSciNet
  13. 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 MathSciNet
  14. 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 MathSciNet
  15. C. Li and Y. Wang, “Numerical algorithm based on Adomian decomposition for fractional differential equations,” Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1672–1681, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. 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 · View at MathSciNet
  17. M. Merdan, “On the solutions fractional Riccati differential equation with modified Riemann-Liouville derivative,” International Journal of Differential Equations, vol. 2012, Article ID 346089, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Yüzbaşı, “Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 6328–6343, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. E. W. Cheney, Introduction to Approximation Theory, AMS Chelsea, Providence, RI, USA, 2nd edition, 1998. View at MathSciNet
  20. M. Alipour, D. Rostamy, and D. Baleanu, “Solving multi-dimensional FOCPs with inequality constraint by BPs operational matrices,” Journal of Vibration and Control, 2012. View at Publisher · View at Google Scholar
  21. D. Rostamy, M. Alipour, H. Jafari, and D. Baleanu, “Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis,” Romanian Reports in Physics, vol. 65, no. 2, pp. 334–349, 2013.
  22. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  23. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006. View at MathSciNet
  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, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. 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
  26. A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473–1481, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  27. Y. Li and N. Sun, “Numerical solution of fractional differential equations using the generalized block pulse operational matrix,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1046–1054, 2011. View at Publisher · View at Google Scholar · View at MathSciNet