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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 752869, 14 pages
http://dx.doi.org/10.1155/2012/752869
Research Article

Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

1Department of Mathematics, Dezhou University, Dezhou 253023, China
2Nonlinear Dynamics and Chaos Group, School of Management, Tianjin University, Tianjin 30072, China

Received 11 November 2011; Revised 25 December 2011; Accepted 30 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Yanqin Liu. 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. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH
  2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, Singapore, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, p. 77, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S. Wang and M. Xu, “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 1087–1096, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. X. J. Xiaoyun and X. M. Yu, “Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media,” International Journal of Non-Linear Mechanics, vol. 41, no. 1, pp. 156–165, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. J. H. Ma and M. Y. Liu, “Exact solutions for a generalized nonlinear fractional Fokker-Planck equation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 515–521, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Y.-Q. Liu and J.-H. Ma, “Exact solutions of a generalized multi-fractional nonlinear diffusion equation in radical symmetry,” Communications in Theoretical Physics, vol. 52, no. 5, pp. 857–861, 2009. View at Publisher · View at Google Scholar
  8. T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616–1621, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3381–3390, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Z. A. Anastassi and T. E. Simos, “New trigonometrically fitted six-step symmetric methods for the efficient solution of the Schrödinger equation,” Communications in Mathematical and in Computer Chemistry, vol. 60, no. 3, pp. 733–752, 2008. View at Zentralblatt MATH
  11. Z. A. Anastassi and T. E. Simos, “Numerical multistep methods for the efficient solution of quantum mechanics and related problems,” Physics Reports, vol. 482/483, pp. 1–240, 2009. View at Publisher · View at Google Scholar
  12. G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. A.-M. Wazwaz and S. M. El-Sayed, “A new modification of the Adomian decomposition method for linear and nonlinear operators,” Applied Mathematics and Computation, vol. 122, no. 3, pp. 393–405, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. 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
  15. A.-M. Wazwaz, “The variational iteration method for analytic treatment of linear and nonlinear ODEs,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 120–134, 2009. View at Publisher · View at Google Scholar
  16. V. S. Ertürk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. A. Al-rabtah, V. S. Ertürk, and S. Momani, “Solutions of a fractional oscillator by using differential transform method,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1356–1362, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. E. Yusufoglu, “Numerical solution of Duffing equation by the Laplace decomposition algorithm,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 572–580, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Y. Khan, “An effective modification of the laplace decomposition method for nonlinear equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1373–1376, 2009. View at Scopus
  20. J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. X. C. Li, M. Y. Xu, and X. Y. 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
  22. S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345–350, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205–209, 2008. View at Zentralblatt MATH
  24. M. A. Noor, “Iterative methods for nonlinear equations using homotopy perturbation technique,” Applied Mathematics & Information Sciences, vol. 4, no. 2, pp. 227–235, 2010. View at Zentralblatt MATH
  25. M. A. Noor, “Some iterative methods for solving nonlinear equations using homotopy perturbation method,” International Journal of Computer Mathematics, vol. 87, no. 1–3, pp. 141–149, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1937–1945, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He's polynomials,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 1963–1967, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1486–1492, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalized KdV equations using he's polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 227–233, 2009. View at Scopus
  30. A. Yildirim, “Application of the homotopy perturbation method for the Fokker-Planck equation,” International Journal for Numerical Methods in Biomedical Engineering, vol. 26, no. 9, pp. 1144–1154, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. M. Khan, M. A. Gondal, and S. Kumar, “A novel homotopy perturbation tranform algorithm for linear and nonlinear system of partial differential equations,” World Applied Sciences Journal, vol. 12, no. 12, pp. 2352–2357, 2011.