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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 924956, 11 pages
http://dx.doi.org/10.1155/2012/924956
Research Article

The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations

1School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
2College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
3Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India

Received 23 October 2012; Revised 18 November 2012; Accepted 18 November 2012

Academic Editor: Igor Andrianov

Copyright © 2012 Jianping Zhao 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. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH
  5. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at Zentralblatt MATH
  6. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  7. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH
  8. V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994. View at Zentralblatt MATH
  9. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH
  10. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong.
  14. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  15. A. H. A. Ali, “The modified extended tanh-function method for solving coupled MKdV and coupled Hirota-Satsuma coupled KdV equations,” Physics Letters. A, vol. 363, no. 5-6, pp. 420–425, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. G. H. Gao, Z. Z. Sun, and Y. N. Zhang, “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions,” Journal of Computational Physics, vol. 231, no. 7, pp. 2865–2879, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008/09. View at Publisher · View at Google Scholar
  19. S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379–387, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. Y. Hu, Y. Luo, and Z. Lu, “Analytical solution of the linear fractional differential equation by Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 220–229, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters. A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. M. A. El-Sayed, S. H. Behiry, and W. E. Raslan, “Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1759–1765, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2199–2208, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476–484, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters. A, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. E. Fan, “Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation,” Physics Letters. A, vol. 282, no. 1-2, pp. 18–22, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. View at Publisher · View at Google Scholar
  30. S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters. A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters. A, vol. 199, no. 3-4, pp. 169–172, 1995. View at Publisher · View at Google Scholar
  32. 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
  33. G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31–48, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters. A, vol. 376, no. 4, pp. 407–411, 2012. View at Publisher · View at Google Scholar
  35. B. Lu, “Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters. A, vol. 376, no. 28-29, pp. 2045–2048, 2012. View at Publisher · View at Google Scholar
  36. G. Jumarie, “Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1444–1450, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. J. H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters. A, vol. 376, no. 4, pp. 257–259, 2012. View at Publisher · View at Google Scholar
  38. J. H. He, “Asymptotic methods for solitary solutions and compacts,” Abstract and applied analysis. In press. View at Publisher · View at Google Scholar
  39. S. E. Esipov, “Coupled Burgers equations: a model of polydispersive sedimentation,” Physical Review E, vol. 52, no. 4, pp. 3711–3718, 1995. View at Publisher · View at Google Scholar · View at Scopus
  40. A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Physica A, vol. 361, no. 2, pp. 394–404, 2006. View at Publisher · View at Google Scholar