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Journal of Applied Mathematics
Volume 2014, Article ID 153706, 9 pages
http://dx.doi.org/10.1155/2014/153706
Research Article

Exact Solutions of the Space-Time Fractional Bidirectional Wave Equations Using the -Expansion Method

College of Mathematics of Honghe University, Mengzi, Yunnan 661100, China

Received 15 March 2014; Accepted 9 June 2014; Published 23 June 2014

Academic Editor: Jin Liang

Copyright © 2014 Wei Li 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. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
  2. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  3. X. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 1, no. 4, pp. 1–225, 2011. View at Google Scholar
  4. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  5. F. Santamaria, S. Wils, E. de Schutter, and G. J. Augustine, “Anomalous diffusion in Purkinje cell dendrites caused by spines,” Neuron, vol. 52, no. 4, pp. 635–648, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters A, vol. 375, no. 3, pp. 309–313, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Zhang, Q. A. Zong, D. Liu, and Q. Gao, “A generalized exp-function method for fractional Riccati differential equations,” Communications in Fractional Calculus, vol. 1, no. 1, pp. 48–51, 2010. View at Google Scholar
  8. A. Bekir, Ö. Güner, and A. C. Cevikel, “Fractional complex transform and exp-function methods for fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 426462, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. 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 MathSciNet
  10. B. Tang, Y. He, L. Wei, and X. Zhang, “A generalized fractional sub-equation method for fractional differential equations with variable coefficients,” Physics Letters A, vol. 376, no. 38-39, pp. 2588–2590, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. F. Meng, “A new approach for solving fractional partial differential equations,” Journal of Applied Mathematics, vol. 2013, Article ID 256823, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. A.-B. Abdel-Salam and E. A. Yousif, “Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method,” Mathematical Problems in Engineering, vol. 2013, Article ID 846283, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 684–693, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. B. Zheng, “(G/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, no. 5, pp. 623–630, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  15. K. A. Gepreel and S. Omran, “Exact solutions for nonlinear partial fractional differential equations,” Chinese Physics B, vol. 21, no. 11, Article ID 110204, 2012. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Wang, X. Li, and J. Zhang, “The (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Akgül, A. Kılıçman, and M. Inc, “Improved (G/G)-expansion method for the space and time fractional foam drainage and KdV equations,” Abstract and Applied Analysis, vol. 2013, Article ID 414353, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. E. M. E. Zayed, Y. A. Amer, and R. M. A. Shohib, “Exact traveling wave solutions for nonlinear fractional partial differential equations using the improved (G/G)-expansion method,” International Journal of Engineering and Applied Science, vol. 4, no. 7, pp. 18–31, 2014. View at Google Scholar
  19. K. A. Gepreel, T. A. Nofal, and F. M. Alotaibi, “Numerical solutions for the time and space fractional nonlinear partial differential equations,” Journal of Applied Mathematics, vol. 2013, Article ID 482419, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. Younis and A. Zafar, “Travelling wave solutions of fractional order coupled Burgers' equations by (G/G)-expansion method,” American Journal of Computational and Applied Mathematics, vol. 3, no. 2, pp. 81–85, 2013. View at Google Scholar
  21. B. Lu, “Bäcklund transformation of fractional Riccati equation and infinite sequence solutions of nonlinear fractional PDEs,” Abstract and Applied Analysis, vol. 2014, Article ID 572052, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. L. Bona and M. Chen, “A Boussinesq system for two-way propagation of nonlinear dispersive waves,” Physica D: Nonlinear Phenomena, vol. 116, no. 1-2, pp. 191–224, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  23. M. Chen, “Exact solutions of various Boussinesq systems,” Applied Mathematics Letters, vol. 11, no. 5, pp. 45–49, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Lee and R. Sakthivel, “New exact travelling wave solutions of bidirectional wave equations,” Journal of Physics, vol. 76, no. 6, pp. 819–829, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. 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 MathSciNet
  26. 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 MathSciNet
  27. Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010. View at Google Scholar · View at MathSciNet
  28. J. He and Z. Li, “Converting fractional differential equations into partial differential equations,” Thermal Science, vol. 16, no. 2, pp. 331–334, 2012. View at Publisher · View at Google Scholar · View at Scopus