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Mathematical Problems in Engineering
Volume 2014, Article ID 104069, 9 pages
http://dx.doi.org/10.1155/2014/104069
Research Article

Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

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

Received 13 February 2014; Accepted 18 May 2014; Published 10 June 2014

Academic Editor: Miguel A. F. Sanjuan

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. View at Publisher · View at Google Scholar · View at MathSciNet
  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. 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
  4. F. Santamaria, S. Wils, E. D. 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 Google Scholar
  5. 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 MathSciNet
  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. M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. 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 · View at MathSciNet
  9. R. W. Ibrahim, “Fractional complex transforms for fractional differential equations,” Advances in Difference Equations, vol. 2012, article 192, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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 · View at MathSciNet
  17. 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 · View at MathSciNet
  18. 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
  19. A. Bekir, 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
  20. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. View at Google Scholar
  21. K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  22. 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
  23. 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
  24. J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,” Quarterly of Applied Mathematics, vol. 9, no. 3, pp. 225–236, 1951. View at Google Scholar · View at MathSciNet
  25. J. M. Burger, “A mathematical model illustrating the theory of turbulence,” in Advances in Applied Mechanics, vol. 1, pp. 171–199, 1948. View at Google Scholar
  26. 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 MathSciNet
  27. T. S. El-Danaf and A. R. Hadhoud, “Parametric spline functions for the solution of the one time fractional Burgers' equation,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4557–4564, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  28. S. M. Choo, S. K. Chung, and Y. J. Lee, “A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient,” Applied Numerical Mathematics, vol. 51, no. 2-3, pp. 207–219, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. E. Gurtin, “Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,” Physica D, vol. 92, no. 3-4, pp. 178–192, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  30. J. Kim, “A numerical method for the Cahn-Hilliard equation with a variable mobility,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 8, pp. 1560–1571, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  31. H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,” Abstract and Applied Analysis, vol. 2013, Article ID 587179, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet