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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 908975, 15 pages
http://dx.doi.org/10.1155/2012/908975
Research Article

Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation

1Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China
3Graduate School, Chinese Academy of Sciences, Beijing 100049, China
4Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China

Received 20 June 2012; Accepted 22 August 2012

Academic Editor: Ramajayam Sahadevan

Copyright © 2012 Hongwei Yang 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.-X. Ma, X. Gu, and L. Gao, “A note on exact solutions to linear differential equations by the matrix exponential,” Advances in Applied Mathematics and Mechanics, vol. 1, no. 4, pp. 573–580, 2009.
  2. S. lie, “Vorlesungen Über Differentialgleichungen mit Bekannten Infinitesinalen Transformationen,” BG Teubner, 1891.
  3. A. G. Johnpillai and C. M. Khalique, “Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1207–1215, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. W.-X. Ma and M. Chen, “Do symmetry constraints yield exact solutions?” Chaos, Solitons & Fractals, vol. 32, no. 4, pp. 1513–1517, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. L. Gandarias and M. S. Bruzon, “Classical and nonclassical symmetries of a generalized Boussinesq equation,” Journal of Nonlinear Mathematical Physics, vol. 5, no. 1, pp. 8–12, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Y. Dong, S. C. Xu, and Z. X. Chen, “On permutation symmetries of hopfield model neural network,” Discrete Dynamics in Nature and Society, vol. 6, no. 2, pp. 129–136, 2001.
  7. W. X. Ma, R. K. Bullough, P. J. Caudrey, and W. I. Fushchych, “Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras,” Journal of Physics A, vol. 30, no. 14, pp. 5141–5149, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. W.-X. Ma and Z.-X. Zhou, “Coupled integrable systems associated with a polynomial spectral problem and their Virasoro symmetry algebras,” Progress of Theoretical Physics, vol. 96, no. 2, pp. 449–457, 1996. View at Publisher · View at Google Scholar
  9. W.-X. Ma and M. Chen, “Direct search for exact solutions to the nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2835–2842, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. P. E. Hydon, “Discrete point symmetries of ordinary differential equations,” The Royal Society of London. Proceedings A, vol. 454, no. 1975, pp. 1961–1972, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. E. Hydon, “How to construct the discrete symmetries of partial differential equations,” European Journal of Applied Mathematics, vol. 11, no. 5, pp. 515–527, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W. X. Ma, “A hierarchy of coupled Burgers systems possessing a hereditary structure,” Journal of Physics A, vol. 26, no. 22, pp. L1169–L1174, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. Nadjafikhah and R. Bakhshandeh-Chamazkoti, “Symmetry group classification for general Burgers' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2303–2310, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. A. Ouhadan and E. H. El Kinani, “Lie symmetries of the equation ut(x,t)+g(u)ux(x,t)=0,” Advances in Applied Clifford Algebras, vol. 17, no. 1, pp. 95–106, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. Nadjafikhah, “Lie symmetries of inviscid Burgers' equation,” Advances in Applied Clifford Algebras, vol. 19, no. 1, pp. 101–112, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Nadjafikhah, “Classification of similarity solutions for inviscid Burgers' equation,” Advances in Applied Clifford Algebras, vol. 20, no. 1, pp. 71–77, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. I. L. Freire, “Note on Lie point symmetries of Burgers Equations,” Anais do CNMAC, 2009.
  18. M. A. Abdulwahhab, A. H. Bokhari, A. H. Kara, and F. D. Zaman, “On the Lie point symmetry analysis and solutions of the inviscid Burgers equation,” Pramana Journal of Physics, vol. 77, pp. 407–414, 2011.
  19. L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982.
  20. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar
  21. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81, Springer, New York, NY, USA, 1989.