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Advances in Mathematical Physics
Volume 2017, Article ID 2825416, 9 pages
https://doi.org/10.1155/2017/2825416
Research Article

Third-Order Conditional Lie–Bäcklund Symmetries of Nonlinear Reaction-Diffusion Equations

1College of Information and Management Science, Henan Agricultural University, Zhengzhou 450046, China
2College of Information Science and Technology, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Keqin Su; moc.liamtoh@usniqek

Received 24 October 2016; Revised 16 January 2017; Accepted 18 January 2017; Published 9 February 2017

Academic Editor: Nikos Mastorakis

Copyright © 2017 Keqin Su and Jie Cao. 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. Z. Zhdanov, “Conditional Lie-Backlund symmetry and reduction of evolution equations,” Journal of Physics A: Mathematical and General, vol. 28, no. 13, pp. 3841–3850, 1995. View at Publisher · View at Google Scholar · View at Scopus
  2. A. S. Fokas and Q. M. Liu, “Nonlinear interaction of traveling waves of nonintegrable equations,” Physical Review Letters, vol. 72, no. 21, pp. 3293–3296, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. V. A. Galaktionov and V. M. Matrosov, “On new exact blow-up solutions for nonlinear heat conduction equations with source and applications,” Differential and Integral Equations, vol. 3, no. 5, pp. 863–874, 1990. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. V. A. Galaktionov, “Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 125, no. 2, pp. 225–246, 1995. View at Publisher · View at Google Scholar · View at Scopus
  5. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York, NY, USA, 1989.
  6. G. W. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” Journal of Applied Mathematics and Mechanics, vol. 18, pp. 1025–1042, 1996. View at Google Scholar
  7. W. I. Fushchych, W. M. Shtelen, and N. I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Naukova Dumka, Kiev, Ukraine, 1989, Translated into English by Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
  8. N. K. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, Mass, USA, 1985.
  9. Q. M. Liu, “Exact interaction of solitary waves for certain nonintegrable equations,” Journal of Mathematical Physics, vol. 37, no. 1, pp. 324–345, 1996. View at Publisher · View at Google Scholar
  10. Q. M. Liu, “Exact solutions to nonlinear equations with quadratic nonlinearity,” Journal of Physics A: Mathematical and General, vol. 34, no. 24, pp. 5083–5088, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. C. Z. Qu, “Group classification and generalized conditional symmetry reduction of the nonlinear diffusion-convection equation with a nonlinear source,” Studies in Applied Mathematics, vol. 99, no. 2, pp. 107–136, 1997. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Z. Qu, “Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry,” IMA Journal of Applied Mathematics, vol. 62, no. 3, pp. 283–302, 1999. View at Google Scholar · View at Scopus
  13. C. Qu, S. Zhang, and R. Liu, “Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source,” Physica D: Nonlinear Phenomena, vol. 144, no. 1-2, pp. 97–123, 2000. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Z. Qu, L. N. Ji, and L. Z. Wang, “Conditional lie Bäcklund symmetries and sign-invariants to quasi-linear diffusion equations,” Studies in Applied Mathematics, vol. 119, no. 4, pp. 355–391, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. L. Ji and C. Qu, “Conditional Lie-Bäcklund symmetries and invariant subspaces to non-linear diffusion equations,” IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), vol. 76, no. 4, pp. 610–632, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. L. Ji and C. Qu, “Conditional Lie-Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations with convection and source,” Studies in Applied Mathematics, vol. 131, no. 3, pp. 266–301, 2013. View at Publisher · View at Google Scholar · View at Scopus
  17. P. Basarab-Horwath and R. Z. Zhdanov, “Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries,” Journal of Mathematical Physics, vol. 42, no. 1, pp. 376–389, 2001. View at Publisher · View at Google Scholar · View at Scopus
  18. R. Z. Zhdanov and A. Y. Andreitsev, “Non-classical reductions of initial-value problems for a class of nonlinear evolution equations,” Journal of Physics A: Mathematical and General, vol. 33, no. 32, pp. 5763–5781, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. R. Z. Zhdanov, “Higher conditional symmetry and reduction of initial value problems,” Nonlinear Dynamics, vol. 28, no. 1, pp. 17–27, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. A. Sergyeyev, “Constructing conditionally integrable evolution systems in (1 + 1) dimensions: a generalization of invariant modules approach,” Journal of Physics A: Mathematical and General, vol. 35, no. 35, pp. 7653–7660, 2002. View at Publisher · View at Google Scholar · View at Scopus
  21. L. N. Ji, C. Z. Qu, and S. Shen, “Conditional Lie-Bäcklund symmetry of evolution system and application for reaction-diffusion system,” Studies in Applied Mathematics, vol. 133, no. 1, pp. 118–149, 2014. View at Publisher · View at Google Scholar · View at Scopus
  22. N. N. Yanenko, “Theory of consistency and methods of integrating systems of nonlinear partial differential equations,” in Proceedings of the 4th All-Union Mathematical Congress, pp. 247–259, Leningrad, Russia, 1964 (Russian).
  23. E. Pucci and G. Saccomandi, “Evolution equations, invariant surface conditions and functional separation of variables,” Physica D, vol. 139, no. 1-2, pp. 28–47, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. V. A. Galaktionov and S. Svirshchevski, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall, London, UK, 2007.
  25. P. J. Olver and P. Rosenau, “The construction of special solutions to partial differential equations,” Physics Letters A, vol. 114, no. 3, pp. 107–112, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. P. J. Olver and P. Rosenau, “Group invariant solutions of differential equations,” SIAM Journal on Applied Mathematics, vol. 47, no. 2, pp. 263–278, 1987. View at Publisher · View at Google Scholar · View at Scopus
  27. P. J. Olver, “Direct reduction and differential constraints,” Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, vol. 444, no. 1922, pp. 509–523, 1994. View at Publisher · View at Google Scholar · View at Scopus
  28. O. V. Kaptsov, “Invariant sets of evolution equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 19, no. 8, pp. 753–761, 1992. View at Publisher · View at Google Scholar · View at Scopus
  29. D. Levi and P. Winternitz, “Non-classical symmetry reduction: example of the boussinesq equation,” Journal of Physics A, vol. 22, no. 15, pp. 2915–2924, 1989. View at Publisher · View at Google Scholar · View at Scopus
  30. L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982.
  31. P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2201–2213, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. O. V. Kaptsov, “B-determining equations: applications to nonlinear partial differential equations,” European Journal of Applied Mathematics, vol. 6, no. 3, pp. 265–286, 1995. View at Publisher · View at Google Scholar · View at Scopus
  33. O. V. Kaptsov, “Linear determining equations for differential constraints,” Sbornik: Mathematics, vol. 189, no. 11-12, pp. 1839–1854, 1998. View at Publisher · View at Google Scholar · View at Scopus
  34. O. V. Kaptsov and I. V. Verevkin, “Differential constraints and exact solutions of nonlinear diffusion equations,” Journal of Physics A: Mathematical and General, vol. 36, no. 5, pp. 1401–1414, 2003. View at Publisher · View at Google Scholar · View at Scopus
  35. V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, Springer, Dordrecht, Netherlands, 1998. View at Publisher · View at Google Scholar
  36. M. Kunzinger and R. O. Popovych, “Generalized conditional symmetries of evolution equations,” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 444–460, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  37. R. O. Popovych, O. O. Vaneeva, and N. M. Ivanova, “Potential nonclassical symmetries and solutions of fast diffusion equation,” Physics Letters A, vol. 362, no. 2-3, pp. 166–173, 2007. View at Publisher · View at Google Scholar · View at Scopus
  38. S. R. Svirshchevskii, “Nonlinear differential operators of first and second order possessing invariant linear spaces of maximal dimension,” Theoretical and Mathematical Physics, vol. 105, no. 2, pp. 1346–1353, 1995. View at Publisher · View at Google Scholar · View at Scopus