Theoretical and Mathematical Physics

, Volume 159, Issue 3, pp 779–786 | Cite as

Nonlocal symmetries and reductions for some ordinary differential equations

Article

Abstract

We derive nonlocal symmetries for ordinary differential equations (ODEs). These symmetries are derived by embedding the ODE in an auxiliary system. Using these symmetries, we find that the order of the ODE can be reduced even if it does not admit point symmetries.

Keywords

conditional symmetry nonlocal symmetry ordinary differential equation 

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References

  1. 1.
    A. González-López, Phys. Lett. A, 133, 190–194 (1988).CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 2, 79–96 (1984).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 15, 161–209 (1989).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. M. Vinogradov and I. S. Krasil’shchik, Sov. Math. Dokl., 29, 337–341 (1984).MATHGoogle Scholar
  5. 5.
    A. M. Vinogradov and I. S. Krasil’shchik, Sov. Math. Dokl., 22, 235–239 (1980).MATHGoogle Scholar
  6. 6.
    G. W. Bluman and G. J. Reid, IMA J. Appl. Math., 40, 87–94 (1988).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. L. Gandarias, E. Medina, and C. Muriel, Nonlinear Anal., 47, 5167–5178 (2001).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. L. Gandarias, E. Medina, and C. Muriel, J. Nonlinear Math. Phys., 9(Suppl. 1), 47–58 (2002).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, J. Sov. Math., 55, 1401–1450 (1991).CrossRefGoogle Scholar
  10. 10.
    P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts in Math., Vol. 107), Springer, New York (1986).MATHGoogle Scholar
  11. 11.
    C. Muriel and J. L. Romero, IMA J. Appl. Math., 66, 111–125 (2001).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. C. Ferraioli, J. Phys. A, 40, 5479–5489 (2007).MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    C. Muriel and J. L. Romero, IMA J. Appl. Math., 72, 191–205 (2007).MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    B. Abraham-Shrauner, K. S. Govinder, and P. G. L. Leach, Phys. Lett. A, 203, 169–174 (1995).MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    B. Abraham-Shrauner, IMA J. Appl. Math., 56, 235–252 (1996).MATHMathSciNetGoogle Scholar
  16. 16.
    B. Abraham-Shrauner, J. Nonlinear Math. Phys., 9(Suppl. 2), 1–9 (2002).CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    A. A. Adam and F. M. Mahomed, IMA J. Appl. Math., 60, 187–198 (1998).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. M. Edelstein, K. S. Govinder, and F. M. Mahomed, J. Phys. A, 34, 1141–1152 (2001).MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    C. Géromini, M. R. Feix, and P. G. L. Leach, J. Phys. A, 34, 10109–10117 (2001).CrossRefMathSciNetGoogle Scholar
  20. 20.
    N. H. Ibragimov, J. Math. Anal. Appl., 318, 742–757 (2006).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© MAIK/Nauka 2009

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de CádizCádizSpain

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