Symmetry Analysis



We have learnt the qualitative analysis of nonlinear systems in previous chapters. Symmetry is an inherent property of natural phenomena as well as man-made devices. Naturally, the concept of symmetry is exploited to study the linear as well as nonlinear problems.


Heat Equation Infinitesimal Generator Determine Equation Invariant Solution Symmetry Analysis 


  1. 1.
    Yaglom, M.: Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century. Birkhäuser (1988)Google Scholar
  2. 2.
    Oberlack, M.: A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328 (2001)CrossRefMATHGoogle Scholar
  3. 3.
    Birkhoff, G.: Hydrodynamics—A Study in Logic, Fact and Similitude. Princeton University Press, Princeton (1960)MATHGoogle Scholar
  4. 4.
    Ovsiannikov, L.V.: Groups and group-invariant solutions of differential equations. Dokl. Akad. Nauk. USSR 118, 439–442 (1958). (in Russian)Google Scholar
  5. 5.
    Ovsiannikov, L.V.: Groups properties of the nonlinear heat conduction equation. Dokl. Akad. Nauk. USSR 125, 492–495 (1958). (in Russian)Google Scholar
  6. 6.
    Ovsiannikov, L.V.: Groups properties of differential equations. Novosibirsk (1962). (in Russian)Google Scholar
  7. 7.
    Ovsiannikov, L.V.: Groups Analysis of Differential Equations. Academic Press, New York (1982)MATHGoogle Scholar
  8. 8.
    Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969)MathSciNetMATHGoogle Scholar
  9. 9.
    Bluman, G.W., Cole, J.D.: Similarity methods for differential equations. Appl. Math. Sci. 13. Springer-Verlag, New York (1974)Google Scholar
  10. 10.
    Kumei, S.: Invariance transformations, invariance group transformation, and invariance groups of sine-Gordon equations. J. Math. Phys. 16, 2461–2468 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kumei, S.: Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV-type equations and nonlinear Schrödinger equations, invariance group transformation. J. Math. Phys. 18, 256–264 (1977)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Olver, P.J.: Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18, 1212–1215 (1977)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Olver, P.J.: Symmetry groups and group invariant solutions of partial differential equations. J. Diff. Geom. 14, 497–542 (1979)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Olver, P.J.: Symmetry and explicit solutions of partial differential equations. App. Num. Math. 10, 307–324 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Riedel, Boston (1985)CrossRefMATHGoogle Scholar
  16. 16.
    Ibragimov, N.H.: Methods of Group Analysis for Ordinary Differential Equations. Znanie Publ, Moscow (1991). (In Russian)MATHGoogle Scholar
  17. 17.
    Olver, P.J.: Applications of Lie Groups To Differential Equations. Springer-Verlag, New York (2000)MATHGoogle Scholar
  18. 18.
    Brian, J.: Cantwell: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2002)Google Scholar
  19. 19.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, Applied Mathematical Sciences 81. Springer-Verlag, New York (1996)Google Scholar
  20. 20.
    Ibragimov, N.H.: Elementary Lie Group Analysis and Ordinary Differential Equations. John Wiley and Sons (1999)Google Scholar
  21. 21.
    Noether, E.: Invariante Variationsprobleme, Nachr. König. Gesell. Wissen. Göttingen, Math.-Phys. KL, 235–257 (1918)Google Scholar
  22. 22.
    Korteweg, J.D., de Vries, G.: On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Philosophical Magazine, Series 5, 39, 422–443 (1895)Google Scholar

Copyright information

© Springer India 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of BurdwanBardhamanIndia

Personalised recommendations