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Symmetries of Second- and Third-Order Ordinary Differential Equations

  • Fritz Schwarz
Conference paper
  • 433 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)

Abstract

In order to apply Lie’s symmetry theory for solving a differential equation it must be possible to identify the group of symmetries leaving the equation invariant. The answer is obtained in two steps. At first a classification of the possible symmetries of equations of the respective order is determined. Secondly a decision procedure is provided which allows to identify the symmetry type within this classification. For second-order equations the answer has been obtained by Lie himself. In this article the complete answer for quasilinear equations of order three is given. An important tool is the Janet base representation for the determining system of the symmetries.

Keywords

Base Type Canonical Variable Algebraic Property Determine System Symmetry Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bluman 1990. Bluman G. W., S. Kumei S.: Symmetries of Differential Equations, Springer, Berlin (1990).Google Scholar
  2. Janet 1920. Janet M.: Les systèmes d’équations aux dérivées partielles, Journal de mathématiques 83, 65–123 (1920).Google Scholar
  3. Kamke 1961. Kamke E.: Differentialgleichungen: Lösungsmethoden und Lösungen, I. Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig (1961).Google Scholar
  4. Killing 1887. Killing W.: Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Mathematische Annalen 31, 252–290, 33, 1–48, 34, 57–122, 36, 161–189 (1887).CrossRefMathSciNetGoogle Scholar
  5. Lie 1883. Lie S.: Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten I, II, III and IV. Archiv for Mathematik VIII, page 187–224, 249–288, 371–458 and IX, page 431–448 respectively (1883) [Gesammelte Abhandlungen, vol. V, page 240–281, 282–310, 362–427 and 432–446].Google Scholar
  6. Lie 1888. Lie S.: Theorie der Transformationsgruppen I, II and III. Teubner, Leipzig (1888). [Reprinted by Chelsea Publishing Company, New York (1970)].Google Scholar
  7. Lie 1891. Lie S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen. Teubner, Leipzig (1891). [Reprinted by Chelsea Publishing Company, New York (1967)].zbMATHGoogle Scholar
  8. Lie 1893. Lie S.: Vorlesungen über continuierliche Gruppen. Teubner, Leipzig (1893). [Reprinted by Chelsea Publishing Company, New York (1971)].Google Scholar
  9. Loewy 1906. Loewy A.: Über vollständig reduzible lineare homogene Differentialgleichungen. Mathematische Annalen 56, 89–117 (1906).CrossRefMathSciNetGoogle Scholar
  10. Olver 1986. Olver P.: Application of Lie Groups to Differential Equations. Springer, Berlin (1986).Google Scholar
  11. Schwarz 1995. Schwarz F.: Symmetries of 2nd and 3rd Order ODE’s. In: Proceedings of the ISSAC’95, ACM Press, A. Levelt, Ed., page 16–25 (1995).Google Scholar
  12. Schwarz 1996. Schwarz F.: Janet Bases of 2nd Order Ordinary Differential Equations. In: Proceedings of the ISSAC’96, ACM Press, Lakshman, Ed., page 179–187 (1996)Google Scholar
  13. Schwarz 2002. Schwarz F.: Algorithmic Lie Theory for Solving Ordinary Differential Equations. To appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Fritz Schwarz
    • 1
  1. 1.FhG, Institut SCAISankt AugustinGermany

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