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.
Supported in part by INTAS Grant 99-0167
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bluman 1990. Bluman G. W., S. Kumei S.: Symmetries of Differential Equations, Springer, Berlin (1990).
Janet 1920. Janet M.: Les systèmes d’équations aux dérivées partielles, Journal de mathématiques 83, 65–123 (1920).
Kamke 1961. Kamke E.: Differentialgleichungen: Lösungsmethoden und Lösungen, I. Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig (1961).
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).
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].
Lie 1888. Lie S.: Theorie der Transformationsgruppen I, II and III. Teubner, Leipzig (1888). [Reprinted by Chelsea Publishing Company, New York (1970)].
Lie 1891. Lie S.: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen. Teubner, Leipzig (1891). [Reprinted by Chelsea Publishing Company, New York (1967)].
Lie 1893. Lie S.: Vorlesungen über continuierliche Gruppen. Teubner, Leipzig (1893). [Reprinted by Chelsea Publishing Company, New York (1971)].
Loewy 1906. Loewy A.: Über vollständig reduzible lineare homogene Differentialgleichungen. Mathematische Annalen 56, 89–117 (1906).
Olver 1986. Olver P.: Application of Lie Groups to Differential Equations. Springer, Berlin (1986).
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).
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)
Schwarz 2002. Schwarz F.: Algorithmic Lie Theory for Solving Ordinary Differential Equations. To appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schwarz, F. (2003). Symmetries of Second- and Third-Order Ordinary Differential Equations. In: Winkler, F., Langer, U. (eds) Symbolic and Numerical Scientific Computation. SNSC 2001. Lecture Notes in Computer Science, vol 2630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45084-X_4
Download citation
DOI: https://doi.org/10.1007/3-540-45084-X_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40554-2
Online ISBN: 978-3-540-45084-9
eBook Packages: Springer Book Archive