Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Lie symmetries and differential Galois groups of linear equations


Authors: W. R. Oudshoorn and M. van der Put
Journal: Math. Comp. 71 (2002), 349-361
MSC (2000): Primary 34A30, 34G34, 34Mxx, 65L99
Published electronically: October 4, 2001
MathSciNet review: 1863006
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.


References [Enhancements On Off] (What's this?)

  • [K] Irving Kaplansky, An introduction to differential algebra, 2nd ed., Hermann, Paris, 1976. Actualités Scientifiques et Industrielles, No. 1251; Publications de l’Institut de Mathématique de l’Université de Nancago, No. V. MR 0460303 (57 #297)
  • [KM] Jorge Krause and Louis Michel, Équations différentielles linéaires d’ordre 𝑛>2 ayant une algèbre de Lie de symétrie de dimension 𝑛+4, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 18, 905–910 (French, with English summary). MR 978467 (89m:58183)
  • [L1] S. Lie, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, Bearbeitet und herausgegeben von Dr. G. Scheffers, Teubner, Leipzig, 1893.
  • [L2] S. Lie, Klassifikation und Integration von gewöhnlicher Differentialgleichungen zwischen $x,y$, die eine Gruppe von Transformationen gestetten I, Math. Ann. 22 (1888), 213-253.
  • [ML] F.M. Mahomed and P.G.L. Leach, Symmetry Lie algebras of $n$th order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), 80-107.
  • [O1] Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734 (88f:58161)
  • [O2] Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276 (96i:58005)
  • [P] Marius van der Put, Galois theory of differential equations, algebraic groups and Lie algebras, J. Symbolic Comput. 28 (1999), no. 4-5, 441–472. Differential algebra and differential equations. MR 1731933 (2001b:12012), http://dx.doi.org/10.1006/jsco.1999.0310
  • [St] Hans Stephani, Differential equations, Cambridge University Press, Cambridge, 1989. Their solution using symmetries. MR 1041800 (91a:58001)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 34A30, 34G34, 34Mxx, 65L99

Retrieve articles in all journals with MSC (2000): 34A30, 34G34, 34Mxx, 65L99


Additional Information

W. R. Oudshoorn
Affiliation: Prinsengracht 275 Den Haag, The Netherlands
Email: woudshoo@sctcorp.com

M. van der Put
Affiliation: Department of Mathematics, P.O. Box 800, 9700 AV, Groningen, The Netherlands
Email: mvdput@math.rug.nl

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01397-7
PII: S 0025-5718(01)01397-7
Received by editor(s): October 13, 1999
Received by editor(s) in revised form: January 24, 2000
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society