Computing

, Volume 34, Issue 2, pp 91–106 | Cite as

Automatically determining symmetries of partial differential equations

  • F. Schwarz
Article

Abstract

A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.

AMS Subject Classifications

22E99 35-04 35 C 05 

Key words

Symmetry groups differential equations 

Automatische Bestimmung von Symmetrien bei partiellen Differentialgleichungen

Zusammenfassung

Es wird ein REDUCE-Programm zur Bestimmung der Symmetrien beliebiger Systeme von partiellen Differentialgleichungen beschrieben. Es kann sowohl interaktiv als auch im Batch-Betrieb verwendet werden. In vielen Fällen findet es die volle Symmetriegruppe vollständig automatisch. In einigen anderen Fällen bleiben einige lineare Differentialgleichungen des bestimmenden Systems übrig, dessen Lösung im Augenblick nicht automatisch gefunden werden kann. Falls sie vom Benutzer eingegeben werden, antwortet das System mit den infinitesimalen Generatoren der Symmetriegruppe.

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References

  1. [1]
    Lie, S.: Differentialgleichungen. Leipzig: 1891. (Reprinted by Chelsea Publ. Co., New York, 1967).Google Scholar
  2. [2]
    Campbell, J. E.: Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups. New York: Chelsea Publ. Col 1903. (Reprinted in 1966 by the same company).Google Scholar
  3. [3]
    Dickson, L. E.: Differential equations from the group standpoint. Ann. Math.25, 287–378 (1924).Google Scholar
  4. [4]
    Cohen, A.: An Introduction to the Lie Theory of One-Parameter Groups. New York: Stechert & Co. 1931.Google Scholar
  5. [5]
    Birkhoff, G.: Hydrodynamics. Princeton: Princeton University Press 1950.Google Scholar
  6. [6]
    Sedov, L. I.: Similarity and Dimensional Methods in Mechanics. London: Infosearch 1959.Google Scholar
  7. [7]
    Ovsjannikov, L. V.: Group Properties of Differential Equations. Novosibirsk: 1962. (Translated by Bluman, G. W.)Google Scholar
  8. [8]
    Hansen, A. G., Similarity Analyses of Boundary Value Problems in Engineering. Englewood Cliffs, N. J.: Prentice-Hall 1964.Google Scholar
  9. [9]
    Ames, W. F.: Nonlinear Partial Differential Equations in Engineering II. New York: Academic Press 1972.Google Scholar
  10. [10]
    Bluman, G. W., Cole, J. D.: Similarity Methods for Differential Equations (Applied Mathematics Series, Vol. 13). Berlin-Heidelberg-New York: 1974.Google Scholar
  11. [11]
    Chester, W.: Continuous transformations and differential equations. J. Inst. Math. Applics.19, 343–376 (1977).Google Scholar
  12. [12]
    Olver, R.: J. Diff. Geometry14, 497–542 (1979).Google Scholar
  13. [13]
    Vladimirov, S. A.: Symmetry Groups of Differential Equations and Relativistic Fields. Moskau: 1979 (in Russian).Google Scholar
  14. [14]
    Barenblatt, G. I.: Similarity, Self-Similarity and Intermediate Asymptotics. New York: Consultants Bureau 1979.Google Scholar
  15. [15]
    Anderson, R. L., Ibragimov, N. H.: Lie Baecklund Transformations in Applications. Philadelphia: SIAM 1979.Google Scholar
  16. [16]
    Sattinger, D. H.: Les symétries des équations et leurs applications dans la mechanique et à la physique. Publications Mathématiques d'Orsay 80.08, 1980.Google Scholar
  17. [17]
    Ovsjannikov, L. V.: Group Analysis of Differential Equations. New York: Academic Press 1982.Google Scholar
  18. [18]
    Hill, J. M.: Solution of Differential Equations by Means of One-parameter Groups. Boston: Pitman 1982.Google Scholar
  19. [19]
    Schwarz, F.: A REDUCE package for determining Lie symmetries of ordinary and partial differential equations. Comp. Phys. Commun.27, 179–186 (1982).Google Scholar
  20. [20]
    Schwarz, F.: Automatically determining symmetries of ordinary differential equations. In: Proceedings of the EUROCAL'83, London, p. 45. Berlin-Heidelberg-New York: Springer 1983.Google Scholar
  21. [21]
    Hearn, A. C.: REDUCE User's Manual. Santa Monica: Rand Corporation 1983.Google Scholar
  22. [22]
    Calogero, F., Degasperis, A.: Spectral Transform and Solutions. Amsterdam: North-Holland 1982.Google Scholar
  23. [23]
    Schwarz, F.: Symmetries of the two-dimensional Korteweg-deVries equation. Jo. Phys. Soc. Japan51, 2387 (1982); Symmetries of SU (2) invariant Yang-Mills theories, Lett. Math. Phys.6, 355 (1982); Lie-symmetries of the von Karman equations, Comp. Phys. Commun.31, 113 (1984).Google Scholar
  24. [24]
    Kamke, E.: Differentialgleichungen, Lösungsmethoden und Lösungen II. Leipzig: Akademische Verlagsgesellschaft 1965.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • F. Schwarz
    • 1
    • 2
  1. 1.Bonn
  2. 2.Institut F 1GMDAugustin 1Federal Republic of Germany

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