Skip to main content

Effective Methods for Systems of Algebraic Partial Differential Equations

  • Chapter
Effective Methods in Algebraic Geometry

Part of the book series: Progress in Mathematics ((PM,volume 94))

Abstract

As a matter of fact, despite the fast progress of computer algebra during the last ten years, only a few steps have been done towards the use of symbolic computers for studying systems of partial differential equations (PDE) ([2], [13]). In particular, one must notice a few modern tentatives for dealing with algebraic PDE through differential algebraic techniques [19] or differential elimination techniques [16], [18] or exterior calculus ([1], Novosibirsk school), but these methods, being absolutely dependent on the coordinate system as they rely on old works ([6], [14], [18]), do not seem to go far inside the intrinsic structure of the system. Despite this point, these methods have been applied with success to control theory during the last five years ([3], [12]).

“God created integers, men made the remaining” (KRONECKER)

…but it is surely the Devil who let them conceive partial differential equations!

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. E. A. Arais, V. P. Sapeev, N. N. Janenko, Realization of Cartan’s method of exterior forms on an electronic computer, Dokl. Akad. Nauk. SSSR 214 (1974). ( Soviet. Math. Dokl., 15 pages 203-205)

    Google Scholar 

  2. J. Davenport, Y. Siret, E. Tournier, “Calcul Formel,” Masson, Paris, 1986.

    Google Scholar 

  3. M. Fliess, Automatique et corps différentiels, Forum Math. 1 (1989).

    Google Scholar 

  4. H. Goldschmidt, Integrability criteria for systems of non-linear partial differential equations, J. Diff. Geometry 1 (1969), 269–307.

    MathSciNet  Google Scholar 

  5. M. Golubitsky, V. Guillemin, “Stable mappings and their singularities,” Springer-Verlag, 1973.

    Google Scholar 

  6. M. Janet, “Leçons sur les systèmes d’équations aux dérivées partielles,” Cahiers scientifiques, Fasc. IV, Gauthier-Villars, Paris, 1929.

    Google Scholar 

  7. E. R. Kolchin, “Differential algebra and algebraic groups,” Academic Press, New-York, 1973.

    MATH  Google Scholar 

  8. A. Kumpera, D. C. Spencer, “Lie equations,” Annals of Math. Studies 73, Princeton University Press, 1972.

    Google Scholar 

  9. J. F. Pommaret, “Systems of partial differential equations and Lie pseudogroups,” New York, 1978.

    Google Scholar 

  10. J. F. Pommaret, “Differential Galois theory,” Gordon and Breach, New York, 1983.

    Google Scholar 

  11. J. F. Pommaret, “Lie pseudogroups and mechanics,” Gordon and Breach, New York, 1988.

    Google Scholar 

  12. J. F. Pommaret, Problémes formels en théorie du contrôle aux dérivées partielles, C. R. Acad. Sc. Paris 308, I (1989), 457–460.

    MathSciNet  MATH  Google Scholar 

  13. R. H. Rand, “Computer algebra in applied mathematics: an introduction to MACSYMA,” Pitman, London, 1984.

    Google Scholar 

  14. J. F. Ritt, “Differential algebra,” Dover, 1950, 1966.

    Google Scholar 

  15. D. J. Saunders, “The geometry of jet bundles,” London Mayhematical Society Lecture Note Series, 142, Cambridge University Press, 1989.

    Google Scholar 

  16. F. Schwarz, The Riquier-Janet theory and its application to non-linear evolution equations, Physica 110 (1984), 243–251.

    Google Scholar 

  17. D. C. Spencer, Overdetermined systems of partial differential equations, Bull. Amer. Math. Soc. 75 (1965), 1–114.

    Google Scholar 

  18. J. M. Thomas, “Systems and roots,” W. Byrd Press, 1962.

    Google Scholar 

  19. Wu Wen-Tsun, On the foundation of algebraic differential geometry, Mathematics-Mechanization research preprints 3 (1989), 1-26, Institute of Systems Science, Academia Sinica, Pekin.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media New York

About this chapter

Cite this chapter

Pommaret, JF., Haddak, A. (1991). Effective Methods for Systems of Algebraic Partial Differential Equations. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_27

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0441-1_27

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6761-4

  • Online ISBN: 978-1-4612-0441-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics