Advertisement

Solving Discontinuous Ordinary Differential Equations

  • Martin von Mohrenschildt
Part of the Progress in Theoretical Computer Science book series (PTCS)

Abstract

In this paper we generalize the basic notations of the Liouville-Ritt-Risch theory of closed-form solutions to discontinuous field extensions. Our aim is to extend the theory of differential fields such that the “classical algorithm” like the Risch structure theorem and the algorithm solving the Risch differential equation can be extended to handle discontinuous extensions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Abe91]
    Aberer, K., Combinatory Differential Fields and Constructive Analysis, ETH-Thesis, 9357, ETH Zürich, (1991).Google Scholar
  2. [An56]
    Andre, J., Über stückweise lineare Differentialgleichungen, die bei Reglungsproblemen auftreten, Arch. Math. 7 (1956), pp. 148–156.MathSciNetCrossRefGoogle Scholar
  3. [Ant73]
    Antosik, P., Theory of Distributions, Elsevier, Amsterdam, (1973).zbMATHGoogle Scholar
  4. [Braun78]
    Braun, M., Differential Equations and their Applications, Springer-Verlag, Berlin, (1978).Google Scholar
  5. [Bro90]
    Bronstein, M., The transcendental Risch differential equation, Jour. Symbol. Comp. 9 (1990), 49–60.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Dav86]
    Davenport, J.H., The Risch differential equation problem, SIAM J. Comput., vol. 15, no. 4 (1986), 903–918.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Eng90]
    Engeler, E., Combinatory differential fields, Theoret. Comput. Sci. 72 (1990), 119–131.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Fil88]
    Filippov, A.F., Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Norwell, MA, (1988).Google Scholar
  9. [Kan83]
    Kanwal, R. P., Generalized Functions: Theory and Technique, Academic Press, New York and San Diego, (1983).zbMATHGoogle Scholar
  10. [Liou33]
    Liouville, J., Sur la détermination des intégrates dont la valeur est algebraique, J. de l’Ecole Poly. 14 (1833).Google Scholar
  11. [Moh94]
    von Mohrenschildt, M., Symbolic Solution of Discontinuous Differential Equations, ETH-Diss No. 10768, ETH Zürich, (1994).Google Scholar
  12. [Ost46]
    Ostrowski, A., Sur l’intégrabilité élémentaire de quelques classes d’expressions, Rev. Roumaine Math. Pures Appl. 6 (1946), 879–887.Google Scholar
  13. [Ri69]
    Risch, R.H., The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969), 167–189.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Ro29]
    Rosenthal, A., Über die Lösung von Systemen gewöhnlicher Differentialgleichungen, Sitzungsberichte Heidelberger Aka. Wiss., (1929).Google Scholar
  15. [Sing90]
    Singer, M.F., Formal solutions of differential equations, J. Symbolic. Comput. 10 (1990), 59–94.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Martin von Mohrenschildt

There are no affiliations available

Personalised recommendations