Advertisement

Symbolic Solutions of First-Order Algebraic ODEs

  • Georg Grasegger
  • Franz WinklerEmail author
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8942)

Abstract

Algebraic ordinary differential equations are described by polynomial relations between the unknown function and its derivatives. There are no general solution methods available for such differential equations. However, if the hypersurface determined by the defining polynomial of an algebraic ordinary differential equation admits a parametrization, then solutions can be computed and the solvability in certain function classes may be decided. After an overview of methods developed in the last decade we present a new and rather general method for solving algebraic ordinary differential equations.

Keywords

Ordinary differential equations General solutions Algebraic curves Algebraic surfaces Rational parametrizations Radical parametrizations 

References

  1. 1.
    Aroca, J.M., Cano, J., Feng, R., Gao, X.-S.: Algebraic general solutions of algebraic ordinary differential equations. In: Kauers, M. (ed.) ISSAC’05. Proceedings of the 30th International Symposium on Symbolic and Algebraic Computation, Beijing, China, pp. 29–36. ACM Press, New York (2005)Google Scholar
  2. 2.
    Feng, R., Gao, X.-S.: Rational general solutions of algebraic ordinary differential equations. In: Gutierrez, J. (ed.) Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 155–162. ACM Press, New York (2004)Google Scholar
  3. 3.
    Feng, R., Gao, X.-S.: A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs. J. Symb. Comput. 41(7), 739–762 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Grasegger, G.: A procedure for solving autonomous AODEs. Technical report 2013–05, Doctoral Program “Computational Mathematics”. Johannes Kepler University Linz, Austria (2013)Google Scholar
  5. 5.
    Harrison, M.: Explicit solution by radicals, gonal maps and plane models of algebraic curves of genus 5 or 6. J. Symb. Comput. 51, 3–21 (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    Huang, Y., Ngô, L.X.C., Winkler, F.: Rational general solutions of trivariate rational differential systems. Math. Comput. Sci. 6(4), 361–374 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Huang, Y., Ngô, L.X.C., Winkler, F.: Rational general solutions of higher order algebraic ODEs. J. Syst. Sci. Complex. 26(2), 261–280 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hubert, E.: The general solution of an ordinary differential equation. In: Lakshman, Y.N. (ed.) Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 189–195. ACM Press, New York (1996)Google Scholar
  9. 9.
    Kamke, E.: Differentialgleichungen: Lösungsmethoden und Lösungen. B. G. Teubner, Stuttgart (1997)Google Scholar
  10. 10.
    Ngô, L.X.C., Sendra, J.R., Winkler, F.: Classification of algebraic ODEs with respect to rational solvability. In: Computational Algebraic and Analytic Geometry. Contemporary Mathematics, vol. 572, pp. 193–210. American Mathematical Society, Providence, RI (2012)Google Scholar
  11. 11.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of first order non-autonomous parametrizable ODEs. J. Symb. Comput. 45(12), 1426–1441 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of parametrizable AODEs. Publicationes Mathematicae Debrecen 79(3–4), 573–587 (2011)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of planar rational systems of autonomous ODEs. J. Symb. Comput. 46(10), 1173–1186 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ritt, J.F.: On algebraic functions, which can be expressed in terms of radicals. Trans. Am. Math. Soc. 24, 21–30 (1924)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ritt, J.F.: Differential Algebra. Colloquium publications, vol. 33. American Mathematical Society, New York (1950)zbMATHGoogle Scholar
  16. 16.
    Schicho, J., Sevilla, D.: Effective radical parametrization of trigonal curves. Computational Algebraic and Analytic Geometry. volume 572 of Contemporary Mathematics, pp. 221–231. American Mathematical Society, Providence, RI (2012)Google Scholar
  17. 17.
    Sendra, J.R., Sevilla, D.: Radical parametrizations of algebraic curves by adjoint curves. J. Symb. Comput. 46(9), 1030–1038 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Sendra, J.R., Sevilla, D.: First steps towards radical parametrization of algebraic surfaces. Comput. Aided Geom. Des. 30(4), 374–388 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sendra, J.R., Winkler, F.: Tracing index of rational curve parametrizations. Comput. Aided Geom. Des. 18(8), 771–795 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational Algebraic Curves, A Computer Algebra Approach. In: Algorithms and Computation in Mathematics, vol. 22. Springer, Heidelberg (2008)Google Scholar
  21. 21.
    Walker, R.J.: Algebraic Curves. Springer, Heidelberg (1978). Reprint of the 1st ed. 1950 by Princeton University PressCrossRefzbMATHGoogle Scholar
  22. 22.
    Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Doctoral Program Computational Mathematics, Research Institute for Symbolic ComputationJohannes Kepler University LinzLinzAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler University LinzLinzAustria

Personalised recommendations