Advertisement

On Symbolic Solutions of Algebraic Partial Differential Equations

  • Georg Grasegger
  • Alberto Lastra
  • J. Rafael Sendra
  • Franz Winkler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8660)

Abstract

In this paper we present a general procedure for solving first-order autonomous algebraic partial differential equations in two independent variables. The method uses proper rational parametrizations of algebraic surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. We will demonstrate in examples that, depending on certain steps in the procedure, rational, radical or even non-algebraic solutions can be found. Solutions computed by the procedure will depend on two arbitrary independent constants.

Keywords

Partial differential equations algebraic surfaces rational parametrizations radical parametrizations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arendt, W., Urban, K.: Partielle Differenzialgleichungen. Eine Einführung in analytische und numerische Methoden. Spektrum Akademischer Verlag, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Lectures on Partial Differential Equations. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Eremenko, A.: Rational solutions of first-order differential equations. Annales Academiae Scientiarum Fennicae. Mathematica 23(1), 181–190 (1998)zbMATHMathSciNetGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Feng, R., Gao, X.S.: A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs. Journal of Symbolic Computation 41(7), 739–762 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Grasegger, G.: Radical Solutions of First Order Autonomous Algebraic Ordinary Differential Equations. In: Nabeshima, K. (ed.) ISSAC 2014: Proceedings of the 39th International Symposium on International Symposium on Symbolic and Algebraic Computation, pp. 217–223. ACM, New York (2014)Google Scholar
  7. 7.
    Huang, Y., Ngô, L.X.C., Winkler, F.: Rational General Solutions of Trivariate Rational Systems of Autonomous ODEs. In: Proceedings of the Fourth International Conference on Mathematical Aspects of Computer and Information Sciences (MACIS 2011), pp. 93–100 (2011)Google Scholar
  8. 8.
    Huang, Y., Ngô, L.X.C., Winkler, F.: Rational General Solutions of Trivariate Rational Differential Systems. Mathematics in Computer Science 6(4), 361–374 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Huang, Y., Ngô, L.X.C., Winkler, F.: Rational General Solutions of Higher Order Algebraic ODEs. Journal of Systems Science and Complexity 26(2), 261–280 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Kamke, E.: Differentialgleichungen: Lösungsmethoden und Lösungen II, Leipzig. Akademische Verlagsgesellschaft Geest & Portig K.-G. (1965)Google Scholar
  12. 12.
    Ngô, L.X.C., Sendra, J.R., Winkler, F.: Birational Transformations on Algebraic Ordinary Differential Equations. Tech. Rep. 12–18, RISC Report Series, Johannes Kepler University Linz, Austria (2012)Google Scholar
  13. 13.
    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 (2012)CrossRefGoogle Scholar
  14. 14.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of first order non-autonomous parametrizable ODEs. Journal of Symbolic Computation 45(12), 1426–1441 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of parametrizable AODEs. Publicationes Mathematicae Debrecen 79(3-4), 573–587 (2011)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Ngô, L.X.C., Winkler, F.: Rational general solutions of planar rational systems of autonomous ODEs. Journal of Symbolic Computation 46(10), 1173–1186 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Schicho, J.: Rational Parametrization of Surfaces. Journal of Symbolic Computation 26(1), 1–29 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Sendra, J.R., Sevilla, D.: First steps towards radical parametrization of algebraic surfaces. Computer Aided Geometric Design 30(4), 374–388 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Georg Grasegger
    • 1
    • 2
  • Alberto Lastra
    • 3
  • J. Rafael Sendra
    • 3
  • Franz Winkler
    • 2
  1. 1.Doctoral Program Computational MathematicsJohannes Kepler University LinzLinzAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler University LinzLinzAustria
  3. 3.Dpto. de Física y MatemáticasUniversidad de AlcaláMadridSpain

Personalised recommendations