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On Symbolic Solutions of Algebraic Partial Differential Equations

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Computer Algebra in Scientific Computing (CASC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8660))

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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.

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References

  1. Arendt, W., Urban, K.: Partielle Differenzialgleichungen. Eine Einführung in analytische und numerische Methoden. Spektrum Akademischer Verlag, Heidelberg (2010)

    Book  MATH  Google Scholar 

  2. Arnold, V.I.: Lectures on Partial Differential Equations. Springer, Heidelberg (2004)

    Book  Google Scholar 

  3. Eremenko, A.: Rational solutions of first-order differential equations. Annales Academiae Scientiarum Fennicae. Mathematica 23(1), 181–190 (1998)

    MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  11. Kamke, E.: Differentialgleichungen: Lösungsmethoden und Lösungen II, Leipzig. Akademische Verlagsgesellschaft Geest & Portig K.-G. (1965)

    Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ngô, L.X.C., Winkler, F.: Rational general solutions of parametrizable AODEs. Publicationes Mathematicae Debrecen 79(3-4), 573–587 (2011)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schicho, J.: Rational Parametrization of Surfaces. Journal of Symbolic Computation 26(1), 1–29 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sendra, J.R., Sevilla, D.: First steps towards radical parametrization of algebraic surfaces. Computer Aided Geometric Design 30(4), 374–388 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Doctoral Program Computational Mathematics, Johannes Kepler University Linz, 4040, Linz, Austria

    Georg Grasegger

  2. Research Institute for Symbolic Computation, Johannes Kepler University Linz, 4040, Linz, Austria

    Georg Grasegger & Franz Winkler

  3. Dpto. de Física y Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Madrid, Spain

    Alberto Lastra & J. Rafael Sendra

Authors
  1. Georg Grasegger
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  2. Alberto Lastra
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  3. J. Rafael Sendra
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  4. Franz Winkler
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, 34132, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Grasegger, G., Lastra, A., Sendra, J.R., Winkler, F. (2014). On Symbolic Solutions of Algebraic Partial Differential Equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-10515-4_9

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10514-7

  • Online ISBN: 978-3-319-10515-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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