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Algebraic and Geometric Analysis of Singularities of Implicit Differential Equations (Invited Talk)

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

We review our recent works on singularities of implicit ordinary or partial differential equations. This includes firstly the development of a general framework combining algebraic and geometric methods for dealing with general systems of ordinary or partial differential equations and for defining the type of singularities considered here. We also present an algorithm for detecting all singularities of an algebraic differential equation over the complex numbers. We then discuss the adaptions required for the analysis over the real numbers. We further outline for a class of singular initial value problems for a second-order ordinary differential equation how geometric methods allow us to determine the local solution behaviour in the neighbourhood of a singularity including the regularity of the solution. Finally, we show for some simple cases of algebraic singularities how there such an analysis can be performed.

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Acknowledgments

This work has been supported by the bilateral project ANR-17-CE40-0036 and DFG-391322026 SYMBIONT. We thank our co-authors on this subject, Markus Lange-Hegermann, Daniel Robertz and Thomas Sturm, for a pleasant collaboration. The first author furthermore thanks the organizers of CASC 2020 for the honor to present an invited talk on this subject.

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Authors and Affiliations

  1. Institut Für Mathematik, Universität Kassel, 34109, Kassel, Germany

    Werner M. Seiler & Matthias Seiß

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  1. Werner M. Seiler
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  2. Matthias Seiß
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Correspondence to Werner M. Seiler .

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Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Seiler, W.M., Seiß, M. (2020). Algebraic and Geometric Analysis of Singularities of Implicit Differential Equations (Invited Talk). In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-60026-6_2

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  • Online ISBN: 978-3-030-60026-6

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