The algebro-geometric method: Solving algebraic differential equations by parametrizations
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- by Sebastian Falkensteiner, Johann J. Mitteramskogler, J. Rafael Sendra and Franz Winkler HTML | PDF
- Bull. Amer. Math. Soc. 60 (2023), 85-122 Request permission
Abstract:
We present a survey of the algebro-geometric method for solving algebraic ordinary differential equations by means of parametrizations of the associated algebraic sets. In particular, we deal with equations of order one, and also systems of algebro-geometric dimension one. Various classes of solutions are treated symbolically, such as rational, algebraic, and power series solutions. We also consider classes of algebraic transformations of the associated algebraic sets preserving the solutions of the differential equations. Two Maple packages, implementing some of these solution methods, are presented.References
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Additional Information
- Sebastian Falkensteiner
- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler Universität Linz, Linz, Austria
- MR Author ID: 1382291
- Email: sebastian.falkensteiner@risc.jku.at
- Johann J. Mitteramskogler
- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler Universität Linz, Linz, Austria
- MR Author ID: 1493628
- ORCID: 0000-0003-3332-5461
- Email: johann.mitteramskogler@risc.jku.at
- J. Rafael Sendra
- Affiliation: Universidad de Alcalá, Dpto. de Física y Matemáticas, Alcalá de Henares, Madrid, Spain
- MR Author ID: 260673
- ORCID: 0000-0003-2568-1159
- Email: rafael.sendra@uah.es
- Franz Winkler
- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler Universität Linz, Linz, Austria
- MR Author ID: 183545
- Email: franz.winkler@risc.jku.at
- Received by editor(s): December 29, 2021
- Published electronically: August 30, 2022
- Additional Notes: The first and third authors were partially supported by the grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MICINN
The second and fourth authors were partially supported by the Austrian Science Fund (FWF) under grant no.\xspace P31327-N32 (Symbolic Solutions of Algebraic Differential Equations (ADE-solve)) - © Copyright 2022 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 60 (2023), 85-122
- MSC (2020): Primary 34A26, 34A09, 34A05, 68W30, 14H50, 14J26
- DOI: https://doi.org/10.1090/bull/1773
- MathSciNet review: 4520777