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On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

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Abstract

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.

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Acknowledgements

The authors would like to thank Gleb Pogudin and François Boulier for useful discussions.

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Author notes

  1. Sebastian Falkensteiner and Yi Zhang are co-first authors.

Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria

    Sebastian Falkensteiner

  2. Department of Applied Mathematics, School of Science, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Yi Zhang

  3. Fractional Calculus, Optimization and Algebra Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Thieu N. Vo

Authors
  1. Sebastian Falkensteiner
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  2. Yi Zhang
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  3. Thieu N. Vo
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Correspondence to Thieu N. Vo.

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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.06; supported by the Austrian Science Fund (FWF): P 31327-N32; the UTD start-up grant: P-1-03246, the Natural Science Foundation of USA grants CCF-1815108 and CCF-1708884; XJTLU Research Development Funding RDF-20-01-12.

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Falkensteiner, S., Zhang, Y. & Vo, T.N. On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations. Mediterr. J. Math. 19, 74 (2022). https://doi.org/10.1007/s00009-022-01984-w

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  • DOI: https://doi.org/10.1007/s00009-022-01984-w

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