Skip to main content
SpringerLink
Log in

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

  • Conference paper
  • First Online:
Analytic, Algebraic and Geometric Aspects of Differential Equations

Part of the book series: Trends in Mathematics ((TM))

  • 979 Accesses

Abstract

This paper is a slightly modified, abridged version of the work (Lastra and Malek, Adv Differ Equ 21:767–800, 2016). It deals with some questions made to the authors during the conference Analytic, Algebraic and Geometric Aspects of Differential Equations, held in Będlewo (Poland) during the second week of September, 2015.

We study analytic and formal solutions related to a singularly perturbed partial differential equation and relate them by means of a multi-level Gevrey order asymptotic behavior, with respect to the perturbation parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext (Springer, New York, 2000), xviii+299 pp.

    Google Scholar 

  2. M. Canalis-Durand, J. Mozo-Fernández, R. Schäfke, Monomial summability and doubly singular differential equations. J. Differ. Equ. 233, 485–511 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Hsieh, Y. Sibuya, Basic Theory of Ordinary Differential Equations. Universitext (Springer, New York, 1999)

    Book  MATH  Google Scholar 

  4. S. Kamimoto, On the exact WKB analysis of singularly perturbed ordinary differential equations at an irregular singular point. Preprint RIMS-1779 (2013)

    Google Scholar 

  5. A. Lastra, S. Malek, J. Sanz, On Gevrey solutions of threefold singular nonlinear partial differential equations. J. Differ. Equ. 255, 3205–3232 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Lastra, S. Malek, On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems. Adv. Differ. Equ. 2015, 200 (2015)

    Article  MATH  Google Scholar 

  7. A. Lastra, S. Malek, Multi-level Gevrey solutions of singularly perturbed linear partial differential equations. Adv. Differ. Equ. 21, 767–800 (2016)

    MathSciNet  MATH  Google Scholar 

  8. S. Malek, On Gevrey functions solutions of partial differential equations with Fuchsian and irregular singularities. J. Dyn. Control Syst. 15, 277–305 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Malek, On the summability of formal solutions for doubly nonlinear partial differential equations. J. Dyn. Control Syst. 18, 45–82 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Tahara, H. Yamazawa, Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations. J. Differ. Equ. 255, 3592–3637 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Alberto Lastra and Stéphane Malek are partially supported by the project MTM2012-31439 of Ministerio de Ciencia e Innovacion, Spain.

Author information

Authors and Affiliations

  1. Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, E-28871, Alcalá de Henares (Madrid), Spain

    Alberto Lastra

  2. Laboratoire Paul Painlevé, University of Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Stéphane Malek

Authors
  1. Alberto Lastra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stéphane Malek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Lastra .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Galina Filipuk

  2. Kumamoto University, Kumamoto, Japan

    Yoshishige Haraoka

  3. Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, Warsaw, Poland

    Sławomir Michalik

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Lastra, A., Malek, S. (2017). Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations. In: Filipuk, G., Haraoka, Y., Michalik, S. (eds) Analytic, Algebraic and Geometric Aspects of Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52842-7_13

Download citation

Publish with us

Policies and ethics

Search

Navigation