Skip to main content
SpringerLink
Log in

Gevrey Order and Summability of Formal Series Solutions of Certain Classes of Inhomogeneous Linear Integro-Differential Equations with Variable Coefficients

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this article, we investigate Gevrey and summability properties of formal power series solutions of certain classes of inhomogeneous linear integro-differential equations with analytic coefficients in a neighborhood of \((0,0)\in \mathbb {C}^{2}\). In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We denote \(\widetilde {f}\) with a tilde to emphasize the possible divergence of the series \(\widetilde {f}\).

  2. A subsector Σof a sector Σ′is said to be a proper subsector and one denotes Σ ⋐Σ′if its closure in \(\mathbb {C} \)is contained in Σ′∪{0}.

  3. Relations (4.1) and (4.2) imply indeed that q − p κ ≤−1 for all \(i\in \mathcal {K}\) and q ∈ Q i .

References

  1. Balser W. Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke. Pacific J Math 1999;188(1):53–63.

    Article  MathSciNet  MATH  Google Scholar 

  2. Balser W. Formal power series and linear systems of meromorphic ordinary differential equations. New York: Springer; 2000.

    MATH  Google Scholar 

  3. Balser W. Multisummability of formal power series solutions of partial differential equations with constant coefficients. J Differ Equ 2004;201(1):63–74.

    Article  MathSciNet  MATH  Google Scholar 

  4. Balser W, Braaksma BJL, Ramis J-P, Sibuya Y. Multisummability of formal power series solutions of linear ordinary differential equations. Asymptot Anal 1991;5(1): 27–45.

    MathSciNet  MATH  Google Scholar 

  5. Balser W, Loday-Richaud M. Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables. Adv Dyn Syst Appl 2009;4 (2):159–77.

    MathSciNet  Google Scholar 

  6. Balser W, Miyake M. Summability of formal solutions of certain partial differential equations. Acta Sci Math (Szeged) 1999;65(3-4):543–51.

    MathSciNet  MATH  Google Scholar 

  7. Balser W, Yoshino M. Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients, Funkcial. Ekvac. 2010;53:411–34.

    Article  MathSciNet  MATH  Google Scholar 

  8. Braaksma BLJ. Multisummability and Stokes multipliers of linear meromorphic differential equations. J Differ Eq 1991;92:45–75.

    Article  MathSciNet  MATH  Google Scholar 

  9. Canalis-Durand M, Ramis JP, Schäfke R, Sibuya Y. Gevrey solutions of singularly perturbed differential equations. J Reine Angew Math 2000;518:95–129.

    MathSciNet  MATH  Google Scholar 

  10. Costin O, Park H, Takei Y. Borel summability of the heat equation with variable coefficients. J Differ Eq 2012;252(4):3076–92.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hibino M. Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I, J Differ Eq 2006;227(2):499–533.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hibino M. On the summability of divergent power series solutions for certain first-order linear PDEs. Opuscula Math 2015;35(5):595–624.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ichinobe K. On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients. J Differ Eq 2014;257(8):3048–70.

    Article  MathSciNet  MATH  Google Scholar 

  14. Loday-Richaud M. Stokes phenomenon, multisummability and differential Galois groups. Ann Inst Fourier (Grenoble) 1994;44(3):849–906.

    Article  MathSciNet  MATH  Google Scholar 

  15. Loday-Richaud M. 2016. Divergent series, summability and resurgence II. Simple and multiple summability, Lecture notes in math., 2154: Springer, Berlin.

  16. Lutz DA, Miyake M, Schäfke R. On the Borel summability of divergent solutions of the heat equation. Nagoya Math J 1999;154:1–29.

    Article  MathSciNet  MATH  Google Scholar 

  17. Malek S. On the summability of formal solutions of linear partial differential equations. J Dyn Control Syst 2005;11(3):389–403.

    Article  MathSciNet  MATH  Google Scholar 

  18. Malek S. On the Stokes phenomenon for holomorphic solutions of integrodifferential equations with irregular singularity. J Dyn Control Syst 2008;14(3):371–408.

    Article  MathSciNet  MATH  Google Scholar 

  19. Malek S. Gevrey functions solutions of partial differential equations with Fuchsian and irregular singularities. J Dyn Control Syst 2009;15(2):277–305.

    Article  MathSciNet  MATH  Google Scholar 

  20. Malgrange B. Sommation des séries divergentes. Expo Math 1995;13:163–222.

    MATH  Google Scholar 

  21. Malgrange B, Ramis J-P. Fonctions multisommables. Ann Inst Fourier (Grenoble) 1992;42:353–68.

    Article  MathSciNet  MATH  Google Scholar 

  22. Martinet J, Ramis J-P. Elementary acceleration and multisummability. Ann Inst H Poincaré, Phys. Théor 1991;54(4):331–401.

    MathSciNet  MATH  Google Scholar 

  23. Michalik S. Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J Dyn Control Syst 2012;18(1):103–33.

    Article  MathSciNet  MATH  Google Scholar 

  24. Miyake M. Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations. J Math Soc Japan 1991;43(2):305–30.

    Article  MathSciNet  MATH  Google Scholar 

  25. Miyake M. Borel summability of divergent solutions of the Cauchy problem to non-Kovaleskian equations. Partial differential equations and their applications. River Edge, NJ: World Sci. Publ.; 1999 . p. 225–239.

    Google Scholar 

  26. Nagumo M. ÜBer das Anfangswertproblem partieller Differentialgleichungen. Jap J Math 1942;18:41–7.

    MathSciNet  MATH  Google Scholar 

  27. Ouchi S. Multisummability of formal solutions of some linear partial differential equations. J Differ Eq 2002;185(2):513–49.

    Article  MathSciNet  MATH  Google Scholar 

  28. Ouchi S. Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields. J Math Soc Japan 2005;57(2): 415–60.

    Article  MathSciNet  MATH  Google Scholar 

  29. Pliś M. E., Ziemian B. Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables. Ann Polon Math 1997;67(1):31–41.

    MathSciNet  MATH  Google Scholar 

  30. Ramis J-P. Les séries k-sommables et leurs applications, in Complex analysis, microlocal calculus and relativistic quantum theory (Proc. Internat. Colloq., Centre Phys., Les Houches, 1979), Lecture Notes in Phys. Berlin: Springer; 1980, Vol. 126, p. 178–199.

  31. Ramis J-P. Théorèmes d’indices Gevrey pour les équations différentielles ordinaires. Mem Amer Math Soc 1984;48:viii+95.

    MATH  Google Scholar 

  32. Remy P. Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients. J Dyn Control Syst 2016;22(4):693–711.

    Article  MathSciNet  MATH  Google Scholar 

  33. Tahara H, Yamazawa H. Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations. J Differ Eq 2013;255(10):3592–637.

    Article  MathSciNet  MATH  Google Scholar 

  34. Yonemura A. Newton polygons and formal Gevrey classes. Publ Res Inst Math Sci 1990;26:197–204.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lycée Les Pierres Vives, 1 rue des Alouettes, F-78 420, Carrières-sur-Seine, France

    Pascal Remy

Authors
  1. Pascal Remy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pascal Remy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Remy, P. Gevrey Order and Summability of Formal Series Solutions of Certain Classes of Inhomogeneous Linear Integro-Differential Equations with Variable Coefficients. J Dyn Control Syst 23, 853–878 (2017). https://doi.org/10.1007/s10883-017-9371-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-017-9371-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation