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On $q$-asymptotics for $q$-difference-differential equations with Fuchsian and irregular singularities

Tom 97 / 2012

Alberto Lastra, Stéphane Malek, Javier Sanz Banach Center Publications 97 (2012), 73-90 MSC: 34K25; 34M25; 34M30; 33E30. DOI: 10.4064/bc97-0-5

Streszczenie

This work is devoted to the study of a Cauchy problem for a certain family of $q$-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series $\hat{X}(t,z)$ solving the problem. Under appropriate conditions, $q$-Borel and $q$-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose $q$-asymptotic expansion in $t$, uniformly for $z$ in the compact sets of $\mathbb{C}$, is $\hat{X}(t,z)$. The small divisors phenomenon owing to the Fuchsian singularity causes an increase in the order of $q$-exponential growth and the appearance of a subexponential Gevrey growth in the asymptotics.

Autorzy

  • Alberto LastraFacultad de Ciencias
    Universidad de Valladolid
    Calle del Doctor Mergelina s/n
    47011 Valladolid, Spain
    e-mail
  • Stéphane MalekUFR de Mathématiques
    Université Lille 1
    Cité Scientifique M2
    59655 Villeneuve d'Ascq Cedex, France
    e-mail
  • Javier SanzFacultad de Ciencias
    Universidad de Valladolid
    Calle del Doctor Mergelina s/n
    47011 Valladolid, Spain
    e-mail

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