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On -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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