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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Lastra, Alberto¹  Malek, Stephane²  Sanz, Javier¹ 

[1] Univ Valladolid, Dept Anal Matemat, Valladolid, Spain

[2] Univ Lille, UFR Math Pures & Appl, F-59655 Villeneuve Dascq, France

关键词: q-Difference-differential equations;    q-Laplace transform;    Formal power series solutions;    q-Cevrey asymptotic expansions;    Small divisors;    Fuchsian and irregular singularities;   

DOI  :  10.1016/j.jde.2012.01.038

来源: Elsevier

【 摘 要 】

We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables (X) over cap (t, z) for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of C, is (X) over cap (t, z). The small divisors' effect is an increase in the order of q-exponential growth and the appearance of a power of the factorial in the corresponding q-Gevrey bounds in the asymptotics. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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