On $q-$Gevrey asymptotics for singularly perturbed $q-$difference-differential problems with an irregular singularity

Details On $q-$Gevrey asymptotics for singularly perturbed $q-$difference-differential problems with an irregular singularity On $q-$Gevrey asymptotics for singularly perturbed $q-$difference-differential problems with an irregular singularity

2011-06-02

arxiv.org/abs/1106.0401v2

Mathematics

Analysis of PDEs

30 pages

Scientific paper

We study a $q-$analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by S. Malek in \cite{malek}. First, we construct solutions defined in open $q-$spirals to the origin. By means of a $q-$Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the $q-$Gevrey asymptotic expansion (of certain type) of the actual solutions.

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