On q−Gevrey Asymptotics for Logarithmic Type Solutions in Singularly Perturbed q−Difference-Differential Equations

Stephane Malek; Alberto Lastra

doi:10.20944/preprints202306.2012.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 28 June 2023 / Approved: 28 June 2023 / Online: 29 June 2023 (03:29:06 CEST)

A family of singularly perturbed q-difference-differential equations under the action of a small complex perturbation parameter is studied. The action of the formal monodromy around the origin is present in the equation, which suggests the construction of holomorphic solutions holding logarithmic terms in both, the formal and the analytic level. We provide both solutions and describe the asymptotic behavior relating them by means of q-gevrey asymptotic expansions of some positive order, with respect to the perturbation parameter. On the way, the development of a space product of Banach spaces in the Borel plane is needed to provide a fixed point for a coupled system of equations.

q-Gevrey asymptotic expansion; monodromy; logarithmic type solution; singularly perturbed equation; formal solution.

Computer Science and Mathematics, Analysis

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On q−Gevrey Asymptotics for Logarithmic Type Solutions in Singularly Perturbed q−Difference-Differential Equations

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