Holomorphic and Singular Solutions of &lt;i&gt;q&lt;/i&gt;-Difference-Differential Equations of Briot-Bouquet Type

2017

Abstract. We study an inhomogeneous linear q-di¤erence di¤erential Cauchy problem, with a complex perturbation parameter e, whose coe‰cients depend holomorphically on e and on time in the vicinity of the origin in C 2 and are bounded analytic on some horizontal strip in C w.r.t the space variable. This problem is seen as a q-analog of an initial value problem recently investigated by the author and A. Lastra in [9]. Here a comparable result with the one in [9] is achieved, namely we construct a finite set of holomorphic solutions on a common bounded open sector in time at the origin, on the given strip above in space, when e belongs to a well selected set of open bounded sectors whose union covers a neighborhood of 0 in C Ã . These solutions are constructed through a continuous version of a q-Laplace transform of some order k b 1 introduced newly in [6] and Fourier inverse map of some function with q-exponential growth of order k on adequate unbounded sectors in C and with exponential decay in the Fourier variable. Moreover, by means of a q-analog of the classical Ramis-Sibuya theorem, we prove that they share a common formal power series (that generally diverge) in e as q-Gevrey asymptotic expansion of order 1=k.

Section: Introductionmentioning

confidence: 99%

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

2017

“…Proof Let 0 ≤ h 1 ≤ ι 1 − 1. Regarding (32) in Theorem 2, Lemma 3 and from usual estimates, we guarantee that for every…”

Section: Asymptotic Expansions For the Analytic Solutions Of The Mainmentioning

confidence: 95%

On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Lastra

2019

Mathematics

Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed q−difference-differential equations in the complex domain are constructed. They stand for a q−analog of the singularly perturbed partial differential equations considered in [15]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter. The appearance of the −1-branch of Lambert W function will be crucial in this respect.

“…In a larger framework, this work is a contribution to the promising and fruitful realm of research in q-difference and q-difference-differential equations in the complex domain. For recent important advances in this area, we mention in particular the works by Tahara and Yamazawa [8][9][10]. Notice that the fields of applications of q-difference equations have also encountered a rapid growth in the last years.…”

Section: Tz T M D U(t Z ϵ)mentioning

confidence: 99%

On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

Abstract and Applied Analysis

2020

A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.