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On Gevrey asymptotics for linear singularly perturbed equations with linear fractional transforms

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine both partial differential operators together with the action of linear fractional transforms. The exotic geometry of the problem in the Borel plane, involving both sectorial regions and strip-like sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey asymptotic expansions. The main results lean on the appearance of domains in the complex plane which remain intimately related to Lambert W function, which turns out to be crucial in the construction of the analytic solutions. On the way, an accurate description of the deformation of the integration paths defining the analytic solutions and the knowledge of Lambert W function are needed in order to provide the asymptotic behavior of the solution near the origin, regarding the perturbation parameter. Such deformation varies depending on the analytic solution considered, which lies in two families with different geometric features.

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Acknowledgements

The authors want to express their gratitude to the referee of the work for the suggestions and comments which helped to improve significantly the work in its presentation and structure.

Funding

G. Chen is supported by a starting research grant from HITSZ. A. Lastra and S. Malek are partially supported by the project PID2019-105621GB-I00 of Ministerio de Ciencia e Innovación, Spain; A. Lastra is partially supported by Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of Comunidad de Madrid (Spain), and Universidad de Alcalá under grant CM/JIN/2019-010, Proyectos de I+D para Jóvenes Investigadores de la Universidad de Alcalá 2019.

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Authors and Affiliations

  1. School of Science, Harbin Institute of Technology (Shenzhen), 518055, Shenzhen, China

    Guoting Chen

  2. Universidad de Alcalá, Departamento de Física y Matemáticas, Alcalá de Henares, Madrid, Spain

    Alberto Lastra

  3. Laboratoire Paul Painlevé, University of Lille 1, 59655, Villeneuve d’Ascq cedex, France

    Stéphane Malek

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  1. Guoting Chen
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  2. Alberto Lastra
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  3. Stéphane Malek
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All authors contributed equally and significantly in this paper and typed, read, and approved the final manuscript.

All authors have made substantial contributions to the conception, design of the work, have drafted the work and substantively revised it.

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All authors have agreed both to be personally accountable for the author’s own contributions and to ensure that questions related to the accuracy or integrity of any part of the work are appropriately investigated, resolved, and the resolution documented in the literature.

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Correspondence to Alberto Lastra.

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Chen, G., Lastra, A. & Malek, S. On Gevrey asymptotics for linear singularly perturbed equations with linear fractional transforms. RACSAM 115, 121 (2021). https://doi.org/10.1007/s13398-021-01064-w

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  • DOI: https://doi.org/10.1007/s13398-021-01064-w

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