# On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

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## Abstract

We consider a family of linear singularly perturbed PDE depending on a complex perturbation parameter \(\epsilon \). As in the former study (Lastra and Malek in J Differ Equ 259(10):5220–5270, 2015) of the authors, our problem possesses an irregular singularity in time located at the origin but, in the present work, it also entangles differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through *iterated* Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by Balser. This construction has a direct consequence on the Gevrey bounds of their asymptotic expansions w.r.t \(\epsilon \) which are shown to increase the order of the leading term which combines both irregular and Fuchsian types operators.

## Keywords

Asymptotic expansion Borel–Laplace transform Fourier transform Initial value problem Formal power series Linear integro-differential equation Partial differential equation Singular perturbation## Mathematics Subject Classification

35R10 35C10 35C15 35C20## Notes

### Acknowledgements

A. Lastra and S. Malek are supported by the Spanish Ministerio de Economía, Industria y Competitividad under the Project MTM2016-77642-C2-1-P.

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