On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities

Stephane Malek
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Stephane Malek: Département de mathématiques, Bât. M2, University of Lille, 59655 Villeneuve d’Ascq CEDEX, France

Mathematics, 2020, vol. 8, issue 6, 1-38

Abstract: A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

Keywords: asymptotic expansion; Borel–Laplace transform; Cauchy problem; formal power series; integro-differential equation; partial differential equation; singular perturbation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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