On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities
- PDF / 984,212 Bytes
- 51 Pages / 439.37 x 666.142 pts Page_size
- 11 Downloads / 197 Views
On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities A. Lastra1
· S. Malek2
Received: 16 January 2019 © Mathematica Josephina, Inc. 2019
Abstract We consider a family of linear singularly perturbed PDE depending on a complex perturbation parameter . 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 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
1 Introduction In this paper, we aim attention at a family of singularly perturbed linear partial differential equations which combines two kinds of differential operators acting on the time variable of so-called irregular and Fuchsian types. The definition of irregular type
B
A. Lastra [email protected] S. Malek [email protected]
1
Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, 28871 Alcalá de Henares, Madrid, Spain
2
Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq cedex, France
123
A. Lastra, S. Malek
operators in the context of PDE can be found in the paper [27] by Mandai and we refer to the excellent textbook [12] by Gérard and Tahara for an extensive study of Fuchsian ordinary and partial differential equations. The problem under study is of the form Q(∂z )u(t, z, ) = R D (∂z ) kδ D (t k+1 ∂t )δ D (t∂t )m D u(t, z, ) + P(z, , t k+1 ∂t , t∂t , ∂z )u(t, z, ) + f (t, z, )
(1)
for vanishing initial data u(0, z, ) ≡ 0, where k, δ D , m D ≥ 1 are integers, Q(X ), R D (X ) stand for polynomials with complex coefficients and P(z, , V1 , V2 , V3 ) represents a polynomial in the arguments V1 , V2 , V3 with holomorphic coefficients w.r.t the perturbation parameter in a vicinity of the origin in C and holomorphic w.r.t the space variable z on a horizontal strip in C of the form Hβ = {z ∈ C/|Im(z)| < β}, for some given β > 0. The forcing term f (t, z, ) relies analytically on near the origin and holomorphically on z on Hβ and defines either an analytic function near 0 or an entire function with (at most) exponential growth of prescribed order w.r.t the time t. Equations involving both irregular and Fuchsian operators are of increasing in
Data Loading...
