nach oben

Journal of Dynamical and Control Systems

Erschienen in:

26.10.2015

Gevrey Order and Summability of Formal Series Solutions of some Classes of Inhomogeneous Linear Partial Differential Equations with Variable Coefficients

verfasst von: Pascal Remy

Erschienen in: Journal of Dynamical and Control Systems | Ausgabe 4/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We investigate Gevrey order and summability properties of formal power series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients and analytic initial conditions. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient k, in a given direction.

Vorheriger Artikel Tangent Hyperplanes to Subriemannian Balls

Nächster Artikel The Phenomenon of Reversal in the Euler–Poincaré–Suslov Nonholonomic Systems

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

We denote \(\widetilde {q}\) with a tilde to emphasize the possible divergence of the series \(\widetilde {q}\).

A subsector Σ of a sector \({\Sigma }^{\prime }\) is said to be a proper subsector and one denotes \({\Sigma }\Subset {\Sigma }^{\prime }\) if its closure in \(\mathbb {C} \) is contained in \({\Sigma }^{\prime }\cup \{0\}\).

Balser W. Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke. Pac J Math. 1999;188(1):53–63.MathSciNetCrossRefMATH

Balser W. Formal power series and linear systems of meromorphic ordinary differential equations. New-York: Springer-Verlag; 2000.MATH

Balser W. Multisummability of formal power series solutions of partial differential equations with constant coefficients. J Diff Equat. 2004;201(1):63–74.MathSciNetCrossRefMATH

Balser W, Loday-Richaud M. Summability of solutions of the heat equation with inhomogeneous thermal conductivity in two variables. Adv Dyn Syst Appl. 2009;4(2):159–177.MathSciNet

Balser W, Miyake M. Summability of formal solutions of certain partial differential equations. Acta Sci Math (Szeged). 1999;65(3–4):543–551.MathSciNetMATH

Balser W, Yoshino M. Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients. Funkcial Ekvac. 2010;53:411–434.MathSciNetCrossRefMATH

Canalis-Durand M, Ramis JP, Schäfke R, Sibuya Y. Gevrey solutions of singularly perturbed differential equations. J Reine Angew Math. 2000;518:95–129.MathSciNetMATH

Costin O, Park H, Takei Y. Borel summability of the heat equation with variable coefficients. J Diff Equat. 2012;252(4):3076–3092.MathSciNetCrossRefMATH

Lutz DA, Miyake M, Schäfke R. On the Borel summability of divergent solutions of the heat equation. Nagoya Math J. 1999;154:1–29.MathSciNetCrossRefMATH

10.

Hibino M. Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I. J Diff Equat. 2006;227(2):499–533.MathSciNetCrossRefMATH

11.

Hibino M. On the summability of divergent power series solutions for certain first-order linear PDEs. Opuscula Math. 2015;35(5):595–624.MathSciNetCrossRefMATH

12.

Ichinobe K. On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients. J Diff Equat. 2014;257(8):3048–3070.MathSciNetCrossRefMATH

13.

Malek S. On the summability of formal solutions of linear partial differential equations. J Dyn Control Syst. 2005;11(3):389–403.MathSciNetCrossRefMATH

14.

Malek S. On the Stokes phenomenon for holomorphic solutions of integrodifferential equations with irregular singularity. J Dyn Control Syst. 2008;14(3):371–408.MathSciNetCrossRefMATH

15.

Malek S. Gevrey functions solutions of partial differential equations with Fuchsian and irregular singularities. J Dyn Control Syst. 2009;15(2):277–305.MathSciNetCrossRefMATH

16.

Malgrange B. Sommation des séries divergentes. Expo Math. 1995;13:163–222.MathSciNetMATH

17.

Michalik S. Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J Dyn Control Syst. 2012;18(1):103–133.MathSciNetCrossRefMATH

18.

Miyake M. Borel summability of divergent solutions of the Cauchy problem to non-Kovaleskian equations, in Partial differential equations and their applications. River Edge: World Scientific Publications; 1999, pp. 225–239.MATH

19.

Nagumo M. Über das Anfangswertproblem partieller Differentialgleichungen. Jap J Math. 1942;18:41–47.MathSciNetMATH

20.

Ouchi S. Multisummability of formal solutions of some linear partial differential equations. J Diff Equat. 2002;185(2):513–549.MathSciNetCrossRefMATH

21.

Pliś ME, Ziemian B. Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables. Ann Polon Math. 1997;67(1):31–41.MathSciNetMATH

22.

Ramis J-P. Les séries k-sommables et leurs applications, in Complex analysis, microlocal calculus and relativistic quantum theory (Proc. Internat. Colloq., Centre Phys., Les Houches, 1979), Lecture Notes in Phys., 126, 178–199. Berlin: Springer; 1980.

23.

Tahara H, Yamazawa H. Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations. J Diff Equat. 2013;255(10):3592–3637.MathSciNetCrossRefMATH

Titel: Gevrey Order and Summability of Formal Series Solutions of some Classes of Inhomogeneous Linear Partial Differential Equations with Variable Coefficients
verfasst von: Pascal Remy
Publikationsdatum: 26.10.2015
Verlag: Springer US
Erschienen in: Journal of Dynamical and Control Systems / Ausgabe 4/2016
Print ISSN: 1079-2724
Elektronische ISSN: 1573-8698
DOI: https://doi.org/10.1007/s10883-015-9301-8

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 4/2016

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

Quenching Phenomenon of Positive Radial Solutions for p-Laplacian with Singular Boundary Flux

KdV Hamiltonian as a Function of Actions

The Phenomenon of Reversal in the Euler–Poincaré–Suslov Nonholonomic Systems

A Blow-up Result in a Viscoelastic System

Hyperbolic Chain Control Sets on Flag Manifolds

Premium Partner

Marktübersichten