On the summability of divergent power series solutions for certain first-order linear PDEs

Hibino, Masaki

doi:10.7494/opmath.2015.35.5.595

Cited by 12 publications

(3 citation statements)

References 9 publications

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“…), nonlinear (see [12][13][14]19,20,23,24,38] etc.) or singular (see [9,10,21,22,37] etc.) partial differential equations or integro-differential equations in two variables or more, allowing thus to formulate many results on Gevrey properties, summability or multisummability.…”

Section: Setting the Problemmentioning

confidence: 99%

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

Remy

2019

J Dyn Control Syst

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In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of p0, 0q P C 2 . In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

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Section: Setting the Problemmentioning

confidence: 99%

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

Remy

2019

J Dyn Control Syst

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“…This construction was extended to more general linear PDEs by W. Balser in [3], under the assumption of adequate extension of the initial data to an infinite sector. More recently, M. Hibino [9] has made some advances in the study of linear first order PDEs. Subsequently, several authors have studied complex heat like equations with It is worth mentioning that, in the previous work [13], the linear part of the equation, ruled by P (t, , ∂ t , ∂ z )u(t, z, ) was assumed to be more general than in the present configuration, admitting an additional term of the form c 0 (t, z, )R(∂ z )u(t, z, ), where c 0 (t, z, ) is given by a certain holomorphic function defined on a product D(0, ρ) × H β × D(0, 0 ).…”

Section: Introductionmentioning

confidence: 99%

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Lastra

Malek

2018

Results Math

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We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables.The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. variable coefficients (see [5,6,21]). The second author [22], both authors [13] and the two authors and J. Sanz [17] have also contributed in this theory. Recently, multisummability of formal solutions of PDEs has also been put forward in different works. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [23] constructed multisummable formal solutions of nonlinear PDEs, coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [24] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [20], G. Lysik constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stokes equations. The authors have obtained results in this direction [14, 15].A recent overview on summability and multisummability techniques under different points of view is displayed in [18].The purpose of the present work is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely, we consider a problem of the formThe elements which conform the nonlinear part P 1 , P 2 are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin, say D(0, 0 ), continuous up to their boundary.Here, D(0, 0 ) stands for the open disc in the complex plane centered at 0, and with positive radius 0 > 0. We write D(0, 0 ) for its closure. Moreover, P stands for some polynomial of six variables, with complex coefficients, and the forcing term f (t 1 , t 2 , z, ) is a holomorphic and bounded function in D(0, ρ) 2 × H β × D(0, 0 ), for some ρ > 0, and where H β stands for the horizontal stripfor some β > 0.The precise configuration of the elements involved in the problem is stated and described in Section 2.2.This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form (1). We first recall some previous advances made in this respect, which motivate the present framework.In [13], we studied under the asymptotic point of view the solutions of certain family of PDEs of the formwhere the elements involved in the problem depend only on one time variable t. Our next aim w...

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“…Let us recall that the theory of summability of the formal solutions of PDEs has been recently intensively developed by such authors as M. Hibino [4]; K. Ichinobe and M. Miyake [7]; K. Ichinobe [5,6]; A. Lastra, S. Malek and J. Sanz [12]; P. Remy [19], H. Tahara and H. Yamazawa [21]; H. Yamazawa and M. Yoshino [23]; M. Yoshino [24,25]; and others.…”

Section: Introductionmentioning

confidence: 99%

The Stokes Phenomenon for Some Moment Partial Differential Equations

Michalik

Tkacz

2018

J Dyn Control Syst

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We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.

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On the summability of divergent power series solutions for certain first-order linear PDEs

Cited by 12 publications

References 9 publications

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

The Stokes Phenomenon for Some Moment Partial Differential Equations

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