Integral Representation for Borel Sum of Divergent Solution to a Certain Non-Kowalevski Type Equation

Ichinobe, Kunio

doi:10.2977/prims/1145476043

Cited by 15 publications

(18 citation statements)

References 10 publications

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“…In this way, we also extend the results of M. Miyake [8], K. Ichinobe [4] and S. Michalik [7] to more general equations.…”

Section: Introductionmentioning

confidence: 51%

“…For such functions we have well defined α-derivative given by (4), which coincides with the Caputo fractional derivative.…”

Section: α-Analytic Functions and α-Derivativesmentioning

confidence: 99%

“…This characterisation was generalised to the equation ∂ p t u−∂ q z u = 0 (with p < q) by M. Miyake [8], to the quasi-homogeneous equations by K. Ichinobe [4] and to some linear partial differential equations by S. Michalik [7].…”

Section: Introductionmentioning

confidence: 99%

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Summability and fractional linear partial differential equations

Michalik

2010

J Dyn Control Syst

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Abstract. We consider the Cauchy problem for the Kowalevskaya type fractional linear partial differential equations in two complex variables with constant coefficients. We show that solution is analytically continued in some directions with exponential growth if and only if the similar properties satisfy the Cauchy data. Applying this result we study the summability of formal power series solution of a Cauchy problem for general non-Kowalevskian linear partial differential equations in normal form with constant coefficients. We obtain the necessary and sufficient conditions for the Borel summability in terms of analytic continuation with an appropriate growth condition of the Cauchy data. Moreover, we show the similar characterisation of Borel summability in the case of non-Kowalevskian fractional equations. IntroductionIn 1999 Lutz, Miyake and Schäfke [5] showed the first result in the theory of summability of formal power series solutions of partial differential equations. They proved that the formal solution to the Cauchy problem for the 1-dimensional homogeneous complex heat equation is 1-summable in a direction d if and only if the Cauchy data ϕ(z) can be analytically continued to infinity in some sectors in directions d/2 and d/2 + π and this continuation is of exponential growth of order at most 2.This characterisation was generalised to the equation ∂ p t u−∂ q z u = 0 (with p < q) by M. Miyake [8], to the quasi-homogeneous equations by K. Ichinobe [4] and to some linear partial differential equations by S. Michalik [7].On the other hand, the sufficient condition for the Borel summability of formal solutions was found by Balser and Miyake [3] (for certain linear PDE with constant coefficients) and by W. Balser [2] (for general linear PDE with constant coefficients). In this last paper W. Balser also posed the conjecture that this sufficient condition for the Borel summability of formal power series solution is also necessary one.In the paper we prove this conjecture in the case of equations in normal form. In this way, we also extend the results of M. Miyake

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“…In this way, we also extend the results of M. Miyake [8], K. Ichinobe [4] and S. Michalik [7] to more general equations.…”

Section: Introductionmentioning

confidence: 51%

“…For such functions we have well defined α-derivative given by (4), which coincides with the Caputo fractional derivative.…”

Section: α-Analytic Functions and α-Derivativesmentioning

confidence: 99%

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Summability and fractional linear partial differential equations

Michalik

2010

J Dyn Control Syst

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“…[3,5,8,10,11]). On the other hand, concerning the summability of formal solutions of a partial differential equation with a singular perturbation parameter we cite [2] and [4].…”

Section: Introductionmentioning

confidence: 99%

Parametric Borel summability for some semilinear system of partial differential equations

Yamazawa

Yoshino

2015

OpMath

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Abstract. In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with n independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.

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“…The problem of summability of formal solutions of linear PDEs was mainly studied under the assumption that the Cauchy data are convergent, see Balser [3], Balser and Loday-Richaud [5], Balser and Miyake [6], Ichinobe [8], Lutz, Miyake and Schäfke [9], Malek [10], Michalik [11,12,13] and Miyake [15].…”

Section: Introductionmentioning

confidence: 99%

Summability of formal solutions of linear partial differential equations with divergent initial data

Michalik

2013

Journal of Mathematical Analysis and Applications

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Abstract. We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent Cauchy data. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t, z). The results are presented in the general framework of moment-PDEs. IntroductionThe problem of summability of formal solutions of linear PDEs was mainly studied under the assumption that the Cauchy data are convergent, see Balser The case of more general initial data was investigated only for the complex heat equation (see Balser [1,4]). In [1] Balser considered the case of entire initial data with an appropriate growth condition and he gave some preliminary results for divergent initial data, too. Next, these results were extended in [4], where a characterisation of summable formal power series solutions of the complex heat equation in terms of properties of divergent Cauchy data was given.The aim of our paper is a generalisation of Balser's results [1, 4] to homogeneous linear partial differential equations with constant coefficients.Namely, we consider the initial value problem for a general linear partial differential equation with constant coefficients in two complex variables (t, z) P (∂ t , ∂ z ) u = 0, ∂ j t u(0, z) = ϕ j (z) (j = 0, . . . , n − 1),2010 Mathematics Subject Classification. 35C10, 35C15, 35E15, 40G10.

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Integral Representation for Borel Sum of Divergent Solution to a Certain Non-Kowalevski Type Equation

Cited by 15 publications

References 10 publications

Summability and fractional linear partial differential equations

Summability and fractional linear partial differential equations

Parametric Borel summability for some semilinear system of partial differential equations

Summability of formal solutions of linear partial differential equations with divergent initial data

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