Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

Balser, Werner

doi:10.1023/b:joth.0000047246.49736.b0

Cited by 36 publications

(73 citation statements)

References 7 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, we consider the Cauchy problem for the non-Kowalevskian linear partial differential equation in two complex variables t and z with constant coefficients P jq/p (∂ z )u(t, z), ∂ n t u(0, z) = ϕ n (z), n = 0, ..., p − 1, (1) where p, q ∈ N, p < q, P jq/p (ξ) are polynomials of degree less than or equal to jq/p (j = 1, . .…”

Section: Introductionmentioning

confidence: 99%

“…Applying B 1,1+1/k to the formal solutionû(t, z) of the initial problem (1) we obtain the associated function v(t, z) satisfying the initial value problem for certain Kowalevskaya type fractional equation related to (1). It means that we can reduce the problem of summability to the study of this new equation.…”

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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Section: Introductionmentioning

confidence: 99%

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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“…The first contribution is rendered by Lutz-Miyake-Schäfke [12], where complex heat equations are dealt with. Balser [2,4], Balser and Miyake [5] and Miyake [14] generalized the result in [12]. InŌuchi [15] also, we can find some interesting results for greatly general linear partial differential equations.…”

Section: Some Remarks On Related Topicsmentioning

confidence: 71%

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

Hibino¹

2015

OpMath

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Abstract. This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.

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“…[7] for the heat equation, [2] for general equations and its references). But the study for equations with variable coefficients has not been developed yet.…”

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confidence: 99%

On k-summability of formal solutions for certain partial differential operators with polynomial coefficients

Ichinobe

Miyake

2015

OpMath

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Abstract. We study the k-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in t. We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates.

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Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

Cited by 36 publications

References 7 publications

Summability and fractional linear partial differential equations

Summability and fractional linear partial differential equations

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

On k-summability of formal solutions for certain partial differential operators with polynomial coefficients

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