Summability and fractional linear partial differential equations

Abstract

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

Analytic Solutions of Moment Partial Differential Equations with Constant Coefficients

“…Repeating the proof of Lemma 6 in [16], we generalise the last result as follows. Now we can state the main result of the paper.…”

“…We also introduce the concept of moment pseudodi¤erential operators, which generalise the pseudodi¤erential operators defined in [15,16]. …”

“…; n À 1). We also give the necessary corrections of our previous papers [15,16,17] concerning summability of formal solutions of linear (fractional) PDEs.…”

“…The inspiration for our study was the paper of Balser and Yoshino [8], where the notion of moment di¤erentiation was introduced and the Gevrey order of formal solutions of general inhomogeneous linear moment PDEs with constant coe‰cients was determined. Finally, let us point out that the summability of formal solutions of homogeneous linear PDEs with constant coe‰cients was studied by Balser [1,3], Balser and Miyake [7], Ichinobe [9], Lutz, Miyake and Schäfke [10], Malek [12], Michalik [13,15,16] and Miyake [19]. The inhomogeneous case was investigated by Balser [4], Balser and Loday-Richaud [6], Balser, Duval and Malek [5] and Michalik [14,17].…”

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

“…This result was extended to more general equations by: Balser [5][6][7], Balser and Malek [8], Balser and Miyake [9], Ichinobe [10], Malek [11], Michalik [12,13] and Miyake [14].…”