Gevrey Order of Formal Power Series Solutions of Inhomogeneous Partial Differential Equations with Constant Coefficients

Abstract

Abstract. In an earlier paper, the first author showed that certain normalized formal solutions of homogeneous linear partial di¤erential equations with constant coe‰cients are multisummable, with a multisummability type that can be determined from a Newton polygon associated with the PDE. In this article, some of the results obtained there are extended in several directions: First of all, arbitrary formal solutions of inhomogeous PDE are considered, and it is shown that, in some sense, they can be computed completely explicitly. Secondly, the Gevrey order of these formal solutions is determined. Finally, formal power series are discussed that, in general, do not satisfy a PDE with constant coe‰cients, but instead may be considered as solutions of singularly perturbed ODE, or integro-di¤erential equations of a certain form.

Analytic and Summable Solutions of Inhomogeneous Moment Partial Differential Equations

“…In this section we recall the notion of moment di¤erential operators constructed by Balser and Yoshino [6] and the concept of moment pseudodi¤erential operators introduced in our previous papers [10,11]. The moment di¤erential operator q m; z is well-defined for every j A OðDÞ.…”

“…Since for every j A O 1=k ðD r Þ and jzj < e < r, wherem mðuÞ :¼ mðu=kÞ, Em m ðz 1=k z 1=k Þ ¼ P y n¼0 z n=k z n=k =m mðnÞ, y A ðÀarg w À p=ð2kÞ; Àarg w þ p=ð2kÞÞ and Þ k jwj¼e means that we integrate k times along the positively oriented circle of radius e. Here the integration in the inner integral is taken over a ray fre iy : r b r 0 g. so (5) holds for the operators lðq m; z Þ defined by (6).…”

“…More generally, for p A N and mðuÞ ¼ Gð1 þ u=pÞ the operator q m; z is closely related to the 1=p-fractional di¤erentiation q 1=p z . The concept of moment di¤erentiation was introduced by Balser and Yoshino [6], who studied the Gevrey order of formal solutions of inhomogeneous moment partial di¤erential equations with constant coe‰cients.…”

“…; n À 1; ð1Þ where Pðl; zÞ is a polynomial of two variables of degree n with respect to l and the inhomogeneityf f ðt; zÞ is a formal power series in both variables of Gevrey order ðs s 1 ;s s 2 Þ A R 2 , and m 1 , m 2 are moment functions of orders s 1 , s 2 respectively. Similarly to [11], we consider the wider class of moment functions than in [6] or [10], which creates the group with respect to multiplication. This extension allows us to study some integro-di¤erential operators as moment di¤erential operators.…”

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

“…The general moment differential equations were introduced by Balser and Yoshino [7], who studied the Gevrey order of formal solutions of such equations. A characterisation of the multisummable formal solutions of moment differential equations in terms of analytic continuation properties and growth estimates of the Cauchy data was established in our previous paper [14] under the assumption of convergence of the Cauchy data.…”