Summability of formal solutions of some singular perturbations problems in differential and moment differential equations

Abstract

In this paper we study the summability of solutions of some general forms of singularly perturbed linear ordinary differential and moment differential equations. We conclude that under some assumptions solutions of these equations are summable. The type of this summability depends on the specific equation. We also show the connection between some singularly perturbed moment ordinary differential equations and some linear moment partial differential equations. We apply this connection to describe summable and multisummable formal solutions of these singularly perturbed moment ordinary differential equations. Main techniques used to show these conclusions are based on Borel transforms, properties of solutions of moment partial differential equations and on the Cauchy integral formula together with integral representations of solutions of such equations.