On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms

Abstract

Abstract This paper is a continuation of the work (Lastra and Malek in J. Differ. Equ. 259(10):5220–5270, 2015) where singularly perturbed nonlinear PDEs have been studied from an asymptotic point of view. Here, the partial differential operators are combined with particular Moebius transforms in the time variable. As a result, the leading term of the main problem needs to be regularized by means of a singularly perturbed infinite order formal irregular operator that allows us to construct a set of genuine solutions in the form of a Laplace transform in time and an inverse Fourier transform in space. Furthermore, we obtain Gevrey asymptotic expansions for these solutions of some order K>1 $K>1$ in the perturbation parameter.

Keywords

• Asymptotic expansion
• Borel–Laplace transform
• Fourier transform
• Initial value problem
• Difference equation
• Formal power series