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

Abstract

This paper is a continuation of the work [14] where parametric Gevrey asymptotics for singularly perturbed nonlinear PDEs has been studied. 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 inverse Fourier transform in space. Furthermore, we obtain Gevrey asymptotic expansions for these solutions of some order K > 1 in the perturbation parameter.

Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

“…We recall some features of the Laplace transform under the action of multiplication by a monomial and differential operators already stated in our foregoing work [9]. A detailed proof of the formulas stated in the forthcoming lemma can be found in the work [10], Lemma 2.…”

“…for all x ≥ 0. According to the sharp bounds reached in Proposition 1 of the paper [10], we can single out a constant K 1 > 0 (depending on the constants γ 2 , γ 3 , k 1 , ν) for which…”

On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

“…Proof. e first two formulas have already been given in our previous works [3,20]. We focus on the third equality.…”

On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

“…hold for all u ∈ D(0, r), all m ∈ R. Consequently, owing to the representation (46) and keeping in mind the Beta function formula (16), it follows…”

“…Proof The first formula follows by mere derivation under the integral symbol and the proof of the second identity is similar to the one given in Lemma 2 of [16] and will not be reproduced here. 2…”