Multi-level Gevrey solutions of singularly perturbed linear partial differential equations

Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter $ε$. This is the continuation of a precedent work by the first author. We construct two families of sectorial meromorphic solutions obtained as a small perturbation in $ε$ of two branches of an algebraic slow curve of the equation in time scale. We show that the nonsingular part of the solutions of each family shares a common formal power series in $ε$ as Gevrey asymptotic expansion which might be different one to each other, in general.

On a $q-$analog of a singularly perturbed problem of irregular type with two complex time variables

Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to …

Strongly regular multi-level solutions of singularly perturbed linear partial differential equations

We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.

Multiscale Gevrey asymptotics in boundary layer expansions for some initial value problem with merging turning points

We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $ε$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $ε$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|ε|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $ε$ …

两个复时间变量问题中的参数形式 Gevrey 渐近展开

Parametric formal Gevrey asymptotic expansions in two complex time variable problems

本文研究复域中含两个复时间变量的奇异扰动偏微分方程的解析与形式解，利用 Borel 平面上的解析延拓克服可和性困难，并发现若干指数衰减率以建立解间的渐近层级。

On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $ε$. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so…

一类线性 q-差分-微分 Cauchy 问题的 Gevrey 与 q-Gevrey 渐近展开比较

Gevrey versus q-Gevrey asymptotic expansions for some linear q-difference-differential Cauchy problem

该研究分析奇异扰动 q-差分-微分 Cauchy 问题解析解的渐近行为，通过调整形式解系数空间范数，并结合路径形变与 Ramis-Sibuya 定理，得到 Gevrey 型及 Gevrey/q-Gevrey 混合型展开。

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

关于某些线性偏$q$-差分-微分方程在两个层级上的参数化$0$-Gevrey渐近展开

On parametric $0$-Gevrey asymptotic expansions in two levels for some linear partial $q$-difference-differential equations

本文研究复域中奇异扰动$q$-差分-微分方程的解析解，获得了具有两个不同层级的新颖渐近表示，并证明了多层级序列Ramis-Sibuya型定理的新版本。