Alberto Lastra

On -Gevrey Asymptotics for Singularly Perturbed -Difference-Differential Problems with an Irregular Singularity

We study a -analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open -spirals to the origin. By means of a -Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the -Gevrey asymptotic expansion (of certain type) of the actual solutions.

A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

On Symbolic Solutions of Algebraic Partial Differential Equations

In this paper we present a general procedure for solving first-order autonomous algebraic partial differential equations in two independent variables. The method uses proper rational parametrizations of algebraic surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. We will demonstrate in examples that, depending on certain steps in the procedure, rational, radical or even non-algebraic solutions can be found. Solutions computed by the procedure will depend on two arbitrary independent constants.

On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

We study a family of singularly perturbed -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One

The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. Ngo and F. Winkler (2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro- Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI: 10.5486/PMD.2015.6032 and it is available at http://dx.doi.org/10.5486/PMD.2015.6032

On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Abstract We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution. The proof leans on a two level novel version of Ramis–Sibuya theorem under Gevrey and q-Gevrey orders.

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

This paper is a slightly modified, abridged version of the work (Lastra and Malek, Adv Differ Equ 21:767–800, 2016). It deals with some questions made to the authors during the conference Analytic, Algebraic and Geometric Aspects of Differential Equations, held in Bedlewo (Poland) during the second week of September, 2015.

On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

On Complex Singularity Analysis for Some Linear Partial Differential Equations in

We investigate the existence of local holomorphic solutions

Boundary layers of an anchored magnetic field in a highly conductive flow

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