Time-periodic solutions with internal and boundary layers to singularly perturbed parabolic problems: Existence, approximation and domain of attraction

奇扰动抛物线问题的具有内部层和边界层的时间周期解：存在性、近似性和吸引域

• 批准号: 259134773
• 负责人: Privatdozent Dr. Lutz Recke
• 金额: --
• 依托单位: Physics Department
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/259134773?language=en
• 关键词: Time; periodic; solutions; internal; boundary

The project concerns boundary value problems for singularly perturbed time-periodic reaction-diffusion-advection equations in 1D.We will develop algorithms of analytic construction of sequences of lower and upper solutions with internal and boundary layers. The positions of the internal layers will periodically move in time, in general. The lower and upper solutions will be used for proving existence of exact solutions with internal and boundary layers, for uniform approximation of them, for verifying their stability and for estimation of their domain of attraction. Moreover, the lower and upper solutions are useful also for creating numerical algorithms for layered time-periodic solutions. The main technical tools are formal asymptotic expansions and using of stretched variables close to the layers. The terms in the asymptotic expansions as well as the additional, modifying terms, which create lower and upper solutions, can be calculated by solving linear inhomogeneous time-periodic parabolic problems on the half axis. operators on the half axis. The inverted linear parabolic partial differential operators have to be order preserving, and they should map exponentially decaying functions into exponentially decaying functions. For proving stability we use the Krein-Rutman Theorem.

该项目涉及一维奇异扰动时间周期反应扩散平流方程的边值问题。我们将开发具有内部层和边界层的下解和上解序列的解析构造算法。一般来说，内层的位置会随时间周期性移动。下解和上解将用于证明具有内部层和边界层的精确解的存在性、对它们的统一逼近、验证它们的稳定性以及估计它们的吸引域。此外，下解和上解对于创建分层时间周期解的数值算法也很有用。主要的技术工具是形式渐近展开和使用靠近层的拉伸变量。渐近展开式中的项以及产生下解和上解的附加修改项可以通过求解半轴上的线性非齐次时间周期抛物线问题来计算。半轴上的运算符。逆线性抛物型偏微分算子必须保持顺序，并且它们应该将指数衰减函数映射为指数衰减函数。为了证明稳定性，我们使用克雷因-鲁特曼定理。

项目成果

Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity

• DOI: 10.1134/s0012266117040103
• 发表时间: 2017-04
• 期刊: Differential Equations
• 影响因子: 0.6
• 作者: N. Nefedov;E. Nikulin

An Implicit Function Theorem and Applications to Nonsmooth Boundary Layers

隐函数定理及其在非光滑边界层中的应用

• DOI: 10.1007/978-3-319-64173-7_7
• 发表时间: 2017
• 作者: V.F. Butuzov;N.N. Nefedov;O.E. Omel’chenko;L. Recke;K.R. Schneider
• 通讯作者: K.R. Schneider

Existence, Asymptotics, Stability and Region of Attraction of a Periodic Boundary Layer Solution in Case of a Double Root of the Degenerate Equation

• DOI: 10.1134/s0965542518120072
• 发表时间: 2018-12
• 期刊: Computational Mathematics and Mathematical Physics
• 影响因子: 0.7
• 作者: V. Butuzov;N. Nefedov;L. Recke;K. Schneider

Time-periodic boundary layer solutions to singularly perturbed parabolic problems

• DOI: 10.1016/j.jde.2016.12.020
• 发表时间: 2016-09
• 作者: L. Recke;V. Butuzov;N. Nefedov

Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes

使用动态自适应网格求解奇异摄动抛物线方程的解析数值方法

• DOI: 10.18255/1818-1015-2016-3-334-341
• 发表时间: 2016
• 期刊: Modeling and Analysis of Information Systems
• 作者: D.V. Lukyanenko;V.T. Volkov;N.N. Nefedov;L. Recke;K.R. Schneider
• 通讯作者: K.R. Schneider