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Reaction diffusion equations on domains with thin layers.

机译：具有薄层的区域上的反应扩散方程。

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摘要

This dissertation is devoted to the study of several types of reaction diffusion equations on domains with thin layers. The common feature is that the thermal tensor or diffusion rate differs significantly in size and/or nature on the large component of the entire domain and on the thin layer. The physical situations include the protection of spacecrafts and turbine engine blades (of metallic nature) by thermal barrier coatings (of ceramic nature), and the effect of a road or a buffer zone in a nature reserve. Such types of PDEs are associated with at least two issues. It is hard to see the effect of the thin layers, and it is numerically challenging to solve them due to the small scales involved. Our resolution here is to study the asymptotic behavior of the solutions to the PDEs and seek the effective boundary conditions (EBCs) on the boundary of the large components, as the thickness of the thin layers shrink. EBCs enable us to see the effect of the thin layers transparently. Furthermore, with the help of EBCs, we can simply solve the PDEs on the large components of the domains, which involve no small scales any more.;We first study the asymptotic behavior and EBCs of the linear heat equation on a body coated by functionally graded material (FGM). The motivation is that, practically, in thermal barrier coatings, there is often coating failure due to the large stress between the ceramic topcoat and the metallic surface, while FGM is meant to replace the sharp interface with a gradient interface that makes a smooth transition from one material to the other. We model the FGM coating by assuming that the material is graded in the normal direction of the boundary of the metallic body and generalize some previous results. In particular, it is shown that there are more options in order to perfectly protect the body due to the introduction of FGM coating.;Next, we investigate the coating problem for the logistic diffusion equation in the context of ecology. Here we interpret the body as a nature reserve, and the coating layer a buffer zone where small diffusion rate occurs. It turns out that the asymptotic behavior of the density function ;Finally, we consider the logistic diffusion equation on the entire plane, including a horizontal strip of width ;+;{-1}right)
机译：本文致力于研究薄层区域上几种类型的反应扩散方程。共同的特征是，在整个区域的大部分和薄层上，热张量或扩散率的大小和/或性质都存在显着差异。物理情况包括通过（陶瓷性质的）热障涂层保护（金属性质的）航天器和涡轮发动机叶片，以及自然保护区中道路或缓冲区的影响。此类PDE与至少两个问题相关。很难看到薄层的效果，并且由于涉及的比例小，在数值上难以解决。我们在这里的解决方案是研究PDE的解的渐近行为，并随着薄层厚度的减小，在大部件的边界上寻找有效边界条件（EBC）。 EBC使我们能够透明地看到薄层的效果。此外，借助EBC，我们可以简单地求解域的大部分上的PDE，这些PDE不再涉及小规模的研究;我们首先研究了线性热方程在被功能覆盖的物体上的渐近行为和EBC分级材料（FGM）。其动机是，实际上，在热障涂层中，由于陶瓷面涂层和金属表面之间的应力过大，经常会导致涂层失效，而FGM则是用梯度界面代替锋利的界面，从而使界面从表面平滑过渡。一种材料给另一种。我们通过假设材料在金属体边界的法线方向上渐变来对FGM涂层建模，并归纳了一些先前的结果。特别是，由于引入了FGM涂层，显示出更多的选择来完美地保护人体。接下来，我们在生态学背景下研究逻辑扩散方程的涂层问题。在这里，我们将身体解释为自然保护区，将涂层解释为发生小扩散速率的缓冲区。事实证明，密度函数的渐近行为；最后，我们考虑了整个平面上的对数扩散方程，包括宽度为; +; {-1}的水平条）

著录项

  • 作者

    Li, Huicong.;

  • 作者单位

    Tulane University School of Science and Engineering.;

  • 授予单位 Tulane University School of Science and Engineering.;
  • 学科 Mathematics.;Applied mathematics.;Theoretical mathematics.
  • 学位 Ph.D.
  • 年度 2015
  • 页码 169 p.
  • 总页数 169
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 物理化学（理论化学）、化学物理学;
  • 关键词

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