n-Widths and Singular Perturbation Problems

奇异摄动偏微分方程用来模拟输运和流体流动的边界层和内部层，以及高频波的传播。由于解的快速变化区域，很难逼近这些方程的解。研究了一些奇摄动的线性偏微分方程解的最佳逼近性质。这项研究将建立在凯洛格和斯泰恩斯最近的一些工作基础上，这些工作处理了一个单一扰动的自伴问题。我们将学习模拟对流和波传播的方程。目标是当问题的数据被限制为具有指定的光滑性时，获得最佳逼近性质(在Kolmogorov意义下)。这些最佳逼近问题的成功解决将导致解的渐近展开，而对数据的光滑性要求最低，并有望得到更好的数值方法来进行这些解的数值计算。一个典型的例子是飞机机翼附近的湍流层。例如，在雷达或超声波中产生的电波传播问题在医疗和军事应用中都是重要的。这些看似不同的问题是相似的，因为有一些现象发生在非常短的距离或时间跨度内。这导致了计算这类系统的行为的困难。该奖项下的研究将研究支配这些系统的方程的基本近似性质。

Singularly perturbed partial differential equations are used to model boundary and interior layers in transport and fluid flow, and in high frequency wave propagation. It is difficult to approximate the solution to these equations because of regions of rapid change in the solution. The best approximation properties of solutions to some singularly perturbed linear partial differential equations will be studied. The research will build on some recent work of Kellogg and Stynes that treats asingularly perturbed self adjoint problem. Equations that model convection and wave propagation will be studied. The goal is to obtain the best approximation properties (in the sense of Kolmogorov) when the data of the problem are restricted to have a specified smoothness. A successful resolution of these best approximation problems will lead to asymptotic expansions of the solution with minimal smoothness requirements on the data, and hopefully to better numerical methods for the numerical computation of these solutions.Physical systems that involve boundary and interior layers are of considerable importance in industrial applications. A typical example is the turbulent layer of air near an aircraft wing. Wave propagation problems, arising e.g. in radar or ultrasound are important in both medical and military applications. These seemingly different problems are similar in that there are phenomena taking place over a very short span of distance or time. This leads to difficulties in calculating the behavior of such systems. The research under this award will study the basic approximation properties of the equations governing these systems.