n-Widths and Singular Perturbation Problems

奇摄动偏微分方程被用来模拟边界层和内部层的运输和流体流动，并在高频波传播。由于解中的快速变化区域，很难近似这些方程的解。本文将研究一类奇摄动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Stynes最近的一些工作，对待奇异摄动自伴问题。将研究模拟对流和波传播的方程。我们的目标是获得最佳的逼近性能（在柯尔莫哥洛夫意义上）时，问题的数据被限制为具有指定的平滑度。一个成功的解决这些最佳逼近问题将导致渐近展开的解决方案与最小的光滑性要求的数据，并希望更好的数值方法，这些solutions.Physical系统的数值计算，涉及边界层和内部层是相当重要的工业应用。 一个典型的例子是飞机机翼附近的空气湍流层。 例如在雷达或超声中出现的波传播问题在医学和军事应用中都是重要的。这些看似不同的问题是相似的，因为有现象发生在一个非常短的距离或时间跨度。这导致计算这种系统的行为的困难。该奖项下的研究将研究控制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.