Asymptotic Expansions, Inverse Problems and Homogenization of Boundary Values

渐进展开、反问题和边界值齐次化

• 批准号: 0072511
• 负责人: Shari Moskow
• 金额: $ 6.25万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072511&HistoricalAwards=false
• 关键词: Asymptotic; Expansions; Inverse; Problems; Homogenization

NSF Award Abstract - DMS-0072511Mathematical Sciences: Asymptotic Expansions, Inverse Problems, and Homogenization of Boundary ConditionsAbstract0072511 MoskowWe will use the technique of asymptotic expansions to solve two types of problems. The first is the accurate detection of the location and shapes of small inhomogeneities from boundary measurements. Asymptotic expansions of the solution with respect to the size of the inhomogeneity will be developed where they are not yet known. A new procedure will be introduced which uses these expansions to find information about the holes. The procedure will be tested for different types of equations and shapes of inhomogeneities. The second problem involves the homogenization of boundary values for equations where the medium has an underlying periodic structure. Asymptotic analysis will aid in finding the limiting or effective equations. This analysis requires examining boundary layer functions which are solutions on a half space with periodic or almost periodic boundary conditions. Many equations that arise from material science and electromagnetics involve some small parameter. This parameter could represent, for example, the diameter of a small imperfection inside an airplane wing. We will analyze mathematically the effects of such imperfections on electric potentials. From this analysis we will develop and test numerically a new method to find the sizes and locations of imperfections. In practice, this method would require inducing electrical currents and measuring the resulting electric potential only on the exterior of the object. We will also analyze the macroscopic behavior of two materials mixed together at the microscopic level. The small parameter in this case represents the size of the microscopic scale. We will use our analysis to model more accurately the propagation of waves through mixed media, for use in oil exploration and materials science.

NSF奖摘要- DMS-0072511数学科学：渐近展开、反问题和边界条件的均匀化 我们将使用渐近展开的技术来解决两类问题。第一个是精确检测的位置和形状的小不均匀性的边界测量。渐近展开的解决方案相对于大小的不均匀性将制定他们还不知道。一个新的程序将被引入，它使用这些扩展来查找有关孔的信息。该程序将测试不同类型的方程和形状的不均匀性。第二个问题涉及方程的边界值的均匀化，其中介质具有潜在的周期性结构。渐近分析将有助于找到极限方程或有效方程。这种分析需要检查边界层功能，这是解决方案的半空间周期性或几乎周期性的边界条件。材料科学和电磁学中的许多方程都涉及到一些小参数。例如，该参数可以表示飞机机翼内的小缺陷的直径。我们将从数学上分析这种缺陷对电势的影响。从这个分析中，我们将开发和测试一种新的方法来找到缺陷的大小和位置。 实际上，这种方法需要感应电流并仅测量物体外部产生的电势。 我们还将在微观水平上分析两种材料混合在一起的宏观行为。在这种情况下，小参数表示微观尺度的大小。 我们将使用我们的分析来更准确地模拟波通过混合介质的传播，用于石油勘探和材料科学。