Asymptotic Expansions, Inverse Problems and Homogenization of Boundary Values

NSF奖摘要-DMS-0072511数学科学：渐近展开、反问题和边界条件的齐次化莫斯科我们将使用渐近展开技术来解决两类问题。首先是从边界测量中准确检测微小不均匀的位置和形状。解关于不均匀的大小的渐近展开式将在它们尚不已知的地方展开。将引入一种新的程序，该程序使用这些展开来查找有关孔的信息。该程序将针对不同类型的方程和不均匀形状进行测试。第二个问题涉及方程边值的齐次化问题，其中介质具有潜在的周期结构。渐近分析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函数，它是具有周期或几乎周期边界条件的半空间上的解。材料科学和电磁学中的许多方程都包含一些小参数。例如，该参数可以表示飞机机翼内部小缺陷的直径。我们将从数学上分析这些缺陷对电势的影响。根据这一分析，我们将开发和测试一种新的方法来找出缺陷的大小和位置。在实践中，这种方法只需要在物体的外部感应电流并测量产生的电势。我们还将从微观层面分析混合在一起的两种材料的宏观行为。本例中的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在混合介质中的传播，用于石油勘探和材料科学。

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.