Mathematical Sciences: Analytical Approaches to Singular Perturbation Problems of Significance in Applications

9404536重要奇异摄动问题的O‘Malley解析方法在应用中的应用。华盛顿大学首席研究员小罗伯特·E·奥马利摘要：这项研究项目寻求非线性奇摄动边值问题的渐近解，这些边值问题涉及边界、激波和其他快速变化的局部区域。这类问题涉及大量的计算和分析挑战，它们的有效解需要使用基于渐近解结构的数值算法和相应的渐近技术的混合渐近/数值方法，这些方法的动机是在困难的模型上进行仔细的数值实验。通过这种方式，我们继续拓宽了我们对渐近匹配这一基本实用概念的理解。这样的数学问题经常出现在整个科学和工程的重要应用中；特别是在材料中的相分离和粗化的亚稳态模型中，以及为生物进化过程提供模式的反应扩散模型中。在这样的文本中使用渐近指数长的时间尺度的需要，对应于做“超越所有阶数的渐近”的数学问题。

9404536 O'Malley Analytical Approaches to Singular Perturbation Problem of Significance in Applications. Robert E. O'Malley, Jr., Principal Investigator University of Washington Abstract: This research project seeks asymptotic solutions to nonlinear singularly perturbed boundary value problems which involve boundary, shock, and other localized regions of rapid change. Such problems involve substantial computational, as well as analytical, challenges, and their effective solution requires hybrid asymptotic/numeric approaches using numerical algorithms based on the structure of asymptotic solutions and corresponding asymptotic techniques motivated by careful numerical experiments on difficult models. In this way, we continue to broaden our understanding of the fundamental practical concept of asymptotic matching. Such mathematical problems occur frequently in important applications throughout science and engineering; in particular, in metastable models for phase-separation and coarsening in materials and in reaction-diffusion models which provide patterns for evolving biological processes. The need to use an asymptotically exponentially-long time scale in such aontexts corresponds to the mathematical issue of doing "asymptotics beyond all orders."