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

小行星9404536 奇异摄动问题的分析方法及其应用。 Robert E.小奥马利主要研究者华盛顿大学摘要：本研究项目旨在渐近解非线性奇摄动边界值问题，涉及边界，冲击，和其他局部地区的快速变化。这样的问题涉及大量的计算，以及分析，挑战，他们的有效解决方案需要混合渐近/数值方法，使用数值算法的基础上的渐近解的结构和相应的渐近技术的激励下仔细的数值实验困难的模型。通过这种方式，我们继续扩大我们的理解的基本实际概念的渐近匹配。 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，特别是在材料的相分离和粗化的亚稳态模型中，以及为生物过程的演变提供模式的反应扩散模型中。在这样的背景下，需要使用一个渐近指数长的时间尺度，这对应于“超越所有阶次的渐近性”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."