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

数学科学：具有应用意义的奇异摄动问题的分析方法

• 批准号: 9404536
• 负责人: Robert O'Malley
• 金额: $ 12.66万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404536&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analytical; Approaches; Singular

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."

9404536 O'Malley分析方法在应用中具有重要意义的单数扰动问题。 小罗伯特·E·奥马利（Robert E. O'Malley，Jr。此类问题涉及实质性的计算以及分析挑战及其有效解决方案，需要基于渐近解决方案的结构和相应的渐近技术，并通过在困难模型上进行仔细的数值实验来激励混合渐近/数字方法。这样，我们将继续扩大对渐近匹配的基本实用概念的理解。 这种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，在材料和反应扩散模型中的相位分​​离和变高的亚稳态模型中，这些模型为不断发展的生物过程提供了模式。在这种情况下使用渐近指数的时间尺度的需求对应于“超出所有订单”的数学问题。