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 具有应用意义的奇异摄动问题的奥马利分析方法。 Robert E. O'Malley, Jr.，首席研究员华盛顿大学 摘要：该研究项目寻求非线性奇扰动边值问题的渐近解，这些问题涉及边界、冲击和其他快速变化的局部区域。此类问题涉及大量的计算和分析挑战，其有效解决方案需要使用基于渐近解结构的数值算法的混合渐近/数值方法以及由对困难模型进行仔细数值实验激发的相应渐近技术。通过这种方式，我们不断拓宽对渐近匹配基本实用概念的理解。 此类数学问题在整个科学和工程的重要应用中经常出现。特别是在材料的相分离和粗化的亚稳态模型以及为不断演变的生物过程提供模式的反应扩散模型中。在此类上下文中使用渐近指数长时间尺度的需要对应于“超越所有阶的渐近”的数学问题。