Mathematical Sciences: Singular Perturbations in Applied Mathematics: Methods and Applications

数学科学：应用数学中的奇异扰动：方法与应用

• 批准号: 8921967
• 负责人: Bernard Matkowsky
• 金额: --
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-15 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8921967&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Perturbations; Applied

The proposed research involves the application of asymptotic methods of singular perturbation type to boundary value problems for ordinary and partial differential equations, integral equations and difference equations. In particular, the problems to be studied center on two types that are relevant to applications: the effects of small random forces on the overall behavior of solutions of deterministic dynamical systems and the evolution of time-dependent solutions of boundary value problems to steady states that possess interior layers. A host of phenomena in nature are described by sets of equations that contain one or more formally small terms. If one were to try solving such equations numerically, for example, one usually runs into difficulty because the small terms can complicate the problems dramatically. An alternative method of solution, the one espoused in this proposal, is to take advantage of the small terms by first simplifying the equations that contain the small terms and then solving the simplified equations. The PI's will use this idea to solve various problems in probability theory and chemical physics.

本文主要研究奇异摄动型渐近方法在常微分方程、偏微分方程、积分方程和差分方程边值问题中的应用。特别是，要研究的问题集中在与应用相关的两种类型：小随机力对确定性动力系统解的整体行为的影响以及具有内层的边值问题的时变解的演化。自然界中的许多现象都是由一组包含一个或多个形式小项的方程来描述的。例如，如果一个人试图用数值方法求解这样的方程，他通常会遇到困难，因为小的项会使问题变得非常复杂。另一种解决方法，也就是本建议所支持的方法，是利用小项的优势，首先简化包含小项的方程，然后求解简化后的方程。PI将使用这个想法来解决概率论和化学物理中的各种问题。