Analytical Approaches to Singular Perturbation Problems of Significance in Applications

9703382 奥马利 华盛顿大学提议继续进行研究项目“奇异摄动问题的分析方法及其应用”。小奥马利作为首席研究员。 研究开发渐近方法来解决常微分方程和偏微分方程的非线性边值问题时，解决方案功能窄边界，冲击，或过渡层区域的快速变化。 研究将强调刚性计算和动态亚稳定性，通过一贯的混合渐近/数值调查。 特别注意将支付渐近指数小的条款，这以前很少受到关注，因为我们现在很清楚，超越所有订单的渐近可能是至关重要的物理意义。 在整个科学和工程中，经常会出现涉及奇摄动边值问题。 最著名的例子是流体动力学边界层。 分析和数值技术必须结合起来，有效地攻击这些重要的问题。 这项工作依赖于一个混合的方法来解决模型问题的应用需求的动机，常微分方程和偏微分方程。

9703382 O'Malley The University of Washington proposes that the research project Analytical Approaches to Singular Perturbation Problems of Significance in Applications be continued, with Professor Robert E. O'Malley, Jr. as the Principal Investigator. The research develops asymptotic methods to solve nonlinear boundary value problems for both ordinary and partial differential equations when solutions feature narrow boundary, shock, or transition layer regions of rapid change. The research will emphasize stiff computation and dynamic metastability, through a consistently hybrid asymptotic/numeric investigation. Special notice will be paid to asymptotically exponentially small terms, which have previously received scant attention, since we are now well aware that asymptotics beyond all orders can be of critical physical significance. Throughout science and engineering, problems often occur which involve singularly perturbed boundary value problems. The best known example is fluid dynamical boundary layers. Analytical and numerical techniques must be combined to effectively attack these important problems. This work rests on a hybrid approach to solving both ordinary and partial differential equations for model problems motivated by application needs.