Analytical Approaches to Singular Perturbation Problems of Significance in Applications

NSF奖摘要- DMS-0103632数学科学：奇异摄动问题的分析方法在应用中的意义0103632奥马利研究将发展渐近方法来解决非线性奇异摄动边值问题的普通和偏微分方程。 一个特别的重点将是重整化方法的系统发展，理论物理学家提出作为一个统一的工具渐近分析。 另一个将继续调查亚稳态动力学的代数，以及指数，渐近。这些问题是相关的，因为它们都涉及长时间渐近性和经典的多尺度方法。渐近方法，如计算，提供了一个重要的途径来找到近似解的非线性问题中出现的重要应用。 这项工作旨在进一步发展这种分析技术，并将其应用于工程和科学。

NSF Award Abstract - DMS-0103632Mathematical Sciences: Analytical Approaches to Singular Perturbation Problems of Significance in Applications0103632O'MalleyThe research will develop asymptotic methods to solve nonlinear singularly perturbed boundary value problems for both ordinary and partial differential equations. A special emphasis will be the systematic development of renormalization methods, which theoretical physicists have proposed as a unified tool for asymptotic analysis. Another will be continued investigation of metastable dynamics for algebraic, as well as exponential, asymptotics. The problems are related since they both deal with long-time asymptotics and the classical method of multiple scales.Asymptotic methods, like computation, provide an important way to find approximate solutions to nonlinear problems arising in significant applications. This work seeks to further develop such analytical techniques and to apply them in engineering and the sciences.