Analytical Approaches to Singular Perturbation Problems of Significance in Applications

NSF奖摘要-DMS-0103632数学科学：应用中有意义的奇摄动问题的分析方法这项研究将发展渐近方法来解决常微分方程和偏微分方程组的非线性奇摄动边值问题。一个特别的重点将是重整化方法的系统发展，理论物理学家已经提出将其作为渐近分析的统一工具。另一项是继续研究亚稳态动力学的代数和指数渐近性。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。渐近性方法与计算一样，为寻找重要应用中出现的非线性问题的近似解提供了一种重要的方法。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.