Analytical Approaches to Singular Perturbation Problems of Significance in Applications

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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.