Mathematical Sciences: Analytical Approaches to Singular Perturbation Problems of Significance in Applications

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9404536 O'Malley Analytical Approaches to Singular Perturbation Problem of Significance in Applications. Robert E. O'Malley, Jr., Principal Investigator University of Washington Abstract: This research project seeks asymptotic solutions to nonlinear singularly perturbed boundary value problems which involve boundary, shock, and other localized regions of rapid change. Such problems involve substantial computational, as well as analytical, challenges, and their effective solution requires hybrid asymptotic/numeric approaches using numerical algorithms based on the structure of asymptotic solutions and corresponding asymptotic techniques motivated by careful numerical experiments on difficult models. In this way, we continue to broaden our understanding of the fundamental practical concept of asymptotic matching. Such mathematical problems occur frequently in important applications throughout science and engineering; in particular, in metastable models for phase-separation and coarsening in materials and in reaction-diffusion models which provide patterns for evolving biological processes. The need to use an asymptotically exponentially-long time scale in such aontexts corresponds to the mathematical issue of doing "asymptotics beyond all orders."