n-Widths and Singular Perturbation Problems

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Singularly perturbed partial differential equations are used to model boundary and interior layers in transport and fluid flow, and in high frequency wave propagation. It is difficult to approximate the solution to these equations because of regions of rapid change in the solution. The best approximation properties of solutions to some singularly perturbed linear partial differential equations will be studied. The research will build on some recent work of Kellogg and Stynes that treats asingularly perturbed self adjoint problem. Equations that model convection and wave propagation will be studied. The goal is to obtain the best approximation properties (in the sense of Kolmogorov) when the data of the problem are restricted to have a specified smoothness. A successful resolution of these best approximation problems will lead to asymptotic expansions of the solution with minimal smoothness requirements on the data, and hopefully to better numerical methods for the numerical computation of these solutions.Physical systems that involve boundary and interior layers are of considerable importance in industrial applications. A typical example is the turbulent layer of air near an aircraft wing. Wave propagation problems, arising e.g. in radar or ultrasound are important in both medical and military applications. These seemingly different problems are similar in that there are phenomena taking place over a very short span of distance or time. This leads to difficulties in calculating the behavior of such systems. The research under this award will study the basic approximation properties of the equations governing these systems.