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Applications of a hybrid asymptotic-numerical method for certain two-dimensional singular perturbation problems

University of British Columbia

This thesis examines a hybrid asymptotic-numerical method for treating two-dimensional singular perturbation problems whose asymptotic solution involves reciprocal logarithms of the small perturbation parameter, ε, in the form [equation]. The purpose of this hybrid asymptotic-numerical method is to treat the slow convergence problems of asymptotic expansions of this form. For the applications that we consider in this thesis, we believe there is sufficient evidence of convergence of these expansions for small enough ε. The hybrid method uses the method of matched asymptotic expansions to exploit the asymptotic structure to reduce the problem to one that is asymptotically related to the original. In general, one must solve this related problem numerically. The hybrid related problem contains the entire infinite logarithmic expansion in its solution, thus removing the necessity of obtaining each coefficient in successive terms individually, as one would have to do using only the method of matched asymptotic expansions. The hybrid solution essentially sums the infinite expansion of reciprocal logarithms and in so doing, improves the accuracy of the solution since the error of the approximation is smaller than any power of ( -1/log ε). An important feature of the hybrid related problem is that it is non-stiff. Thus, it does not suffer from the difficulty in solving the original problem numerically of resolving the rapidly varying scale structure. Another advantage of the hybrid method solution is that the parameter dependence of the problem is reduced from that of the original. The reduction in parameter dependence means that the hybrid method solution is less computationally intensive than a full numerical solution. We show that singular perturbation problems containing infinite logarithmic expansions arise in a wide variety of contexts. Four chapters of this thesis are dedicated to the detailed application of the hybrid method to such singular perturbation problems occurring in fluid flow in a straight pipe with a core, skeletal tissue oxygenation from capillary systems, heat transfer convected from small cylindrical objects, and low Reynolds number fluid flow past a cylinder that is asymmetric to the uniform free-stream. Following the detailed analysis of these four problems, we remark on possible extensions to the general framework of applicable problems. For example, we discuss applications in black body radiation, multi-body low Reynolds number fluid flow, vibration of thin plates with small holes or concentrated masses, localized non-linear reactions on catalytic surfaces, low frequency scattering of light, diffusion of a chemical species out of an almost impervious container, and steady-state current flow from microelectrodes.

Thesis/Dissertation

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2009-07-03

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http://hdl.handle.net/2429/10036

Mathematics

University of British Columbia

UBCV

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