Layer-type Problems. Ordinary Differential Equations
This classical and historically important example was introduced by Friedrichs (1942), primarily to illustrate that Prandtl’s matching principle for boundary-layer equations in fluid dynamicst† made good sense in spite of its seemingly paradoxical nature. It has been extensively discussed and generalized in the literature. We shall use the same form as Lagerstrom and Casten (1972), but the discussion will be different. Also, since the understanding of singular perturbation problems has developed greatly during the past thirty-five years, our purpose will be different from that of Friedrichs’. Rather than using the example to show that Prandtl was right (incidentally, Prandtl’s ideas about higher-order approximations were not correct), we use it as an introductory problem whose solution is simple to obtain and has a very simple structure. It is useful for illustrating various basic ideas and techniques. In subsequent sections of this chapter we show how successively more sophisticated ideas must be used and how the structure of the solutions becomes increasingly more complex. Some of these ideas will be introduced here because they are easily grasped for this problem, even though not really necessary here. The significance and usefulness of these ideas will be apparent in subsequent problems.
KeywordsGauge Function Outer Solution Outer Limit Interior Layer Matched Asymptotic Expansion
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