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Even if a differential equation cannot be solved explicitly, we can often obtain useful information about the solutions by computing functions, called analytic approximations, that are close to the actual solutions. In this chapter, we give a brief account of some of the most successful methods of finding analytic approximations, namely perturbation methods. The idea is to identify a portion of the problem that is small in relation to the other parts and to take advantage of the size difference to reduce the original problem to one or more simpler problems. This fundamental idea has been a cornerstone of applied mathematics for over a century since it has proven remarkably successful for attacking an array of challenging and important problems. In order to introduce some of the basic terminology and to illustrate the idea of perturbation methods, let’s begin with an elementary example.
KeywordsBoundary Layer Periodic Solution Perturbation Method Erential Equation Singular Perturbation
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