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Singular Perturbation Theory Math and Analyt Technique w. by R.S. Johnson

By R.S. Johnson

The idea of singular perturbations has developed as a reaction to the necessity to locate approximate strategies (in an analytical shape) to complicated difficulties. usually, such difficulties are expressed when it comes to differential equations which include a minimum of one small parameter, they usually can come up in lots of fields: fluid mechanics, particle physics and combustion strategies, to call yet 3.

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E. for any choice of the functions a and Thus we must extend the definition of an asymptotic expansion to accommodate this: Definition (asymptotic expansion with a parameter 2) We write (cf. 42) for x = O(1) and every where as It is clear that the separable case is simply a special version of this more general definition; let us investigate an example which incorporates such a term. 8 One more example of Consider the function for with x = O(1) we obtain because the term is exponentially small. 53), cannot be uniformly valid!

E. the two cases x = a ± R) must be investigated via individual and special calculations. Here, we will employ the same philosophy, namely, to apply our procedure in the case p = 0, and note the results; they may, or may not, prove useful. In the event, it will transpire that the results are fundamentally important, and lead to a very significant property of asymptotic expansions. 8 THE MATCHING PRINCIPLE Again, we suppose that we have two asymptotic expansions, one valid for x = O(1) and one for exactly as described in the previous section.

In order to generate a better approximation, we may use any appropriate method. For example, we could invoke the familiar procedure of iteration, so we may write with Then we obtain and so on (but note that iteration may not generate a correct asymptotic expansion at a given order in It is clear from this approach that a complete representation of the root will be obtained if we use the asymptotic sequence and so an alternative is to seek this form directly—and this is more in keeping with the ideas of perturbation theory.

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