The Devil ’ s Invention : Asymptotic , Superasymptotic and Hyperasymptotic Series ?
نویسنده
چکیده
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a ‘hyperasymptotic’ approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration. Mathematics Subject Classifications (1991): 34E05, 40G99, 41A60, 65B10.
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