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The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

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Abstract

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.

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Boyd, J.P. The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series. Acta Applicandae Mathematicae 56, 1–98 (1999). https://doi.org/10.1023/A:1006145903624

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