The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series |
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Authors: | John P Boyd |
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Institution: | (1) University of Michigan, Ann Arbor, MI, 48109, U.S.A. |
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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 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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Keywords: | perturbation methods asymptotic hyperasymptotic exponential smallness |
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