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.
CITATION STYLE
Boyd, J. P. (1999). The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series. Acta Applicandae Mathematicae, 56(1), 1–98. https://doi.org/10.1023/A:1006145903624
Mendeley helps you to discover research relevant for your work.