Singular perturbation solution of a class of singular integral equations

Willis, J. R.; Nemat-Nasser, S.

doi:10.1090/qam/1079917

Authors: J. R. Willis and S. Nemat-Nasser
Journal: Quart. Appl. Math. 48 (1990), 741-753
MSC: Primary 45E99; Secondary 73M25
DOI: https://doi.org/10.1090/qam/1079917
MathSciNet review: MR1079917
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Abstract: A formal method is developed for finding asymptotic solutions to a class of strongly singular integral equations containing a small parameter, $\varepsilon$. The class has relevance to the analysis of microcrack growth in reinforced ceramics. The method makes use of the asymptotic matching principle of Van Dyke. Its application is mechanical and it appears to allow, in principle, the construction of asymptotic solutions to any order. Consistency to order $\varepsilon$ is demonstrated for the general case and a solution correct to order ${\varepsilon ^2}$ is constructed for a particular example, previously studied only to leading order.

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