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Asymptotic analysis of the Cooke-Noble integral equation

Published online by Cambridge University Press:  24 October 2008

L. R. F. Rose
L. R. F. Rose
Affiliation:
Aeronautical Research Laboratories, Melbourne 3207, Australia
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Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, let

where p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example with

where Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:

where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 2 , September 1982 , pp. 293 - 306
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

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