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# Asymptotic analysis of the Cooke-Noble integral equation

## Extract

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, ∞).

## References

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(1)Abramowitz, M. and Stegun, I. A.Handbook of mathematical functions (Dover, 1965) p. 487, no. 11.4.41.
(2)Anderssen, R. S., De Hoog, F. R. and Rose, L. R. F.Explicit solutions for a class of dual integral equations. Proc. Roy. Soc. Edin. (in press, 1982).
(3)Bleistein, N. and Handelsman, R. A.Asymptotic expansions of integrals, chap. 4 (Holt, Rinehart and Winston, 1975).
(4)Cooke, J. C.A solution of Tranter's dual integral equations problem. Quart. J. Mech. App. Math. 9 (1956), 103110.
(5)Davies, B.Integral transforms and their applications, §§ 12, 14 (Springer-Verlag, 1978).
(6)England, A. H. and Green, A. E.Some two-dimensional punch and crack problems in classical elasticity. Proc. Cambridge Phil. Soc. 59 (1963), 489500.
(7)Erdelyi, A. (ed.). Higher transcendental functions, vol. 1, p. 68 (McGraw-Hill, 1953).
(8)Jones, D. S.Diffraction at high frequencies by a circular disc. Proc. Cambridge Phil. Soc. 61 (1965), 223245.
(9)Handelsman, R. A. and Lew, J. S.Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Applies. 35 (1971), 405433.
(10)Noble, B.Certain dual integral equations. J. Math. Phys. 37 (1958), 128136.
(11)Noble, B.The solution of Bessel function dual integral equations by a multiplying-factor method. Proc. Cambridge Phil. Soc. 59 (1963), 351362.
(12)Rose, L. R. F.A cracked plate repaired by bounded reinforcements. Int. J. Fracture 18 (1982), 135144.
(13)Sneddon, I. N.Mixed boundary valve problems in potential theory, chap. 4 (North Holland, 1966).
(14)Sneddon, I. N. and Lowengrub, M.Crack problems in the classical theory of elasticity (Wiley, 1969).
(15)Watson, G. N.A treatise on the theory of Bessel functions (Cambridge, 1944).
(16)Zabreyko, P. P. et al. Integral equations: a reference text, chap. 2, §6 (Noordhoff, 1975).

# Asymptotic analysis of the Cooke-Noble integral equation

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