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The Quarterly Journal of Mechanics and Applied Mathematics 1983 36(3):419-436; doi:10.1093/qjmam/36.3.419
© 1983 by Oxford University Press
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EXACT SOLUTIONS OF CERTAIN DUAL INTEGRAL EQUATIONS AND THEIR ASYMPTOTIC PROPERTIES

L. R. F. ROSE and F. R. DE HOOG

( Aeronautical Research Laboratories Melbourne 3207, Australia
Division of Mathematics and Statistics C.S.I.R.O., Canberra, Australia )

It is noted that a set of mixed boundary-value problems for the two-dimensional Laplace's equation, which can be solved exactly by complex-variable techniques, constitutes a useful source of exact solutions of dual integral equations. Three particular cases, which can be interpreted as anti-plane crack problems, are considered in detail. The solutions exhibit widely different asymptotic properties as a -> {infty} , where 2a is the crack length, although all three solutions approach the same limiting form as a -> 0. A correlation is noted between the asymptotic properties of these solutions and those of the weight function appearing in the corresponding dual integral equations. The implications of this correlation for arbitrary weight functions and the difficulties encountered in an attempt to develop a rigorous asymptotic analysis for the limit a -> {infty} are briefly discussed.


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