Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T22:28:24.386Z Has data issue: false hasContentIssue false
  • English
  • Français

The elementary solution of dual integral equations†

Published online by Cambridge University Press:  18 May 2009

Ian N. Sneddon
Ian N. Sneddon
Affiliation:
Department of Mathemetics, The University Glasgow, W.2.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

When the theory of Hankel transforms is applied to the solution of certain mixed boundary value problems in mathematical physics, the problems are reduced to the solution of dual integral equations of the type

where α and ν are prescribed constants and f(ρ) is a prescribed function of ρ [1]. The formal solution of these equations was first derived by Titchmarsh [2]. The method employed by Titchmarsh in deriving the solution in the general case is difficult, involving the theory of Mellin transforms and what is essentially a Wiener-Hopf procedure. In lecturing to students on this subject one often feels the need for an elementary solution of these equations, especially in the cases α = ± 1, ν = 0. That such an elementary solution exists is suggested by Copson's solution [3] of the problem of the electrified disc which corresponds to the case α = –l, ν = 0. A systematic use of a procedure similar to Copson's has in fact been made by Noble [4] to find the solution of a pair of general dual integral equations, but again the analysis is involved and long. The object of the present note is to give a simple solution of the pairs of equations which arise most frequently in physical applications. The method of solution was suggested by a procedure used by Lebedev and Uflyand [5] in the solution of a much more general problem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 3 , January 1960 , pp. 108 - 110
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

1.Sneddon, I. N., Fourier transforms (McGraw-Hill, New York, 1951), 275, 456, 488.Google Scholar
2.Titchmarsh, E. C., Introduction to the theory of Fourier integrals, 2nd edn (Clarendon Press, Oxford, 1948), 334.Google Scholar
3.Copson, E. T., On the problem of the electrified disc, Proc. Edinburgh Math. Soc, 8 (1947), 14–19.CrossRefGoogle Scholar
4.Noble, B., Certain dual integral equations, J. Math. Phys. 37 (1958), 128–136.CrossRefGoogle Scholar
5.Lebedev, N. N. and Uflyand, Ya. C., The axisymmetric contact problem for an elastic layer, Prik. Math. Mech. 22 (1958), 320–326.Google Scholar
6.Watson, G. N., A treatise on the theory of Bessel functions, 2nd edn (Cambridge University Press, 1944), 405.Google Scholar
7.Whittaker, E. T. and Watson, G. N., Modern analysis, 4th edn (Cambridge University Press, 1927), 229.Google Scholar