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Solution of some boundary value problems in potential theory

Published online by Cambridge University Press:  26 February 2010

K. B. Ranger
K. B. Ranger
Affiliation:
Department of Mathematics, University of Toronto, Toronto, 181, Canada.
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Extract

Various methods have been developed for solutions of boundary value problems involving discs of finite radius and spherical caps. A recent account of this work is described in the book by Sneddon [1]. In the present paper a simple method is presented for the solutions of potential problems for the electrified disc and spherical cap by reducing the axially symmetric boundary value problems to a corresponding problems for the two-dimensional Laplace equation. The essence of the method is to employ integral operators which map two-dimensional harmonic functions into axially symmetric potentials and are closely related to the integral transformations given in [3]. In particular it is shown how the mixed boundary value problems for the disc and spherical cap are mapped into Dirichlet problems for the two-dimensional Laplace equation in the half plane and interior of the unit circle respectively. In both cases a standard Green's function approach is applied to determine the solution of the two-dimensional problem. Williams [2] demonstrated how the potential problem for the lens can be found using a similar method. It is noted that Rostovtsev [5] Mossakovskii [4] and Heins [7] have used techniques similar to that presented in this paper.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 98 - 101
Copyright
Copyright © University College London 1970

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References

1.Sneddon, I. N., Mixed boundary value problems in potential theory (John Wiley and Sons Inc., New York, 1966).Google Scholar
2.Williams, W. E., Quart. J. Mech. App. Math., 19 (1961), 443.Google Scholar
3.Ranger, K. B., J. Math. Mech., 14 (1965), 383.Google Scholar
4.Mossakovskii, V. I., Prikl. Mat. Mokh. (English Trans.) 18 (1954).Google Scholar
5.Rostovtsev, N. A., Prikl. Mat. Mokh. (English Trans.), 23 (1959), 1143.Google Scholar
6.Morse, P. and Feshbach, H., Methods of theoretical physics part I (McGraw Hill Book Company, 1953), p. 813.Google Scholar
7.Heins, A., Bull. Amer. Math. Soc, 71 (1965), 787.CrossRefGoogle Scholar