Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-12T02:23:23.358Z Has data issue: false hasContentIssue false
  • English
  • Français

On some dual series equations and their application to electrostatic problems for spheroidal caps

Published online by Cambridge University Press:  24 October 2008

W. D. Collins
W. D. Collins
Affiliation:
King's CollegeNewcastle Upon Tyne
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is concerned with dual series equations of the form where is the associated Legendre function of degree n + m and order −m of the first kind, Hn is a known coefficient of n, f(θ) and g(θ) are given functions of the variable θ and the equations are to be solved for the unknown coefficients An. In hydrodynamics, elasticity and electromagnetism such equations occur in many problems involving a spherical or spheroidal boundary when different conditions are prescribed on different parts of the boundary. These dual series equations are analogous to the dual integral equations which arise in the application of Hankel transforms to problems involving plane or paraboloidal boundaries, the theory and applications of these latter equations having been discussed by several writers amongst whom are Sneddon, Noble and Lebedev(1), (2), (3). The main result of this paper is that the solution of the dual series equations (1·1) and (1·2) for m a positive integer or zero can be reduced to the solution of a Fredholm integral equation of the second kind in one independent variable. The restriction that m be a positive integer or zero covers those dual series equations which arise in three-dimensional boundary-value problems but excludes those which arise in two-dimensional ones where m is . These latter equations have, however, been discussed in a recent paper by Tranter (13).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 2 , April 1961 , pp. 367 - 384
Copyright
Copyright © Cambridge Philosophical Society 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Sneddon, I. N., Fourier transforms (New York, 1951).Google Scholar
(2)Noble, B. J., Math. Phys. 37 (1958), 128–36.CrossRefGoogle Scholar
(3)Lebedev, N. N., C.R. Acad. Sci. U.R.S.S. (N.S.), 114 (1957), 513–16.Google Scholar
(4)MacRobert, T. M., Spherical harmonics (London, 1947).Google Scholar
(5)Erdelyi, A., et al. Higher transcendental functions, vol. II (New York, 1953).Google Scholar
(6)Erdelyi, A., et al. Higher transcendental functions, vol. I (New York, 1953).Google Scholar
(7)Collins, W. D., Proc. Camb. Phil. Soc. 55 (1959), 377–9.CrossRefGoogle Scholar
(8)Copson, E. T., Proc. Edinb. Math. Soc. (2), 8 (1947–1950), 14–19.CrossRefGoogle Scholar
(9)Lamb, H., Hydrodynamics, 6th ed. (Cambridge, 1953).Google Scholar
(10)Hobson, E. W., Spherical and ellipsoidal harmonics (Cambridge, 1931).Google Scholar
(11)Byrd, P. F., and Friedman, M. D., Handbook of elliptic integrals (Berlin, 1954).Google Scholar
(12)Nicholson, J. W., Phil. Trans. A, 224 (1924), 49–93.Google Scholar
(13)Tranter, C. J., Proc. Glasgow Math. Ass. 4 (1957), 49–57.CrossRefGoogle Scholar