Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T11:18:53.185Z Has data issue: false hasContentIssue false
  • English
  • Français

On the solution of some axisymmetric boundary value problems by means of integral equations, II: further problems for a circular disc and a spherical cap

Published online by Cambridge University Press:  26 February 2010

W. D. Collins
W. D. Collins
Affiliation:
King's College, Newcastle upon Tyne
Get access

Extract

Green and Zerna [1] have given a method of determining the electrostatic potential due to a circular disc maintained at a given axisymmetric potential, their method depending on the solution of a Volterra integral equation of the first kind and being a generalization of a method given by Love [2] for the determination of the electrostatic potential due to two equal co-axial circular discs maintained at constant potentials. In a recent paper [3], henceforth referred to as Part I, the author applied this method to the corresponding problem for a hollow spherical cap and also to the determination of the Stokes' stream-functions for perfect fluid flows past a cap and a disc. The method consists of expressing the potential as the real part of a complex integral of a real variable t, the integrand involving an unknown function g(t). The boundary condition on the disc or cap gives a Volterra integral equation of the first kind for g(t), it being possible to solve this equation and hence determine the potential by integration.

Type
Research Article
Information
Mathematika , Volume 6 , Issue 2 , December 1959 , pp. 120 - 133
Copyright
Copyright © University College London 1959

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

1.Green, A. E. and Zerna, W., Theoretical Elasticity (Oxford, 1954), 172.Google Scholar
2.Love, E. R., Quart. J. Mechs. and Applied Math., 2 (1949), 428–451.CrossRefGoogle Scholar
3.Collins, W. D., Quart. J. Mechs. and Applied Math. 12 (1959), 232–241.CrossRefGoogle Scholar
4.Lamb, H., Hydrodynamics, 6th edn. (Cambridge, 1953), 164.Google Scholar
5.Copson, E. T., Proc. Edinburgh Math. Soc. (2), 8 (1947–50), 14–19.CrossRefGoogle Scholar
6.Terazawa, K., Phil. Trans. Royal Soc. A, 217 (1917–18), 35–50.Google Scholar
7.Love, A. E. H., Phil. Trans. Royal Soc. A, 228 (1928–29), 377–420.Google Scholar
8.Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals (Springer-Verlag, 1954).Google Scholar
9.Collins, W. D., Mathematika, 2 (1955), 42–47.CrossRefGoogle Scholar
10.Mikhlin, S. G., Integral Equations (Pergamon, 1957).CrossRefGoogle Scholar
11.Sternberg, W. J. and Smith, T. L., Theory of Potential and Spherical Harmonics (University of Toronto, 1944), 294.CrossRefGoogle Scholar
12.Trieomi, F. G., Integral Equations (Interscience, 1957), 52.Google Scholar