Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T18:15:31.496Z Has data issue: false hasContentIssue false
  • English
  • Français

XXI—The Whittaker-Hill Equation and the Wave Equation in Paraboloidal Co-ordinates

Published online by Cambridge University Press:  14 February 2012

F. M. Arscott
F. M. Arscott
Affiliation:
University of Surrey.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the Helmholz equation is separable, and each of the separated equations is reducible to Hill's equation with three terms (the Whittaker-Hill equation). The properties of solutions of this equation are developed sufficiently to make possible the formal solution of simple boundary-value problems for paraboloidal surfaces, principally for the case k2 < 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 67 , Issue 4 , 1967 , pp. 265 - 276
Copyright
Copyright © Royal Society of Edinburgh 1967

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 to Literature

Arscott, F. M., 1964 a. Periodic Differential Equations, Pergamon Press, Oxford.Google Scholar
Arscott, F. M., 1964 b. Paraboloidal co-ordinates and Laplace's equation, Proc. Roy. Soc. Edinb. A., 66, 129–139.Google Scholar
Arscott, F. M., 1964 c. Two-parameter eigenvalue problems in differential equations, Proc. Lond. Math. Soc., 14, 459–70.CrossRefGoogle Scholar
Campbell, R., 1955. Theorie generale de l'equation de Mathieu, Masson, Paris.Google Scholar
Ince, E. L., 1923. A linear differential equation with periodic coefficients, Proc. Lond. Math. Soc., 23, 56–74.Google Scholar
Ince, E. L., 1926. The real zeros of a linear differential equation with periodic co efficients, Proc Lond. Math. Soc., 25, 53–8.CrossRefGoogle Scholar
Klotter, K. and Kotowski, G., 1943. Uber die Stabilität der Lösungen Hillscher Differentialgleichungen, Z. Angew. Math. Mech., 23, 149–255.CrossRefGoogle Scholar
Magnus, R. and Winkler, S., 1958. The co-existence problem for Hill's-equation, N. Y. Univ. Inst. Math. Sci. Res. Rep., BR-26.Google Scholar
Magnus, R. and Winkler, S., 1961. Hill's equation, Part II, N.Y. Univ. Inst. Math. Sci. Res. Rep., BR-38.Google Scholar
Perron, O., 1957. Die Lehre von den Kettenbrüchen. Teubner, Stuttgart.Google Scholar
Urwin, Kathleen M., 1964. Integral equations for paraboloidal wave functions (I), Quart. J. Math., 15, 309–315.Google Scholar
Urwin, Kathleen M., 1965. Integral equations for paraboloidal wave functions (II), Quart. J. Math., 16, 257–262.CrossRefGoogle Scholar
Urwin, Kathleen M., 1967. Perturbation solutions of the Whittaker-Hill equation, J. Inst. Maths Applic, 3.Google Scholar