Advertisement

Applied Mathematics and Mechanics

, Volume 5, Issue 2, pp 1151–1162 | Cite as

New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

  • Dong Ming-de
Article
  • 47 Downloads

Abstract

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and Möglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution.

Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species ℰci(sna), ℰsi (i=1,2,3,4) including the well known Lamé functions Eci(sna), Esi(sna) as special cases. This is effected by deriving two integro-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann's idea of P function, we introduce D function to express their transformation properties.

Keywords

Mathematical Modeling Wave Function Integral Equation Mathematical Physic Industrial Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    Erdelyi, A., Higher Transcendental Functions (Bateman Manuscript Project), Vol. 1–3, (1953–1955).Google Scholar
  2. (2).
    Whittaker, E. T and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, (1940).Google Scholar
  3. (3).
    Hobson, E. W., Theory of Spherical and Ellipsoidal Harmonics, Cambridge Univ Press, (1931).Google Scholar
  4. (4).
    Möglich, H., Beugunsercheinungen an Körpern von Ellipsoidischer Gestalt, Ann. d. Phys. 83, (1927), 609–734.Google Scholar
  5. (5).
    Arscott, F. M., (a) Periodic Differential Equations, Pergamon Press, (1964).Google Scholar
  6. (5)b.
    , A New Treatment of the Ellipsoidal Wave Equations, Proc. Lond. Math. Soc. 33, (1959), 21–50.Google Scholar
  7. (6).
    Malurkar, Ellipsoidal Wave Functions. Ind. J. Phys., 9, (1935), 45–80.Google Scholar
  8. (7).
    Dong Mind-de, Poincare's Problem of Irregular Integrals Lecture Notes (unpublished), (1981).Google Scholar

Copyright information

© HUST Press 1984

Authors and Affiliations

  • Dong Ming-de
    • 1
  1. 1.Institute of Theoretjcal PhysicsAcademia SinicaBeijing

Personalised recommendations