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- (7).Dong Mind-de, Poincare's Problem of Irregular Integrals Lecture Notes (unpublished), (1981).Google Scholar