On the numerical construction of ellipsoidal wave functions

Authors:
F. M. Arscott, P. J. Taylor and R. V. M. Zahar

Journal:
Math. Comp. **40** (1983), 367-380

MSC:
Primary 65D20; Secondary 33A60, 65H10, 65P05

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679452-6

MathSciNet review:
679452

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Abstract: The ellipsoidal wave equation, which is the most general equation derived by separation of the Helmholtz equation in confocal coordinates, presents unusual computational difficulties, and its solutions, despite their importance for physical applications, have not hitherto been effectively computed.

This paper describes a successful technique, which involves the solution of a four-term recursion and the simultaneous handling of two eigenparameters.

**[1]**F. M. Arscott,*Perturbation solutions of the ellipsoidal wave equation*, Quart. J. Math. Oxford Ser. (2)**7**(1956), 161–174. MR**0094484**, https://doi.org/10.1093/qmath/7.1.161**[2]**F. M. Arscott,*A new treatment of the ellipsoidal wave equation*, Proc. London Math. Soc. (3)**9**(1959), 21–50. MR**0104837**, https://doi.org/10.1112/plms/s3-9.1.21**[3]**F. M. Arscott,*Periodic Differential Equations*, Pergamon Press, New York, 1964.**[4]**F. M. Arscott,*Neumann-series solutions of the ellipsoidal wave equation*, Proc. Roy. Soc. Edinburgh Sect. A**67**(1963/1965), 69–77. MR**0187579****[5]**F. M. Arscott and I. M. Khabaza,*Tables of Lamé polynomials*, A Pergamon Press Book, The Macmillan Co., New York, 1962. MR**0152680****[6]**F. M. Arscott, R. Lacroix, and W. T. Shymanski,*A three-term recursion and the computation of Mathieu functions*, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978) Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979, pp. 107–115. MR**541916****[7]**F. M. Arscott and B. D. Sleeman,*High-frequency approximations to ellipsoidal wave functions*, Mathematika**17**(1970), 39–46. MR**0270003**, https://doi.org/10.1112/S0025579300002680**[8]**Robert Campbell,*Sur la vibration d’un haut-parleur elliptique*, C. R. Acad. Sci. Paris**228**(1949), 970–972 (French). MR**0029454****[9]**W. Gautschi,*Zur Numerik rekurrenter Relationen*, Computing (Arch. Elektron. Rechnen)**9**(1972), 107–126 (German, with English summary). MR**0312714****[10]**B. A. Hargrave and B. D. Sleeman,*Uniform asymptotic expansions for ellipsoidal wave functions. I. High frequency solutions of the ellipsoidal wave equations*, J. Inst. Math. Appl.**14**(1974), 31–40. MR**0419891****[11]**G. W. Hill,*On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon*, Acta Math.**8**(1886), no. 1, 1–36. MR**1554690**, https://doi.org/10.1007/BF02417081**[12]**E. L. Ince,*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**0010757****[13]**E. L. Ince, "Tables of the elliptic-cylinder functions,"*Proc. Roy. Soc. Edinburgh*, v. 52, 1932, pp. 335-423.**[14]**E. L. Ince, "Zeros and turning-points of the elliptic-cylinder functions,"*Proc. Roy. Soc. Edinburgh*, v. 52, 1932, pp. 424-433.**[15]**E. Mathieu, "Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique,"*J. Math. Pures Appl.*, v. 13, 1868, pp. 137-203.**[16]**R. V. M. Zahar, "A mathematical analysis of Miller's algorithm,"*Numer. Math.*, v. 27, 1977, pp. 427-447.**[17]**R. V. M. Zahar,*Recurrence techniques for a differential eigenvalue problem*, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978) Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979, pp. 479–485. MR**541940**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679452-6

Article copyright:
© Copyright 1983
American Mathematical Society