Solution of Eigenvalue Problems for Spheroidal Wave Equation, Lame Equation, and Ellipsoidal Wave Equation

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The authors have conducted the research in detail on the eigenvalue problems of infinite complex band matrices, gaining the proof of the computability of approximated eigenvalues as well as its accurate error estimate formula, by an extremely natural truncation method. For more details, refer to1. Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada, The Eigenvalue Problem for Infinite Compact Complex Symmetric Matrices with Application to the Numerical Computation of Complex Zeros of J_0(z) -iJ_1(z) and of Bessel Functions J_m(z) of Any Real Order m, Linear Algebra and Its Applications, Vol. 194(1993), pp. 35-702. Y. Ikebe, N. Asai, Y. Miyazaki, and D. Cai, The Eigenvalue Problem for Infinite Complex Symmetric Tridiagonal Matrices with Application, Linear Algebra and Its Applications, Vol. 241-243(combined volume) (1996), pp.599-618On the other hand, transforming the three-dimensional wave equation Δw + p^2w = 0 by the spheroidal coordinates (there are two kinds, o … More r prolate type and oblate type), and executing the method of separation of variables give w"-tanθ・w' + (λ-μ^2 sec^2θ+γcos^2θ)w = 0, which is widely called spheroidal wave equations. The formulation, error estimate, as well as their visualization were attempted, by matrix method, as an application of the numerical commutation mentioned above.Alternatively, in analogy with the solution of Mathieu differential equation, which produced considerably significant results concerning the computation of eigenvalues up to the present, the authors handled the solution of Lame equation w"(z) ― (a + bk^2 sn^2 z)W(Z) = 0, also from the standpoint of the application stated earlier for its eigenvalues such that periodic Lame function has the period of 2π or 4π. Again, formulation, error estimate along with their visualization were obtained. Lastly, let us note that an ongoing project for solving the eigenvalues of ellipsoidal wave equation is also under investigation, whose related literature is seldom found. Less