Abstract
The method presented and studied in [1, 2] for solving self-adjoint multiparameter spectral problems for weakly coupled systems of ordinary differential equations is based on marching with respect to a parameter introduced into the problem. Although the method is formally applicable to systems of ordinary differential equations with singularities, its direct use for the numerical solution of the problem indicated in this paper’s title is limited. A modification of the method is proposed that applies to the computation of various, including high-frequency, acoustic oscillations in both nearly spherical and strongly prolate spheroids.
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REFERENCES
A. A. Abramov and V. I. Ul’yanova, “A method for solving self-adjoint multiparameter spectral problems for weakly coupled sets of ordinary differential equations,” Comput. Math. Math. Phys. 37 (5), 552–557 (1997).
A. A. Abramov and V. I. Ul’yanova, “A Method for solving self-adjoint multiparameter spectral problems for systems of equations with singularities,” Comput. Math. Math. Phys. 38 (10), 1566–1570 (1998).
A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” IPN Prog. Rep. 42 (162), 1–51 (2005).
A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery nodes: Part I. Basics,” IEEE J. Sel. Topics Quantum Electron. 12, 3–14 (2006).
M. L. Gorodetsky, Fundamentals of the Theory of Optical Microresonators (Fiz. Fak., Mosk. Gos. Univ., Moscow, 2010) [in Russia].
A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, T. V. Pak, and B. S. Pariiskii, “Evaluation of prolate spheroidal function by solving the corresponding differential equations,” USSR Comput. Math. Math. Phys. 24 (1), 1–11 (1984).
A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, and T. V. Levitina, “Computation of radial wave functions for spheroids and triaxial ellipsoids by the modified phase function method,” USSR Comput. Math. Math. Phys. 31 (2), 25–42 (1991).
T. V. Levitina and E. J. Brändas, “Computational techniques for prolate spheroidal wave functions in signal processing,” J. Comput. Methods Sci. Eng. 1, 287–313 (2001).
P. Amodio, T. Levitina, G. Settanni, and E. B. Weinmüller, “On the calculation of the finite Hankel transform eigenfunctions,” J. Appl. Math. Comput. 43 (1–2), 151–173 (2013).
I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976) [in Russian].
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
H. Volkmer, Multiparameter Eigenvalue Problems and Expansion Theorems (Springer-Verlag, Berlin, 1988).
N. B. Konyukhova, S. E. Masalovich, and I. B. Staroverova, “Computation of rapidly oscillating eigenfunctions of a continuous spectrum and their improper integrals,” Comput. Math. Math. Phys. 35 (3), 287–302 (1995).
P. Amodio, A. Arnold, T. Levitina, G. Settanni, and E. B. Weinmüller, “On the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids,” Appl. Math. Comput. [to appear].
M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Nauka, Moscow, 1983; Springer, Berlin, 1993).
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The author is deeply grateful to the reviewer for careful reading of the manuscript and numerous valuable comments.
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Translated by I. Ruzanova
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Levitina, T.V. Computation of Eigenfrequencies of an Acoustic Medium in a Prolate Spheroid by a Modified Abramov Method. Comput. Math. and Math. Phys. 60, 1642–1655 (2020). https://doi.org/10.1134/S0965542520100103
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DOI: https://doi.org/10.1134/S0965542520100103