A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator \( \overline{A}=A+\overline{B} \) such that the eigenvalue problem for Ā is supposed to be simpler than for A: The algorithm for this eigenvalue problem is based on the homotopy idea and, for a given eigenpair number, recursively computes a sequence of approximate eigenpairs that converges to the exact eigenpair with a superexponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The case of multiple eigenvalues of the operator Ā is emphasized. Examples of eigenvalue problems for the high-order ordinary differential operators are presented to support the theory.
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References
L. D. Akulenko and S. V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, Chapman & Hall/CRC, Boca Raton, etc. (2005).
M. Armstrong, Basic Topology, Springer, Berlin (1979).
R. Ballarini, The Da Vinci–Euler–Bernoulli Beam Theory?, Mech. Eng. Mag. Online, April 18, 2003, No. 07, http://www.memagazine.org/contents/current/webonly/webex418.html.
B. I. Bandyrskii, I. Gavrilyuk, I. I. Lazurchak, and V. L. Makarov, “Functional-discrete method (FD-method) for matrix Sturm–Liouville problems,” CMAM, 5, No. 4, 362–386 (2005).
B. Bandyrskij, V. Makarov, and O. Ukhanev, “FD-method for the Sturm–Liouville problem. Exponential convergence rate,” J. Numer. Appl. Math., 85, No. 1, 1–60 (2000).
N. N. Bogoliouboff and N. M. Kryloff, “Sopra il metodo dei coefficienti constanti (metodo dei tronconi) per l’integrazione approssimata delle equazioni differenziali della fisica matematica,” Boll. Unione Mat. Ital., 7, No. 2, 72–77 (1928).
I. P. Gavrilyuk, A. V. Klimenko, V. L. Makarov, and N. O. Rossokhata, “Exponentially convergent parallel algorithm for nonlinear eigenvalue problems,” IMA J. Numer. Anal., No. 27, 818–838 (2007).
I. P. Gavrilyuk, V. L. Makarov, and A. M. Popov, “Superexponentially convergent parallel algorithm for eigenvalue problems for the fourth order ODE’s,” J. Numer. Appl. Math., 100, No. 1, 60–81 (2010).
I. P. Gawrilyuk and V. Makarov, “Superexponentially convergent parallel algorithm for eigenvalue problems in Hilbert spaces,” in: V. Kleiza, et al. (editors), Proc. Internat. Conf. “Different. Equat. Their Appl. (DETA-2009),’ ’Kaunas (2009), pp. 86–92.
M. Gordienko and V. Samoylenko, “Mathematical researches of M. M. Bogoliubov of time period 1923–1932,” Visn. Kyiv. Nats. Univ. im. T. Shevchenko. Mat. Mekh., No. (2)30, 25–32 (2013).
L. Greenberg, “An oscillation method for fourth-order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues,” SIAM J. Math. Anal., 22, No. 4, 1021–1042 (1991).
S. Han, H. Benaroya, and T. Wei, “Dynamics of transversely vibrating beams using four engineering theories,” J. Sound Vibrat., 225, No. 5, 935–988 (1999).
E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, I. Gew¨ohnliche Differentialgleichungen, Teubner, Stuttgart (2002).
A. N. Kolmogorov and S. V. Fomin, Reelle Funktionen und Funktionalanalysis, VEB Deutscher Verlag der Wissenschaften, Berlin (1975).
S. G. Krein, et al., Functional Analysis [in Russian], Nauka, Moscow (1972).
V. L. Makarov, “FD-method–exponential convergence rate,” J. Comput. Appl. Math., No. 82, 69–74 (1997).
M. A. Naimark, “Linear differential operators I. Elementary theory of linear differential operators, George G. Harrap & Company, 1967.
M. A. Naimark, Linear Differential Operators I. Elementary Theory of Linear Differential Operators [in Russian], Nauka, Moscow (1969).
S. Pruess, “Estimating the eigenvalues of Sturm–Liouville problems by approximating the differential equation,” SIAM J. Numer. Anal., 10, 55–68 (1973).
S. Pruess and C. Fulton, “Mathematical software for Sturm–Liouville problems,” ACM Trans. Math. Software, 19, No. 3, 360–376 (1993).
J. Pryce, Numerical Solution of Sturm–Liouville Problems, Oxford Univ. Press, Oxford, UK (1993).
M. Roseau, Vibrations in Mechanical Systems, Springer-Verlag, Berlin (1987).
A. Zettl, Sturm–Liouville theory, Amer. Math. Soc. (2005).
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Published in Neliniini Kolyvannya, Vol. 18, No. 3, pp. 332–356, July–September, 2015.
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Gavrilyuk, I.P., Makarov, V.L. & Romanyuk, N.M. Superexponentially Convergent Algorithm for an Abstract Eigenvalue Problem with Applications to Ordinary Differential Equations. J Math Sci 220, 273–300 (2017). https://doi.org/10.1007/s10958-016-3184-4
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DOI: https://doi.org/10.1007/s10958-016-3184-4