Abstract
The paper constructs and analyzes a combination difference scheme for numerical determination of the eigenvalues of the Laplace operator. The proposed scheme uses the two-sided (from above and from below) properties of variational-difference and ordinary difference schemes for the eigenvalue problem of the Laplace operator in convex domains. The half-sum of the two schemes in convex domains gives an O(h4) approximation to the exact eigenvalue. A summation representation formula is constructed as an implementation of the ten-point difference scheme.
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References
I. N. Lyashenko, Eigenvalue Problems for Second-Order Partial Finite-Difference Equations [in Russian], Kiev (1970).
I. N. Lyashenko and Kh. M. Meredov, Numerical Solution of Some Spectral Problems of Oscillation Theory [in Russian], Kiev (1978).
I. N. Lyashenko, Kh. M. Meredov, and A. Embergenov, "Two-sided difference method for numerical determination of the eigenvalues of the Laplace operator," Vychisl. Prikl. Mat., No. 40, 98–101 (1980).
I. N. Lyashenko, Kh. M. Meredov, and A. Embergenov, "Analysis of the variational-difference method for determination of eigenvalues of the Laplace operator, 1, 2," Izv. Akad. Nauk Turkmen. SSR, No. 2, 4–9 (1984); No. 4, 12–16 (1984).
Additional information
Kiev University. Nukus University. Turkmen Teachers College. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 55–60, 1991.
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Lyashenko, I.N., Embergenov, A. & Meredov, K. A combination difference scheme for the eigenvalue problem of the Laplace operator. J Math Sci 72, 3091–3094 (1994). https://doi.org/10.1007/BF01259477
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DOI: https://doi.org/10.1007/BF01259477