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Vestnik Yuzhno-Ural'skogo Universiteta. Seriya Matematicheskoe Modelirovanie i Programmirovanie, 2013, Volume 6, Issue 4, Pages 15–25
(Mi vyuru100)
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This article is cited in 5 scientific papers (total in 6 papers)
Mathematical Modelling
A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators
S. I. Kadchenko Magnitogorsk State University, Magnitogorsk, Russian Federation
Abstract:
Based on the methods of
regularized traces and Bubnov–Galerkin's method a new method for
the solution of inverse problems is developed in spectral
characteristics perturbed self-adjoint operators. Simple formulas
for calculating the eigenvalues of discrete operators without the
roots of the corresponding secular equation are found. Computation
of eigenvalues of a perturbed self-adjoint operator can be started
with any of their numbers, regardless of whether the previous
numbers of eigenvalues are known or not. Numerical calculations
for eigenvalues of the Sturm–Liouville's operator show that the
proposed formulas for large numbers of eigenvalues give more
accurate results than the Bubnov–Galerkin's method. In addition,
the obtained formulas allow us to calculate the eigenvalues of
perturbed self-adjoint operator with very large numbers, where the
use of the Bubnov–Galerkin's method becomes difficult. It can be
used in problems of hydrodynamic stability theory, if you want to
find signs of the real or imaginary parts of the eigenvalues with
large numbers.
An integral Fredholm equation of the first kind, restoring the value of the perturbing operator in the nodal points of the sample, is obtained.
The method is tested on inverse problems for the
Sturm–Liouville's problem. The results of numerous calculations
have shown its computational efficiency.
Keywords:
the inverse spectral problem; perturbation theory; discrete and self-adjoint operators; eigenvalues; eigenfunctions; incorrectly formulated problems.
Received: 11.05.2013
Citation:
S. I. Kadchenko, “A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 6:4 (2013), 15–25
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https://www.mathnet.ru/eng/vyuru100 https://www.mathnet.ru/eng/vyuru/v6/i4/p15
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Abstract page: | 278 | Full-text PDF : | 106 | References: | 41 | First page: | 2 |
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