The bisection method for symmetrie eigenvalue problems with a parameter entering nonlinearly

Даутов, Р. З.; Lyashko, A. D.; Solov’ev, S. I.

doi:10.1515/rnam.1994.9.5.417

Cited by 27 publications

(8 citation statements)

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“…Nonlinear eigenvalue problems arise in various fields of science and technology [16][17][18][19][20][21][22][23]. Approximate methods for solving eigenvalue problems with monotone and nonmonotone dependence on the spectral parameter were studied in [24][25][26][27][28][29][30][31][32][33][34]. The theoretical basis for the study of nonlinear spectral problems is results obtained for linear eigenvalues problems [35][36][37][38][39][40][41][42][43][44].…”

Section: Problem Statementmentioning

confidence: 99%

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Желтухин

Solov’ev

et al. 2014

Lobachevskii J Math

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A condition for the existence of a minimum eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem.

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Section: Problem Statementmentioning

confidence: 99%

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Желтухин

Solov’ev

et al. 2014

Lobachevskii J Math

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“…The error of the finite difference method for solving differential eigenvalue problems with nonlinear dependence on the spectral parameter was investigated in [1,15]. For nonlinear differential spectral problems, the finite element method was studied in [16][17][18][19] based on the use general results in the linear case [20][21][22][23]. Approximate methods for solving applied nonlinear boundary value problems and variational inequalities have been investigated in the papers [24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Investigation of eigenvibrations of a loaded bar

Samsonov

Solov’ev

2018

MATEC Web Conf.

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The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

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“…Nonlinear eigenvalue problems arise in various applications [1][2][3]. Numerical methods for solving matrix eigenvalue problems with nonlinear dependence on the spectral parameter were constructed and investigated in the papers [4][5][6][7][8][9][10][11][12][13]. Mesh methods for solving differential nonlinear eigenvalue problems were studied in the papers [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Eigenvibrations of a bar with load

Samsonov

́ev

2017

MATEC Web Conf.

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Abstract. The differential eigenvalue problem describing eigenvibrations of an elastic bar with load is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

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The bisection method for symmetrie eigenvalue problems with a parameter entering nonlinearly

Cited by 27 publications

References 0 publications

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Investigation of eigenvibrations of a loaded bar

Eigenvibrations of a bar with load

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