Finite element approximation with numerical integration for differential eigenvalue problems

Solov’ev, S. I.

doi:10.1016/j.apnum.2014.02.009

Cited by 30 publications

(11 citation statements)

References 18 publications

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“…Then the error estimates of Theorem 3 are proved with using results from [19][20][21][22][23]. This completes the proof of the theorem.…”

Section: Variational Statement Of the Problemsupporting

confidence: 48%

“…Mesh methods for solving differential nonlinear eigenvalue problems were studied in [14][15][16]. The theoretical basis for the study of nonlinear eigenvalue problems is results obtained for linear eigenvalues problems [17][18][19][20][21][22][23]. In the papers [24][25][26][27][28][29][30], numerical methods for solving applied nonlinear boundary value problems and variational inequalities have been studied.…”

Section: Introductionmentioning

confidence: 99%

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Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

́ev

Chebakova

2017

MATEC Web Conf.

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Abstract. The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Error estimates for the approximate eigenvalue and the corresponding approximate positive eigenfunction are proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.

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“…Then the error estimates of Theorem 3 are proved with using results from [19][20][21][22][23]. This completes the proof of the theorem.…”

Section: Variational Statement Of the Problemsupporting

confidence: 48%

Section: Introductionmentioning

confidence: 99%

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

́ev

Chebakova

2017

MATEC Web Conf.

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“…Эта задача будет использована в дальнейшем для исследования форм потери устойчивости пластины и нахождения критической нагрузки, при которой наступает потеря устойчивости оболочки.. Будут разработаны приближенные методы решения указанной задачи на основе разработанных в [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70] подходов. При этом для нахождения критических нагрузок будут использованы спектральные задачи с нелинейным вхождением параметра [71][72][73][74][75][76], а также методы исследования форм потери устойчивости многослойных конструкций. [77,78].…”

Section: заключениеunclassified

Properties of the Frechet derivative of the operator of the geometrically nonlinear problem of bending of a three-layer plate and statement of a nonlinear spectral problem

Badriev¹,

Buyanov²,

Makarov³

2019

SDTE

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РАЗДЕЛ XVI. МАТЕМАТИКА Бадриев И.Б., Буянов В.Ю., Макаров М.В. Свойства производной Фреше оператора геометрически нелинейной задачи об изгибе трехслойной пластины и постановка нелинейной спектральной задачи Казанский (Приволжский) федеральный университет (Россия, Казань

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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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Finite element approximation with numerical integration for differential eigenvalue problems

Cited by 30 publications

References 18 publications

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

Properties of the Frechet derivative of the operator of the geometrically nonlinear problem of bending of a three-layer plate and statement of a nonlinear spectral problem

Investigation of eigenvibrations of a loaded bar

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