The convergence rate of approximate methods in the eigenvalue problem when the parameter appears non-linearly

Vainikko, Gennadi; Karma, Otto

doi:10.1016/0041-5553(74)90166-9

Cited by 15 publications

(18 citation statements)

References 2 publications

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“…Under the corresponding conditions, this sequence converges to the operator A ∈L(E,V ) . Note that the definitions of convergence of the operators A n ∈L(E n ,V n ) to A ∈L(E,V ) can be found in [5].…”

Section: Nonlinear Two-parameter Spectral Problemmentioning

confidence: 99%

“…The operators p n are called connecting [5,15]. Choosing the spaces E n , n = 1, 2,…, and the system of connecting operators p n : E → E n , we perform a discretization of the initial problem (24), i.e., we approximate the operator function A (λ 1 , λ 2 ) by the approximate operator functions A n (λ 1 , λ 2 ) , n ∈N , defined in the respective finite-dimensional spaces.…”

Section: Nonlinear Two-parameter Spectral Problemmentioning

confidence: 99%

“…(22) to the exact solutions, we consider a nonlinear two-parameter problem on the general operator level, which enables us to use results on the nonlinear one-parameter spectral problem known from the literature [4,5].…”

Section: Nonlinear Two-parameter Spectral Problemmentioning

confidence: 99%

See 2 more Smart Citations

Synthesis of Radiating Systems with Flat Aperture According to a Prescribed Power Directivity Pattern. І. Finding the Set of Bifurcation Points of the Solutions

Савенко

2015

J Math Sci

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We study the problem of synthesis of a radiating system with flat aperture according to a prescribed power directivity pattern. We consider a special case where the field in the aperture is linearly polarized. It is shown that this class of problems is characterized by the nonuniqueness and bifurcation of solutions. The problem of finding the set of bifurcation points is reduced to the Cauchy problem for a differential equation of the first order. We construct numerical algorithms and present numerical examples of finding the bifurcation lines of solutions of the main equations of synthesis.

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Section: Nonlinear Two-parameter Spectral Problemmentioning

confidence: 99%

Section: Nonlinear Two-parameter Spectral Problemmentioning

confidence: 99%

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Synthesis of Radiating Systems with Flat Aperture According to a Prescribed Power Directivity Pattern. І. Finding the Set of Bifurcation Points of the Solutions

Савенко

2015

J Math Sci

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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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“…Galerkin approximations to general holomorphic differential eigenvalue problems of the form (4.12) have been analyzed by numerous authors. We mention here only [31 [17], [181 [34]. Galerkin approximations for general, holomorphic pseudo-differential eigenvalue problems have been analyzed, e.g., in [36].…”

Section: The Convergence Of the Boundary Layer Exponents And The Bounmentioning

confidence: 99%

Boundary layers of hierarchical beam and plate models

Schwab

Wright

1995

J Elasticity

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The boundary behavior of a family of hierarchical models of linearly elastic, isotropic plates is studied. The hierarchical models are obtained by spectral semidiscretization of the displacement fields in the transverse direction and strain energy projection. The well known Reissner-Mindlin model is contained in the hierarchy as a special case. A decomposition of the boundary layers of any model in the hierarchy into a bending and a torsion layer, both of which are model dependent, is given. It is shown further that the bending and torsion layers arise as Galerkin approximations of certain (nonlinear) eigenvalue problems in the plate cross section with the subspaces used to derive the hierarchical model. It is shown that the bending layers converge, as the order of the plate model tends to infinity, to the so-called Papkovich functions on an elastic strip.The regularity of the solution on polygonal plates is investigated for the whole hierarchy of plate models and shown to equal the regularity of the plane elasticity problem.The mathematical problem of plate modeling in elasticity has a history of more than 150 years, starting with the derivation of the biharmonic equation by Germain [11] and Kirchhoff [19] to explain plate bending phenomena. For a long time, this remained the most widely used model until Reissner [30] and later Mindlin [26] suggested the more sophisticated model which carries their names. Contrary to the Germain-Kirchhoff model, it consists of a singularly perturbed, coupled system of three second order differential equations. The derivations of these models were based upon intuitive mechanical principles and a mathematical justification of the Germain-Kirchhoff model as zero thickness limit of the three-dimensional elasticity problem was done only much later (see [9] and the references there).With the advent of modern digital computers, the Finite Element Method (FEM) and the need for highly accurate and efficient solutions of finite thickness plate problems, the use of a flexible hierarchy of models of variable order and complexity became feasible and was suggested and analyzed in a

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The convergence rate of approximate methods in the eigenvalue problem when the parameter appears non-linearly

Cited by 15 publications

References 2 publications

Synthesis of Radiating Systems with Flat Aperture According to a Prescribed Power Directivity Pattern. І. Finding the Set of Bifurcation Points of the Solutions

Synthesis of Radiating Systems with Flat Aperture According to a Prescribed Power Directivity Pattern. І. Finding the Set of Bifurcation Points of the Solutions

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

Boundary layers of hierarchical beam and plate models

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