A successive quadratic approximations method for nonlinear eigenvalue problems

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.

Section: Introductionmentioning

confidence: 99%

Investigation of eigenvibrations of a loaded bar

Samsonov

Solov’ev

2018

“…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

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.

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

Section: Introductionmentioning

confidence: 99%

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

́ev

Chebakova

2017

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.