• 대한수학회지
• /
• 제31권3호
• /
• pp.477-492
• /
• 1994

For the eigenvalue problem of $Au = \lambda u$ where A is considered as a semi-bounded self-adjoint operator on a Hilbert space, we are used to apply two complentary methods finding upper bounds and lower bounds to the eigenvalues. The most popular method for finding upper bounds may be the Rayleigh-Ritz method which was developed in the 19th century while a method for computing lower bounds may be the method of intermediate eigenvalue problems which has been developed since 1950's. In the method of intermediate eigenvalue problems (IEP), we consider the original operator eigenvalue problem as a perturbation of a simpler, resolvable, self-adjoint eigenvalue problem, called a base problem, that gives rough lower bounds.

• 대한수학회지
• /
• 제43권1호
• /
• pp.87-98
• /
• 2006

In this paper, a class of constrained inverse eigenvalue problem for antisymmetric matrices and their optimal approximation problem are considered. Some sufficient and necessary conditions of the solvability for the inverse eigenvalue problem are given. A general representation of the solution is presented for a solvable case. Furthermore, an expression of the solution for the optimal approximation problem is given.

• 대한수학회논문집
• /
• 제38권3호
• /
• pp.925-935
• /
• 2023

In this paper, we consider a boundary value problem for a sixth order difference equation. We prove the monotone behavior of the eigenvalue of the problem as the coefficients in the difference equation change values and the existence of a positive solution for a class of problems.

• Structural Engineering and Mechanics
• /
• 제88권5호
• /
• pp.493-500
• /
• 2023

The multiparameter eigenvalue method can be used to solve the damped finite element model updating problems. This method transforms the original problems into multiparameter eigenvalue problems. Comparing with the numerical methods based on various optimization methods, a big advantage of this method is that it can provide all possible choices of physical parameters. However, when solving the transformed singular multiparameter eigenvalue problem, the proposed method based on the generalised inverse of a singular matrix has some computational challenges and may fail. In this paper, more details on the transformation from the dynamic model updating problem to the multiparameter eigenvalue problem are presented and the structure of the transformed problem is also exposed. Based on this structure, the rigorous mathematical deduction gives the upper bound of the number of possible choices of the physical parameters, which confirms the singularity of the transformed multiparameter eigenvalue problem. More importantly, we present a row and column compression method to overcome the defect of the proposed numerical method based on the generalised inverse of a singular matrix. Also, two numerical experiments are presented to validate the feasibility and effectiveness of our method.

• Korean Journal of Mathematics
• /
• 제26권3호
• /
• pp.467-481
• /
• 2018

In this paper we present a postprocessing scheme for the Raviart-Thomas mixed finite element approximation of the second order elliptic eigenvalue problem. This scheme is carried out by solving a primal source problem on a higher order space, and thereby can improve the convergence rate of the eigenfunction and eigenvalue approximations. It is also used to compute a posteriori error estimates which are asymptotically exact for the $L^2$ errors of the eigenfunctions. Some numerical results are provided to confirm the theoretical results.

• Korean Journal of Mathematics
• /
• 제28권3호
• /
• pp.587-601
• /
• 2020

In this paper we propose a way of enhancing eigenvalue approximations with the Bank-Weiser error estimators for the P1 and P2 conforming finite element methods of the Laplace eigenvalue problem. It is shown that we can achieve two extra orders of convergence than those of the original eigenvalue approximations when the corresponding eigenfunctions are smooth and the underlying triangulations are strongly regular. Some numerical results are presented to demonstrate the accuracy of the enhanced eigenvalue approximations.

• 대한수학회지
• /
• 제58권4호
• /
• pp.835-847
• /
• 2021

We study behavior of numerical solutions for a nonlinear eigenvalue problem on ℝⁿ that is reduced from a dispersion managed nonlinear Schrödinger equation. The solution operator of the free Schrödinger equation in the eigenvalue problem is implemented via the finite difference scheme, and the primary nonlinear eigenvalue problem is numerically solved via Picard iteration. Through numerical simulations, the results known only theoretically, for example the number of eigenpairs for one dimensional problem, are verified. Furthermore several new characteristics of the eigenpairs, including the existence of eigenpairs inherent in zero average dispersion two dimensional problem, are observed and analyzed.

• 대한수학회논문집
• /
• 제33권1호
• /
• pp.317-323
• /
• 2018

We study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact manifolds with zero boundary condition. In particular, we obtain some geometric estimates for the first eigenvalue.

• Structural Engineering and Mechanics
• /
• 제39권4호
• /
• pp.541-558
• /
• 2011

This paper presents an analysis of stochastic eigenvalue problem of plane bar structures. Particular attention is paid to the effect of spatial variations of the flexural properties of the structure on the first four eigenvalues. The problem of spatial variations of the structure properties and their effect on the first four eigenvalues is analyzed in detail. The stochastic eigenvalue problem was solved independently by stochastic finite element method (stochastic FEM) and Monte Carlo techniques. It was revealed that the spatial variations of the structural parameters along the structure may substantially affect the eigenvalues with quite wide gap between the two extreme cases of zero- and full-correlation. This is particularly evident for the multi-segment structures for which technology may dictate natural bounds of zero- and full-correlation cases.

• 한국소음진동공학회:학술대회논문집
• /
• 한국소음진동공학회 2005년도 춘계학술대회논문집
• /
• pp.117-119
• /
• 2005

For large structural eigen-analysis, the whole structure is divided into some partitioned structures and through synthesis of partitioned structural model the eigen-data of structure can be obtained. In that case, eigenvalue problem consists of semidefinite mass matrix form because of displacement constraint condition. In this work the eigenvalue problem is considered by means of several method, determinant search and null space reduction method.