Numerical Solution Of Eigenvalue Problems Research Articles

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

The Extension of the Monte Carlo Method for Neutron Transfer Problems Calculating to the Problems of Quantum Mechanics

There are several methods of numerical solution of eigenvalue problems by the Monte Carlo method, which are used in the calculation of nuclear reactors. This paper is devoted to the investigation of the possibility of using such methods for solving the stationary Schrodinger equation. The latter equation can easily be transformed into the form of an integral equation of the first kind, very similar to those integral equations that arise in problems of nuclear power. The Monte Carlo method for this form of the stationary Schrodinger equation looks very attractive, since it naturally parallels and is very convenient for calculations on multiprocessor systems. In addition, in this case it is necessary to operate with functions defined on a large-dimensional space. This is also natural for the Monte Carlo method. It is described how the methods long used for the calculation of nuclear reactors are transformed for this case. The main problem is that the wave function of fermions changes its sign under a permutation of identical particles, and can not be nonnegative. The proposed approach is significantly different from the known methods of applying the Monte Carlo method to quantum mechanical problems. In this paper, several examples of the successful application of the proposed new method are given.

Linearization techniques for band structure calculations in absorbing photonic crystals

SUMMARYBand structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency‐dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatial discretization using an interior penalty discontinuous Galerkin method, we employ a recently developed linearization technique to deal with the resulting rational matrix eigenvalue problems. In particular, the efficient implementation of Krylov subspace methods for solving the linearized eigenvalue problems is investigated in detail. Numerical experiments demonstrate that our new approach is considerably cheaper in terms of memory and computing time requirements compared with the naive approach of turning the rational eigenvalue problem into a polynomial eigenvalue problem and applying standard linearization techniques. Copyright © 2011 John Wiley & Sons, Ltd.

Some theorems about spectrum and finite element approach for eigenvalue problems for elastic bodies with voids

The paper concerns eigenvalue problems for elastic bodies with voids in contact with massive rigid plane punches. The linear theory of elastic materials with voids according to the Cowin–Nunziato model is used. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical and “porous” boundary conditions and the modulus of elastic solid with voids change, are established. A finite element method is proposed for numerical solution of eigenvalue problems for elastic media with voids. Some effective block algorithms for finite element eigenvalue problems with partial coupling are described. Numerical experiments are presented for determining the first eigenfrequencies of an axisymmetric elastic body with voids.

A numerical procedure to solve the multichannel Schrödinger eigenvalue problem

We discuss the numerical solution of eigenvalue problems for systems of (regular) coupled Schrödinger equations. Using a high order CPM (abbreviation for piecewise Constant (reference potential) Perturbation Method) in a shooting procedure, eigenvalues can be computed accurately. A generalization of the Prüfer method for scalar Sturm–Liouville problems makes the whole procedure more robust and allows us to specify the required eigenvalue by its index.

Phase Diagrams of One-Dimensional Commensurate-Incommensurate Transition Model with Triple-Well Interactions

We generalize the Frenkel-Kontorov model to the Frenkel-Kontorova-Devonshire model in which the interaction is the triple-well potential. By use of the effective potential method, numerical solutions of eigenvalue problem are used to work out the exact phase diagrams of a triple-well potential W and a piecewise parabolic potential V. According to the winding number ω and the rotation number Ω, we analyze the periodicity of the phase diagram and find some complex but regular phase structures. The properties of the phase structures are closely related to the period of the external potential D.

Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: a paradigm in three dimensions.

We present a methodology for the efficient numerical solution of eigenvalue problems of full three-dimensional elasticity for thin elastic structures, such as shells, plates and rods of arbitrary geometry, discretized by the finite element method. Such problems are solved by iterative methods, which, however, are known to suffer from slow convergence or even convergence failure, when the thickness is small. In this paper we show an effective way of resolving this difficulty by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA). As an example, we present an algorithm for computing the minimal eigenvalue of a thin elastic plate and we show both theoretically and numerically that it is robust with respect to both the thickness and discretization parameters, i.e. the convergence does not deteriorate with diminishing thickness or mesh refinement. This robustness is sine qua non for the efficient computation of large-scale eigenvalue problems for thin elastic structures.

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Numerical solution of eigenvalue problems for Hamiltonian systems

This paper discusses the numerical solution of eigenvalue problems for Hamiltonian systems of ordinary differential equations. Two new codes are presented which incorporate the algorithms described here; to the best of the author’s knowledge, these are the first codes capable of solving numerically such general eigenvalue problems. One of these implements a new new method of solving a differential equation whose solution is a unitary matrix. Both codes are fully documented and are written inPfort-verifiedFortran 77, and will be available in netlib/aicm/sl11f and netlib/aicm/sl12f.

Regularity and Numerical Solution of Eigenvalue Problems with Piecewise Analytic Data

This paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems. A detailed and systematic numerical study of these approximations is presented, together with an analysis of the numerical results in light of the theoretical results. The specific aim is to assess the reliability of the theoretical results, which are of an asymptotic nature, as a guide to practical computations, which may take place in the pre-asymptotic phase, and to look for characteristic features of the numerical results that are not completely explained by known theoretical results.

Numerical solution of eigenvalue problems using the compound matrix method

An algorithm based on the compound matrix method is presented for solving difficult eigen-value problems. Details are given for systems of ordinary differential equations of fourth-order that are valid over connected domains coupled through interfacial conditions. As an example we examine the linear stability of two superposed fluids in plane Poiseuille flow and consider both interfacial and shear modes. The Orr-Sommerfeld system that describes linear stability is solved for a selected range of parameters.

Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines method

We discuss a hybrid approach which uses the Tau Method in combination with the Method of Lines and treat a number of eigenvalue problems defined by partial differential equations with constant and variable coefficients, on rectangular or circular domains and with the eigenvalue parameters entering in the equation or in the boundary conditions. We obtain results of considerable accuracy which compare favourably with those published in the very recent literature and obtained by using advanced formutions of the boundary integral equation method or the multigrid method.

Interpolation, differentiation and solution of eigenvalue problems in more than one dimension

The extension to more than one dimension is outlined of recent results providing a convenient finite-dimensional matrix representation of the differential operator and indicating its usefulness for the numerical solution of eigenvalue problems.

Numerical solution of differential eigenvalue problems with an operational approach to the Tau method

A technique for the numerical solution of eigenvalue problems defined by differential equations, based on an operational approach to the Tau method recently proposed by the authors, is shown to be equivalent to a method of Chaves and Oritz. The technique discussed here leads to an algorithmic formulation of remarkable simplicity and to numerical results of high accuracy. It requires no shooting and can deal with complex multipoint boundary conditions and a nonlinear dependence on the eigenvalue parameter.

Asymptotische Entwicklungen für Eigenwerte und Eigenvektoren bei der Approximation parameternichtlinearer Eigenwertaufgaben

In order to apply extrapolation processes to the numerical solution of eigenvalue problems depending nonlinear on a parameter the existence of asymptotic expansions for eigenvalues and eigenvectors is studied. At the end of the paper some numerical examples are given.

Numerical solution of eigenvalue problems in control and communication

A new numerical method for solving Fredholm integral equations of the third kind over finite intervals is presented. The integral operator is represented by a square matrix which is computed by means of a two-dimensional series expansion of the kernel function with respect to a complete orthonormal set. The method is especially useful in computing the Karhunen-Loèvo expansion of a stochastic process and the best approximation of a function of two variables by a sum of products of functions of a single variable. Here the Karhunen-Loève expansion of a well-known stochastic process is considered. These techniques can be applied to deterministic and stochastic control systems.

Numerical solution of eigenvalue problems for nuclear-theory integro-differential equations

Numerical solution of eigenvalue problems for nuclear-theory integro-differential equations

О несколвких принципиальных вопросах задач о цоБственных значениях отосительно систем Дифференциальных уравнений. I

In the course of the numerical solution of Eigenvalue-problems connected with linear differential equations, the so-called encompassing theorems are of great importantce. We generalize in this paper the most important results of the theory, especially the encompassing-theorems, to the case of systems of differential equations. We expound further a method of perturbation which can be well applied also in the practice for the solution of Eigenvalue-problems connected with the systems. The problem acrose in connection with the fixation of the number of critical self-oscillation of turbine shovels.

The numerical solution of eigenvalue problems in which the eigenvalue problems in which the eigenvalue parameter appears nonlinearly, with an application to differential equations

The numerical solution of eigenvalue problems in which the eigenvalue problems in which the eigenvalue parameter appears nonlinearly, with an application to differential equations