Eigenvalue Problems For Equations Research Articles

A streamline approach to two-dimensional steady non-ideal detonation: the straight streamline approximation

Detonations in non-ideal explosives tend to propagate significantly below the nominal one-dimensional detonation speed. In these cases, multi-dimensional effects within the reaction zone are important. A streamline-based approach to steady-state non-ideal detonation theory is developed. It is shown in this study that, given the streamline shapes, the two-dimensional problem reduces to an ordinary differential equation eigenvalue problem along each streamline, the solution of which determines the local shock shape that, in turn, leads to the solution of the detonation speed as a function of charge diameter. A simple approximation of straight but diverging streamlines is considered. The results of the approximate theory are compared with those of high-resolution direct numerical simulations of the problem. It is shown that the straight streamline approximation is remarkably predictive of highly non-ideal explosive diameter effects. It is even predictive of failure diameters. Given this predictive capability, one potential use of the method is in the determination of rate law parameters by fitting to data from unconfined rate stick experiments. This is illustrated by using data for ammonium nitrate fuel oil explosives.

Acceleration of a two-grid method for eigenvalue problems

This paper provides a new two-grid discretization method for solving partial differential equation or integral equation eigenvalue problems. In 2001, Xu and Zhou introduced a scheme that reduces the solution of an eigenvalue problem on a finite element grid to that of one single linear problem on the same grid together with a similar eigenvalue problem on a much coarser grid. By solving a slightly different linear problem on the fine grid, the new algorithm in this paper significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Numerical examples are also provided to demonstrate the efficiency of the new method.

On a class of generalized eigenvalue problems and equivalent eigenvalue problems that arise in systems and control theory

Systems and control theory has long been a rich source of problems for the numerical linear algebra community. In many problems, conditions on analytic functions of a complex variable are usually evaluated by solving a special generalized eigenvalue problem. In this paper we develop a general framework for studying such problems. We show that for these problems, solutions can be obtained by either solving a generalized eigenvalue problem, or by solving an equivalent eigenvalue problem. A consequence of this observation is that these problems can always be solved by finding the eigenvalues of a Hamiltonian (or discrete-time counterpart) matrix, even in cases where an associated Hamiltonian matrix, cannot (normally) be defined. We also derive a number of new compact tests for determining whether or not a transfer function matrix is strictly positive real. These tests, which are of independent interest due to the fact that many problems can be recast as SPR problems, are defined even in the case when the matrix D + D ∗ is singular, and can be formulated without requiring inversion of the system matrix A .

Very-high-precision solutions of a class of Schrödinger type equations

We investigate a method to solve a class of Schrödinger type equation eigenvalue problems numerically to very high precision P (from thousands to a million of decimals). The memory requirement, and the number of high-precision algebraic operations, of the method scale essentially linearly with P when only eigenvalues are computed. However, since the algorithms for multiplying high-precision numbers scale at a rate between P 1.6 and P log P log log P , the time requirement of our method increases somewhat faster than P 2 .

Оценивание норм оператора в задачах на собственные значения для уравнений с разрывными операторами

Оценивание норм оператора в задачах на собственные значения для уравнений с разрывными операторами

Product quasi-interpolation method for weakly singular integral equation eigenvalue problem

A discrete method of accuracy O( h m ) is constructed to solve the integral equation eigenvalue problem λ ϕ ( x ) = ∫ − 1 1 K ( x , y ) ϕ ( y ) d y , where K( x, y) = log| x − y| or K( x, y) = | x − y| − α , 0 < α < 1, ϕ( x) being the eigenfunction associated with the eigenvalue λ. The method is based first on improving the boundary behavior of the exact solution ϕ( x) with the help of a change of variables, and second on the product integration method based on a discrete spline quasi-interpolant (abbr. dQI) of order m.

Weighted eigenvalue problems for Hessian equations

In this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving Hessian operators. In particular we consider a singular weight, which behaves like a Hardy potential and we prove the existence of weak eigenfunctions.

Application of Periodized Harmonic Wavelets towards Solution of Eigenvalue Problems for Integral Equations

This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.

On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities

We study the existence of nonzero solutions of the eigenvalue problem for equations of elliptic type with discontinuous nonlinearities. An upper bound for the value of the bifurcation parameter in these problems is obtained.

Characterization of the critical magnetic field in the Dirac–Coulomb equation

We consider a relativistic hydrogenic atom in a strong magnetic field. The ground-state level depends on the strength of the magnetic field and reaches the lower end of the spectral gap of the Dirac–Coulomb operator for a certain critical value, the critical magnetic field. We also define a critical magnetic field in a Landau level ansatz. In both cases, when the charge Z of the nucleus is not too small, these critical magnetic fields are huge when measured in tesla, but not so big when the equation is written in dimensionless form. When computed in the Landau level ansatz, orders of magnitude of the critical field are correct, as well as the dependence in Z. The computed value is however significantly too big for a large Z, and the wavefunction is not well approximated. Hence, accurate numerical computations involving the Dirac equation cannot systematically rely on the Landau level ansatz. Our approach is based on a scaling property. The critical magnetic field is characterized in terms of an equivalent eigenvalue problem. This is our main analytical result, and also the starting point of our numerical scheme.

Trapped modes in elastic plates, ocean and quantum waveguides

Trapped modes and bound states occur in waveguides in a wide variety of physical situations, here we consider one class: the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. We consider the possibility of trapped modes in otherwise uniform elastic/ocean/quantum waveguides, that have a perturbation in thickness, or in curvature. These are all mathematically connected, so their study can be unified, and the same asymptotic procedure can be applied to them all. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility, or otherwise, of trapping. An asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. A mathematical explanation and physical argument for the existence, or otherwise, of trapped modes, depending on different situations, is given. The asymptotic approach leads to an ordinary differential equation eigenvalue problem and its solutions are compared both with numerics, for the full governing equations, and with a further simplification that gives highly accurate estimates for the lowest eigenvalue.

Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals

A wavelet-based method was developed to compute elastic band gaps of one-dimensional phononic crystals. The wave field was expanded in the wavelet basis and an equivalent eigenvalue problem was derived in a matrix form involving the adaptive computation of integrals of the wavelets. The method was then applied to a binary system. For comparison, the elastic band gaps of the same one-dimensional phononic crystals computed with the wavelet method and the well-known plane wave expansion (PWE) method are both presented in this paper. The numerical results of the two methods are in good agreement while the computation costs of the wavelet method are much lower than that of PWE method. In addition, the adaptability of wavelets makes the method possible for efficient band gap computation of more complex phononic structures.

Eigenvalue Methods for Unimolecular Rate Calculations with Several Products

When the calculation of a unimolecular reaction rate constant is cast in the form of a master equation eigenvalue problem, the magnitude of that rate is often smaller than the rounding error of the trace of the corresponding reaction matrix. Here, a previously published procedure (Pritchard, H. O. J. Phys. Chem. A 2004, 108, 5249-5252) for solving this problem is extended to the case of more than one reaction product. An Appendix notes the occurrence of avoided crossings between eigenvalues of the master equation in reversible, in mixed reversible-irreversible, and in multiwell unimolecular reaction calculations.

Numerical Analysis of Plasmon Resonances in Nanoparticles Based on Fast Multipole Method

A novel technique for the numerical analysis of plasmon resonances by using the fast multipole method (FMM) is presented. This approach is based on the solution of the eigenvalue problem for boundary integral equations, which can be naturally implemented by using the FMM. Numerical examples that highlight the efficiency of the fast multipole implementation are reported

A scheme for the evaluation of dominant time-eigenvalues of a nuclear reactor

This paper presents a scheme to obtain the fundamental and few dominant solutions of the prompt time eigenvalue problem (referred to as α-eigenvalue problem) for a nuclear reactor using multi-group neutron diffusion theory. The scheme is based on the use of an algorithm called Orthomin(1). This algorithm was originally proposed by Suetomi and Sekimoto [Suetomi, E., Sekimoto, H., 1991. Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations. Ann. Nucl. Energy 18 (4), 205–227] to obtain the fundamental K-eigenvalue ( K-effective) of nuclear reactors. Recently, it has been shown that the algorithm can be used to obtain the further dominant K-modes also. Since α-eigenvalue problem is usually more difficult to solve than the K-eigenvalue problem, an attempt has been made here to use Orthomin(1) for its solution. Numerical results are given for realistic 3-D test case.

Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals

A wavelet-based method is developed to calculate elastic band gaps of two-dimensional phononic crystals. The wave field is expanded in the wavelet basis and an equivalent eigenvalue problem is derived in a matrix form involving the adaptive computation of integrals of the wavelets. The method is applied to a binary system. We first compute the band gaps of Au cylinders in an epoxy host. The advantages of the wavelet-based method are discussed in regard with the well-known plane-wave expansion method. Then the method is used to compute the band gaps of Au cylinders in a soft rubber with elastic constant ${10}^{6}$ times lower than that of Au. The convergence tests show that the wavelet-based method can reduce the Gibbs effect to a certain extent. These advantages make it possible for easy calculations of band structures of mixed solid-fluid phononic crystals where the traditional plane wave method encounters difficulties. In addition, the adaptability of wavelets makes the method possible for efficient band gap computations of more complex phononic structures.

Time-sliced path integrals with stationary states

The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Feynman's original development. In the time-slicing approximation the quantum mechanical evolution will generally not have any stationary states. We look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states without having to perform a continuum limit. When the stationary states are postulated to be solutions of a second-order ordinary differential equation (ODE) eigenvalue problem it is found that the potential is required to be a solution of a particular first-order ODE. Similarly, when the stationary states are postulated to be solutions of a second-order ordinary difference equation (O$\Delta$E) eigenvalue problem the potential is required to be a solution of a particular first-order O$\Delta$E. The classical limits (which are at times very nontrivial) are integrable maps.

New applications of Orthomin(1) algorithm for K-eigenvalue problem in Reactor Physics

This paper presents some new investigations on the use of Orthomin(1) algorithm for the solution of K-eigenvalue problem in Nuclear Reactor Physics. The algorithm was first used by Suetomi and Sekimoto [Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations, Annals of Nuclear Energy, 18(4) (1991) 205–227.] to find fundamental mode solution of K-eigenvalue problem based on multi-group neutron diffusion theory. It was shown to be an efficient alternative to the conventional methods based on power iterations. The use of this algorithm needs software development to evaluate product of coefficient matrices with vectors. Here, it is shown that the algorithm can also be applied to the K-eigenvalue equation cast in terms of fission matrix. This opens up the possibility of using conventional well-established diffusion codes to implement Orthomin(1) so that new software is not needed. Apart from this, a remarkable feature of Orthomin(1) brought out in this paper is its utility to find higher mode solutions of the K-eigenvalue problem. This procedure is radically different from the methods commonly used for finding higher K-modes. It may also be of interest for finding higher mode eigensolutions in other scientific fields.

Positive solutions of an elliptic system depending on two parameters

We consider the Dirichlet problem associated to a $2\times 2$ semilinear elliptic system of equations in a bounded domain of $\mathbb R^n$. This system depends on two parameters $(\eta,\mu)$ and the nonlinear terms are of power type. The existence of solutions for such a problem in the plane $(\eta,\mu)$ is discussed. Our results are natural extensions of those obtained for the corresponding nonlinear eigenvalue problem for elliptic scalar equations.

Eigenvalue Methods in Unimolecular Rate Calculations

When the calculation of a unimolecular reaction rate constant is cast in the form of a master equation eigenvalue problem, the magnitude of that rate is often smaller than the rounding error of the trace of the corresponding reaction matrix. Two available methods to overcome this cancellation problem are examined, and it is shown that one of them, the Nesbet procedure, can fail if the master equation relaxation matrix is improperly normalized, or when some time-saving computational approximations are used.