Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Solving Fully Fuzzy Nonlinear Eigenvalue Problems of Damped Spring-Mass Structural Systems Using Novel Fuzzy-Affine Approach

The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

CONVERGENCE ANALYSIS ON SS-HOPM FOR BEC-LIKE NONLINEAR EIGENVALUE PROBLEMS

Shifted symmetric higher-order power method(SS-HOPM)has been proved e ective in solving the nonlinear eigenvalue problem oriented from the Bose-Einstein Condensation(BEC-like NEP for short)both theoretically and numerically.However,the convergence of the sequence generated by SS-HOPM is based on the assumption that the real eigenpairs of BEC-like NEP are nite.In this paper,we will establish the point-wise convergence via Lojasiewicz inequality by introducing a new related sequence.

THE SINGULAR SECOND ORDER NONLINEAR EIGENVALUE PROBLEM WITH INFINITELY MANY POSITIVE SOLUTIONS

In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.

Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

In this paper,we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional.We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

A Uniformly Convergent Numerical Method for Singularly Perturbed Nonlinear Eigenvalue Problems

In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemistry.We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint.Matched asymptotic approximations for the problem are presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution.A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem.Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems.Finally,the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported.

POSITIVE SOLUTIONS TO FOURTH-ORDER THREE-POINT NONLINEAR EIGENVALUE PROBLEM

In this paper, we investigate a fourth-order three-point nonlinear eigenvalue problem. By the Krasnosel’skii’s fixed-point theorem in cone, some new results on the existence of positive solutions are obtained.

Best Estimates of Weighted Eigenvalues of One-dimensional p-Laplacian

In this paper, we determine the infimum and the supremum of the Dirich-let eigenvalues λn(p) (n = 1,2,…)of the problem t∈ ?[0,T], where 1 < p < ∞, and the weights p are nonnegative and are subject to conditions p(t)dt = M and max(e[0,T] p(t) = H. It is also explained for whatweights p the infimum and the supremum will be attained.

NONLINEAR RANK-ONE MODIFICATION OF THE SYMMETRIC EIGENVALUE PROBLEM

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.

RECURSIVE INTEGRAL METHOD FOR THE NONLINEAR NON-SELFADJOINT TRANSMISSION EIGENVALUE PROBLEM

The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission eigenvalue problem is discretized by the Mor- ley element. For the resulting quadratic eigenvalue problem, a recursive integral method is used to compute real and complex eigenvalues in prescribed regions in the complex plane. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Dynamic properties of structures with dampers modelled using fractional order derivatives

The focus of this paper is on determination of the dynamic parameters of structural systems with viscoelastic (VE) dampers described by Maxwell rheological models. Such parameters could be obtained after solving the appropriately defined nonlinear eigenvalue problem for frames with VE dampers. The solution to the nonlinear eigenvalue problem is obtained by equating to zero the determinant of the considered system of equations. Apart from complex conjugate eigenvalues, the real ones occurred when dampers that are described by the classic Maxwell model, are also determined.

Semilinear Elliptic Resonant Problems at Higher Eigenvalue with Unbounded Nonlinear Terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.

Periodic Eigenvalues of One-Dimensional p-Laplacian with Indefinite Weights

Rotation numbers are used in this paper to study the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic weight which changes sign. The analysis proves that for any nonnegative integer n, ρ -1(n/2) is the union of two closed intervals, one of which lies in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] + and the other in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] -, and the endpoints of these intervals yield the corresponding periodic and anti-periodic eigenvalues.

POSITIVE SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENCE EQUATIONS

In this paper we apply a cone theoretic fixed-point theorem proved by Krasnosel＇skii to obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations. We consider the case that the nonlinear term satisfies asvrnntoticallv linear growth.

Band Structure Calculations of Dispersive Photonic Crystals in 3D using Holomorphic Operator Functions

We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function.The N´ed´elec edge elements are employed to discretize the operators,where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier.The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements.The spectral indicator method is then applied to compute the discrete eigenvalues.Numerical examples are presented demonstrating the effectiveness of the proposed method.

Analysis and Efficient Solution of Stationary Schrodinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field

In this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number±ℓ.We show,that each of those pair/quadruple of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions,which are satisfied for our examples and the common InAs/GaAs heterojunction.Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40%of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

两端具有弹性支承双索浅拱模型的内共振分析

在以往关于斜拉桥非线性动力学的研究中,边界条件并没有被合适地模拟.为了更准确地探索斜拉桥的非线性动力学特性,论文考虑斜拉桥的初始构型,建立了两端具有弹性支承的双索浅拱模型.首先,将浅拱根据拉索的数量分为三段,对模型的面内特征值问题进行求解,并将求得的分段函数作为浅拱的试函数.其次,对浅拱发生主共振时全局模态和局部模态间的1:1:1内共振进行研究.采用伽辽金方法得到系统的常微分方程,并通过多尺度法进行求解.采用牛顿-拉夫逊方法得到调谐方程的稳定平衡解,并给出不同竖向刚度下模型的频率/力响应曲线,从而对模型的非线性行为进行分析.最后,基于以上研究得出一些有意义的结论.