Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

Inequalities Among Eigenvalues of Left-definite Sturm-Liouville Problems

There are well-known inequalities among eigenvalues of right-definite Sturm- Liouville problems. In this paper, we study left-definite regular self-adjoint Sturm-Liouville problems with separated and coupled boundary conditions. For any fixed equation, we establish a sequence of inequalities among the eigenvalues for different boundary conditions, which is both theoretical and computational importance.

An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems

In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.

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.

Oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions via Prufer transformation

A class of Sturm-Liouville problems with discontinuity is studied in this paper.The oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions are obtained.The main method used in this paper is based on Prufer transformation,which is different from the classical ones.Moreover,we give two examples to verify our main results.

Eigenvalue Computation of Regular 4th Order Sturm-Liouville Problems

In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.

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.

SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

In this article,we consider a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions.By constructing some new Lagrange-type identities and two fundamental functions,we obtain not only the existence,the simplicity,and the interlacing properties of the real eigenvalues,but also the oscillation properties,orthogonality of the eigenfunctions,and the expansion theorem.Finally,we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems.

Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

In this paper,an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated.Liouville transformation is applied to change the problem into an equivalent minimization problem.Finite element method is proposed and the convergence for the finite element solution is established.A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem.A global convergence for the algorithm in the continuous case is proved.A few numerical results are given to depict the efficiency of the method.

Structured Eigenvalue Problems in Electronic Structure Methods from a Unified Perspective

In(relativistic)electronic structure methods,the quaternion matrix eigenvalue problem and the linear response(Bethe-Salpeter)eigenvalue problem for excitation energies are two frequently encoun-tered structured eigenvalue problems.While the former problem was thoroughly studied,the later problem in its most general form,namely,the complex case without assuming the positive definiteness of the electronic Hessian,was not fully understood.In view of their very similar mathematical structures,we examined these two problems from a unified point of view.We showed that the identification of Lie group structures for their eigenvectors provides a framework to design diagonalization algorithms as well as numerical optimizations techniques on the corresponding manifolds.By using the same reduction algorithm for the quaternion matrix eigenvalue problem,we provided a necessary and sufficient condition to characterize the different scenarios,where the eigenvalues of the original linear response eigenvalue problem are real,purely imaginary,or complex.The result can be viewed as a natural generalization of the well-known condition for the real matrix case.

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.

Inverse Eigenvalue Problems for a Structure with Linear Parameters

The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

Bai et al.proposed the multistep Rayleigh quotient iteration(MRQI)as well as its inexact variant(IMRQI)in a recent work(Comput.Math.Appl.77:2396–2406,2019).These methods can be used to effectively compute an eigenpair of a Hermitian matrix.The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector.In this paper,we show that these two conditions can be merged into a relaxed one,so the convergence conditions in these theorems can be weakened,and the resulting convergence theorems are applicable to a broad class of matrices.In addition,we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors,leading to rigorous convergence theories for both the MRQI and the IMRQI.

SUFFICIENT CONDITIONS FOR THE SOLUBILITY OF ADDITIVE INVERSE EIGENVALUE PROBLEMS

1 IntroductionLet Rn×n be the set of all n×n real matrices.Rn=Rn×1.Cn×ndenotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(aij),and n distinct real numbers λ1,λ2,…,λn,find a real n×n diagonal matrix

THE SUFFICIENT CONDITIONS FOR SOLVABILITY OF ALGEBRAIC INVERSE EIGENVALUE PROBLEMS

Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenvalue problems,solvability,sufficient conditions.

THE UNSOLVABILITY OF GENERALIZED INVERSE EIGENVALUE PROBLEMS ALMOST EVERYWHERE

In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywhere.Then adopting the method used in [14],we present some sufficient conditions such that the generalized inverse eigenvalue problems are unsohable almost everywhere.

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is stud- ied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special ease is also given.

Left-definite boundary conditions of Sturm-Liouville problems when the interval shrinks to a point

On the condition that the interval of the problem shrinks to a point, we investigated the separated boundary conditions Sα,β of left-definite Sturm-Liouville problem, and answered the following question： Is there a co ∈ J such that Sα,β is always left-definite or semi-left-definite for the Sturm-Liouville equation for each c ∈ （a, co）？

On Open Problems of Nonnegative Inverse Eigenvalues Problem

In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.