Inverse Problem for a Fourth-Order Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems are proved for two inverse problems for a fourth-order differential operator with nonseparated boundary conditions. The first of the problems, which has technical applications, is the problem of identification of a differential equation and two boundary conditions, and the second problem is the problem of identification of a differential equation and four boundary conditions. One of two data sets is used as the spectral data of the problem. The first data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectral data of a system of three problems, and the second data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectra of ten boundary value problems.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Boundary layer analysis for the eigenvalues of hanging rod

The motion of a naturally straight inextensible flexible elastic hanging rod is formulated and then linearized about the straight solution. To solve this equation by separation of variables, an eigenvalue problem is derived. When the stiffness of the rod is small, the eigenvalue equation is a singular perturbation problem. This paper is devoted to solving this eigenvalue problem by boundary layer analysis when the stiffness is suitably small, especially on the analytic approximate solutions of the first several eigenvalues and eigenfunctions. The first three eigenvalues are also compared with the numerical results computed by a finite difference method. The excellent agreement shows the efficiency of the boundary layer analysis.     

带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.     

On the Asymptotic and Numerical Analyses of Exponentially III-Conditioned Singularly Perturbed Boundary Value Problems

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation.     

Spectral properties of the family of even order differential operators with a summable potential

A boundary value problem for a higher order differential operator with separated boundary conditions is considered. The asymptotics of solutions of the corresponding differential equation for large values of the spectral parameter is studied. The indicator diagram of the equation for the eigenvalues is studied. The asymptotic behavior of eigenvalues and the formula for calculation of eigenfunctions of the studied operator is obtained in different sectors of the indicator diagram.     

Nonlinear spectral problem for a singular system of ordinary differential equations with a nonlocal condition

We consider a nonlinear spectral problem for a system of ordinary differential equations defined on an unbounded half-line and supplemented with a nonlocal condition specified by a Stieltjes integral. We suggest a numerically stable method for finding the number of eigenvalues lying in a given bounded domain of the complex plane and for the computation of these eigenvalues and the corresponding eigenfunctions. Our approach uses a simpler (with uncoupled boundary conditions) auxiliary boundary value problem for the same equation.     

一类正则Sturm—Liouville问题特征的渐近分析

考虑[0,π]上一类带一般分离型边界条件的正则Sturm—Liouville问题特征的渐近表示,利用Frechet导数,对特征值进行精细的分析,清楚地给出了方程系数q（x）及边界条件中常数cot α,col β对特征值的影响．使结论更具一般化．     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

边界条件含特征参数的奇异不连续Sturm-Liouville算子的渐近特征

研究有限区间内一类边界条件含特征参数的不连续奇异Sturm-Liouville问题.利用函数论和算子理论的方法,证明该问题的自伴性,得到其特征值的相关性质,基本解及其特征值的渐近公式.     

Inverse Problem for a Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems for the solution of an inverse problem for a fourth-order differential operator with nonseparated boundary conditions are proved. The spectral data for the problem is specified as the spectrum of the problem itself (or its three eigenvalues) and the spectral data of a system of three problems.     

On the spectrum of a multipoint boundary value problem for a fourth-order equation

We study the spectral properties of a multipoint boundary value problem for a fourth-order equation that describes small deformations of a chain of rigidly connected rods with elastic supports. We study the dependence of the spectrum of the boundary value problem on the rigidity coefficients of the supports. We show that the spectrum of the boundary value problem splits into two parts, one of which is movable under changes of the rigidity coefficients and the other remains fixed. As the rigidity coefficients grow, the eigenvalues corresponding to the movable part of the spectrum grow as well; moreover, the double degeneration of some eigenvalues is possible.     

On the Multiplicity of Eigenvalues of a Fourth-Order Differential Operator on a Graph

Differential Equations - We study the properties of eigenvalues of a boundary value problem for a fourth-order differential equation on a geometric graph modeling the elastic deformations of a...     

A modification of the invariant imbedding method for a singular boundary value problem

We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.     

Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale

In this study, we consider a boundary value problem generated by a second-order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem.     

The dirichlet problem for a loaded mixed-type equation in a rectangular domain

We establish necessary and sufficient conditions for the unique solvability of the first boundary problem for a loaded equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We obtain a solution to the stated problem as the sum of the eigenfunction series for the corresponding one-dimensional problem with respect to eigenvalues. We prove the stability of the solution with respect to boundary functions.     

Eigenvalue Asymptotics for the Non-Selfadjoint Operator Induced by a Parameter-Elliptic Multi-Order Boundary Problem

In this paper we consider a boundary problem for a parameter-elliptic, multi-order system of differential equations defined over a bounded region in ${\mathbb{R}^n}$ and under Dirichlet boundary conditions. In addition, the problem is considered under limited smoothness assumptions. Information is then derived concerning the asymptotic behaviour of the eigenvalues of the Hilbert space operator, in general non-selfadjoint, induced by the boundary problem under null boundary conditions.     

Spectral Problems for the Dirac System with Spectral Parameter in Local Boundary Conditions

We consider a spectral boundary value problem in a 3-dimensional bounded domain for the Dirac system that describes the behavior of a relativistic particle in an electromagnetic field. The spectral parameter is contained in a local boundary condition. We prove that the eigenvalues of the problem have finite multiplicities and two points of accumulation, zero and infinity and indicate the asymptotic behavior of the corresponding series of eigenvalues. We also show the existence of an orthonormal basis on the boundary consisting of two-dimensional parts of the four-dimensional eigenfunctions.