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.     

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.     

Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, an efficient technique based on the Chebyshev spectral collocation method for computing the eigenvalues of fourth-order Sturm–Liouville boundary value problems is proposed. The excellent performance of this scheme is illustrated through four examples. Numerical results and comparison with other methods are presented.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.     

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

1IntroductionIt is well-known that the Sturmian Theory is an important aid in solving many problemsin mathematical physics.Therefore this theory is one of the most actual and extensivelydeveloped field in spectral analysis of boundary-value problems of St…     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.     

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.     

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.     

Inverse boundary spectral problem for non–selfadjoint maxwells–s equations with incomplete data

The inverse boundary spectral problem for selfadjoint Maxwell–s equations is to reconstruct unknown coefficient functions in Maxwell– equations from the knowledge of the boundary spectral data, i.e. fromt eh eigenvalues and the boudnary value of the eigenfunctions. Since the spectrum of non–selfadjoint Maxwell–s operator consists of normal eigenvalues and an interval, the complete boundary spectral data can be defind only in a very complicated way. In this article we show that the coefficients can be reconstructed from incomplete data, that is, from the large eigenvalues and the boundary values of the generalized eigenfunctions. Particularly, we do not need the nfinit–dimensional data corresponding to the non–discrete spectrum.     

Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions

This paper is concerned with periodic and antiperiodic boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues of these two different boundary value problems is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. In addition, a representation of solutions of a nonhomogeneous linear equation with initial conditions is given.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

We study first-order nonlinear planar Hamiltonian boundary value problems on time scales. Estimates on lower bounds for the eigenvalues of the problems are established by way of the Lyapunov inequality method. Our results are interpreted to nonlinear differential and difference planar Hamiltonian boundary value problems. As a special case, an estimate on lower bounds for eigenvalues of half-linear dynamic equations is obtained which generalizes and improves the existing ones to nonlinear Hamiltonian systems. Based on the main results, we establish existence and uniqueness of solutions of a related linear boundary value problem.     

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.     

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.     

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.     

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.     

Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Differential-Operator Equation with Spectral Parameter Quadratically Occurring in the Boundary Condition

The asymptotic behavior of eigenvalues of a boundary value problem for a secondorder differential-operator equation in a separable Hilbert space on a finite interval is studied for the case in which the same spectral parameter occurs linearly in the equation and quadratically in one of the boundary conditions. We prove that the problem has a sequence of eigenvalues converging to zero.     

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.     

Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube

We consider spectral boundary value problems of Steklov, Neumann, and Dirichlet types for second-order elliptic operators with ε-periodic coefficients in a perforated cube; the coefficients of the differential equations are assumed to satisfy some symmetry conditions. Complete asymptotic expansions with respect to the small parameter ε are constructed for eigenvalues and eigenfunctions of the said problems. Bibliography: 24 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 51–88, 1994.