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.     

Spectral analysis of nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrödinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem are given.     

Spectral analysis of nonselfadjoint Schr(o)dinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schr(o)dinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator,and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schr(o)dinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schr(o)dinger boundary value problem are given.     

On Counting the Number of Eigenvalues in the Right Half-Plane for Spectral Problems Connected with Hyperbolic Systems. II. Differential Equations

This article is an immediate continuation of [1]. Solution of the Lyapunov equation leads to a boundary value problem for the first-order hyperbolic equations in two variables with data on the boundary of the unit square. In general, the problems of this kind are not normally solvable. We prove that the boundary value problems in question possess the Fredholm property under some conditions.     

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.     

Zakharov–Shabat Spectral Transform on the Half-Line

The Zakharov–Shabat inverse spectral problem is constructed for a potential with support on the half-line and with a boundary value at the origin. This prescribed value is shown to produce a Jost solution with an essential singularity at large values of the spectral parameter; this requires particular attention when solving the related Hilbert boundary value problem. The method is then used to illustrate the sine-Gordon equation (in the light cone) and is discussed using a singular limit of the stimulated Raman scattering equations.     

Solvability of a Boundary Value Problem for Second-Order Elliptic Differential Operator Equations with a Spectral Parameter in the Equation and in the Boundary Conditions

In a Hilbert space H, we study noncoercive solvability of a boundary value problem for second-order elliptic differential-operator equations with a spectral parameter in the equation and in the boundary conditions in the case where the leading part of one of the boundary conditions contains a bounded linear operator in addition to the spectral parameter. We also illustrate applications of the general results obtained to elliptic boundary value problems.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

Reconstruction of an Impedance in Two-dimensions from Spectral Data

The two-dimensional spectral inverse problem involves the reconstruction of an unknown coefficient in an elliptic partial differential equation from spectral data, such as eigenvalues. Projection of the boundary value problem and the unknown coefficient onto appropriate vector spaces leads to a matrix inverse problem. Unique solutions of this matrix inverse problem exist provided that the eigenvalue data is close to the eigenvalues associated with the analogous constant coefficient boundary value problem. We discuss here the application of such a technique to the reconstruction of an impedance p in the boundary value problem $$ \eqalign{ -\nabla (\,p \nabla u) = \lambda p u \hbox {\quad in R} \cr u = 0 \hbox {\quad on R}}$$ where R is a rectangular domain. The matrix inverse problem, although nonstandard, is solved by a fixed-point iterative method and an impedance function p * is constructed which has the same m lowest eigenvalues as the unknown p . Numerical evidence of the success of the method will be presented.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

Spectral Analysis for an Indefinite Singular Sturm-Liouville Problem

Direct and inverse problems of spectral analysis are studied for an indefinite singular boundary value problem coming from astrophysics. We establish properties of the spectrum, prove completeness and expansion theorems and investigate the inverse problem of recovering the differential equation from the given spectral characteristics.     

On an inhomogeneous problem for parabolic-hyperbolic equation

We study a boundary value problem for an inhomogeneous parabolic-hyperbolic equation with a noncharacteristic type change line. Boundary conditions of the first kind are posed on characteristics in the parabolic and hyperbolic parts of the domain where the equation is given, and a condition of the third kind is posed on the noncharacteristic part of the boundary in the parabolic part. First, we study the solvability of an inhomogeneous initial–boundary value problem for a parabolic equation.     

非线性三阶常微分方程的非线性三点边值问题解的存在性

基于上下解方法,在一定条件下,得到了一类带有非线性混合边界条件的三阶常微分方程的非线性三点边值问题解的存在性,作为上述存在性结果的应用,在推论中给出了一类三阶非线性微分方程三点边值问题解的存在性.     

Spectral problem for a triple differentiation operator with asymmetric weight

We study the spectrum of a class of two-point boundary value problems for an ordinary differential operator of triple differentiation with weight. We show that the spectrum of a problem with asymmetric weight and with periodic boundary conditions fills the entire complex plane. We present an example of a problem with asymmetric weight to which one cannot assign a given spectrum by changing only one of the boundary conditions.     

Fredholm property of boundary value problems for a fourth-order elliptic differential-operator equation with operator boundary conditions

In a Hilbert space H, we study the Fredholm property of a boundary value problem for a fourth-order differential-operator equation of elliptic type with unbounded operators in the boundary conditions. We find sufficient conditions on the operators in the boundary conditions for the problem to be Fredholm. We give applications of the abstract results to boundary value problems for fourth-order elliptic partial differential equations in nonsmooth domains.     

The construction of solutions to pseudoparabolic equations in noncylindrical domains

We consider the first initial boundary value problem for pseudoparabolic equations in a domain with time dependent boundaries. Sufficient conditions on the coefficients of the equation and the boundary of the region are given in order that this initial boundary value problem has a unique solution.     

The Fokas-Lenells equation on the finite interval

Using the Fokas unified method, we consider the initial boundary value problem for the Fokas-Lenells equation on the finite interval. We present that the Neumann boundary data can be explicitly expressed by Dirichlet boundary conditions prescribed, and extend the idea of the linearizable boundary conditions for equations on the half line to Fokas-Lenells equation on the finite interval.     

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Green's function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.