Eigenvalue Dependent Boundary Conditions Research Articles

Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

Fourth order problems with the differential equation y^{(4)}-(gy')'=lambda ^2y, where gin C^1[0,a] and a>0, occur in engineering on stability of elastic rods. They occur as well in aeronautics to describe the stability of a flexible missile. Fourth order Birkhoff regular problems with the differential equation y^{(4)}-(gy')'=lambda ^2y and eigenvalue dependent boundary conditions are considered. These problems have quadratic operator representations with non-self-adjoint operators. The first four terms of the asymptotics of the eigenvalues of the problems as well as those of the eigenvalues of the problem describing the stability of a flexible missile are evaluated explicitly.

Least-Squares Spectral Methods for ODE Eigenvalue Problems

We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

A q-Dirac boundary value problem with eigenparameter-dependent boundary conditions

We study a boundary value problem for the q-Dirac equation and eigenvalue-dependent boundary conditions. We introduce a self-adjoint operator in a suitable Hilbert space and illustrate the boundary value problem as a spectral problem for this operator. We investigate the properties of the eigenvalues and vector-valued eigenfunctions. We construct Green?s function.

Stability results on radial porous media and Hele-Shaw flows with variable viscosity between two moving interfaces

Abstract We perform a linear stability analysis of three-layer radial porous media and Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change of variables results in an eigenvalue problem that has time-dependent coefficients and eigenvalue-dependent boundary conditions. We study this eigenvalue problem and find upper bounds on the spectrum. We also give a characterization of the eigenvalues and prescribe a measure for which the eigenfunctions form an orthonormal basis of the corresponding $L^2$ space. This allows for any initial perturbation of the interfaces and viscosity profile to be easily expanded in terms of the eigenfunctions by using the inner product of the $L^2$ space, thus providing an efficient method for simulating the growth of the perturbations via the linear theory. The limit as the viscosity gradient goes to zero is compared with previous results on multi-layer radial flows. We then numerically compute the eigenvalues and obtain, among other results, optimal profiles within certain classes of functions.

Dependence of eigenvalues of fourth-order boundary value problems with transmission conditions

A general fourth-order regular ordinary differential equation with eigenvalue dependent boundary conditions and transmission conditions are considered. We prove that the eigenvalues depend continuously and smoothly on the coefficients of the differential equation and on the boundary and transmission matrices. We provide as well formulas for the derivatives with respect to each of these parameters.

Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions

In this paper, we consider the operator L generated in L₂(R₊) by the differential expression l(y)=-y′′+q(x)y,x∈R₊:=[0,∞) and the boundary condition ((y′(0))/(y(0)))=α₀+α₁λ+α₂λ², where q is a complex valued function and α_{i}∈C,[mbox] \mbox{\:} i=0,1,2α₂. We have proved that spectral expansion of L in terms of the principal functions under the condition q∈AC(R₊), lim_{x→∞}q(x)=0, sup[e^{e√x}|q′(x)|] 0 taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.

Spectral properties of Sturm–Liouville system with eigenvalue-dependent boundary conditions

In this paper, we consider the boundary value problem [Formula: see text][Formula: see text] where λ is the spectral parameter and [Formula: see text] is a Hermitian matrix such that [Formula: see text] and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

Sixth order differential operators with eigenvalue dependent boundary conditions

We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper half-plane, except for finitely many eigenvalues on the negative imaginary axis.

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.

Darboux transformations and the factorization of generalized Sturm–Liouville problems

A version of the Darboux transformation is explored for Sturm-Liouville problems with eigenvalue-dependent boundary conditions, from differential-equation and operator-theoretic viewpoints. Some of the literature on Darboux's transformation is related in a historical introduction.

Boundary value problems with eigenvalue depending boundary conditions

AbstractWe investigate some classes of eigenvalue dependent boundary value problems of the form equation image where A ⊂ A+ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ0, Γ1 are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent PK (à – λ)–1|K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary value problems for elliptic partial differential operators on bounded domains

For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and self-adjoint extensions A Θ of A as restrictions of an operator or relation T which is a core of the adjoint A ∗ . This concept is applied to second order elliptic partial differential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated.

Spectral problems for operator matrices

We study spectral properties of 2 × 2 block operator matrices whose entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components. We investigate closability in the product space, essential spectra and generation of holomorphic semigroups. Application is given to several models governed by ordinary and partial differential equations, for example containing delays, floating singularities or eigenvalue dependent boundary conditions.

Spectral problems for operator matrices

AbstractWe study spectral properties of 2 × 2 block operator matrices whose entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components. We investigate closability in the product space, essential spectra and generation of holomorphic semigroups. Application is given to several models governed by ordinary and partial differential equations, for example containing delays, floating singularities or eigenvalue dependent boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Eigenvalues and eigenfunctions of discontinuous Sturm--Liouville problems with eigenparameter-dependent boundary conditions

We extend some fundamental spectral properties of classic regular Sturm--Liouville problems to discontinuous boundary-value problems with eigenvalue-dependent boundary conditions. We suggest a new approach for investigation of such type discontinuous problems.

Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

TRANSIENT RESPONSE OF ONE-DIMENSIONAL DISTRIBUTED SYSTEMS: A CLOSED FORM EIGENFUNCTION EXPANSION REALIZATION

The exact and closed form transient response of general one-dimensional distributed dynamic systems subject to arbitrary external, initial and boundary disturbances is determined. Non-self-adjoint operators characterizing damping, gyroscopic and circulatory effects, and eigenvalue-dependent boundary conditions are considered. Through introduction of augmented operators, a closed form modal expansion of the displacement and internal forces of the distributed system is derived. The eigenfunction expansion is realized in a spatial state formulation, which systematically yields exact eigensolutions, eigenfunction normalization coefficients and modal co-ordinates. The proposed method is illustrated on a cantilever beam with end mass, viscous damper and spring.

Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flow-duct acoustics

When the boundary conditions at the two endpoints of a second-order differential system depend explicitly upon the eigenvalues such that the system becomes non-self-adjoint, a generalized condition of orthogonality which includes endpoint terms can be developed. The generalized orthogonality condition is used to determine modal coefficients for the expansion of arbitrary functions in series of eigenfunctions. The method is applied to the particular case of acoustic wave propagation in a rectangular duct with a uniform mean-flow profile and walls with finite acoustic impedance. The ability of the eigenfunction expansion to converge to a plane-wave acoustic pressure profile is demonstrated under a variety of flow, frequency, and wall-impedance conditions.