scholarly journals Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

2020 ◽  
Vol 22 (1) ◽  
pp. 48-70
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Boundary Value ◽  
Weight Functions ◽  
Order Differential Operator ◽  
Eighth Order ◽  
Entire Family ◽  
Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

Adjoint Interior-Point Boundary Conditions for Linear Differential Operators

1977 ◽  
Vol 20 (4) ◽  
pp. 447-450 ◽  
Author(s):  
Robert Neff Bryan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Interior Point ◽  
Differential Operators ◽  
Boundary Value ◽  
Point Boundary ◽  
Linear Differential Operators ◽  
Linear Differential

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line

2016 ◽  
Vol 5 (3) ◽  
Author(s):  
Imed Bachar ◽  
Habib Mâagli
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Half Line ◽  
Continuous Solutions ◽  
Second Order Differential Equations

AbstractWe are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problemsubject to the boundary conditions

scholarly journals Spectral theory of some non-selfadjoint linear differential operators

2013 ◽  
Vol 469 (2154) ◽  
pp. 20130019 ◽  
Author(s):  
B. Pelloni ◽  
D. A. Smith
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Differential Operators ◽  
Boundary Value ◽  
Bounded Interval ◽  
Linear Differential Operators ◽  
Linear Differential

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂ t q ±i Sq =0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

scholarly journals A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Zheyan Zhou ◽  
Jianhe Shen
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Degree Theory ◽  
Mixed Boundary Conditions ◽  
Second Order ◽  
Order Differential Operator ◽  
Uniqueness Of Solutions

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered,Lx=f(t,x,x′),t∈(a,b),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a)in whichLis a formally self-adjoint second-order differential operator. Under appropriate assumptions onL,f, andg, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.

scholarly journals On the asymptotic behavior of the spectrum of a sixth-order differential operator, whose potential is the delta function

2020 ◽  
Vol 22 (3) ◽  
pp. 280-305
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Delta Function ◽  
Discontinuous Coefficients ◽  
Sixth Order ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Order Differential Operator ◽  
Indicator Diagram ◽  
Dirac Delta

In this paper we propose a new method for studying differential operators with discontinuous coefficients.We consider a sequence of sixth-order differential operators with piecewise-smooth coefficients. The limit of the sequence of these operators’ potentials is the Dirac delta function. The boundary conditions are separated. To correctly determine solutions of differential equations with discontinuous coefficients at the points of discontinuity, “gluing” conditions are required. Asymptotic solutions were written out for large values of the spectral parameter, with the help of them the “gluing” conditions were studied and the boundary conditions were investigated. As a result, we derive an eigenvalues equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalues equation, which is a regular hexagon, is investigated. In various sectors of the indicator diagram, the method of successive approximations has been used to find the eigenvalues asymptotics of the studied differential operators. The limit of the asymptotic of the spectrum determines the spectrum of the sixth-order operator, whose potential is the delta function.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

Sign in / Sign up

Export Citation Format

Share Document