scholarly journals Eigenfunctions and asymptotic behavior of Eigenvalues to the given boundary value problem with Eigenparameter in the boundary conditions

2015 ◽  
Vol 18 (1) ◽  
pp. 179-190
Author(s):  
Aryan Ali Mohammed ◽  
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
The Given

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.

scholarly journals Parametrization for some boundary value problems of interpolation type

2009 ◽  
Vol 43 (1) ◽  
pp. 229-242
Author(s):  
Miklós Rontó ◽  
Natalia Shchobak
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Analytic Form ◽  
Successive Approximations ◽  
Two Dimensional ◽  
Linear Boundary ◽  
Scalar Parameter ◽  
The Given

Abstract We obtain some results concerning the investigation of two-dimensional non-linear boundary value problems of interpolation type. We show that it is useful to reduce the given boundary value problem, using an appropriate substitution, to a parametrized boundary value problem containing some unknown scalar parameter in the boundary conditions. To study the transformed parametrized problem, we use a method which is based upon special types of successive approximations constructed in an analytic form.

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

The M-matrix inverse problem for the Sturm—Liouville equation on graphs

2009 ◽  
Vol 139 (4) ◽  
pp. 775-796 ◽  
Author(s):  
Sonja Currie ◽  
Bruce A. Watson
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Inverse Spectral Problem ◽  
Boundary Value ◽  
Unitary Equivalence ◽  
Matrix Inverse ◽  
Normal Boundary ◽  
Sturm Liouville ◽  
The Given

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

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.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

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