scholarly journals Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Rauf Kh. Amırov ◽  
A. Adiloglu Nabıev
Keyword(s):  
Inverse Problems ◽  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Weyl Function ◽  
Integral Representations ◽  
Asymptotic Formulas ◽  
Linearly Independent ◽  
Quadratic Pencil ◽  
Sturm Liouville

In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.

scholarly journals Inverse spectral problems for energy-dependent Sturm-Liouville equations with δ-interaction

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 2935-2946 ◽  
Author(s):  
Manaf Manafov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Spectral Characteristics ◽  
Weyl Function ◽  
Integral Representations ◽  
Inverse Spectral Problems ◽  
Spectral Problems ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Energy Dependent

In this study, inverse spectral problems for a energy-dependent Sturm-Liouville equations with ?-interaction on a finite interval are considered. Some useful integral representations for the solutions of the considered equation have been derived and using these, properties of the spectral characteristics of the boundary value problem are investigated. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

scholarly journals Inverse problems for a conformable fractional Sturm–Liouville operator

2020 ◽  
Vol 28 (6) ◽  
pp. 775-782
Author(s):  
İbrahi̇m Adalar ◽  
Ahmet Sinan Ozkan
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Type Theorem ◽  
Weyl Function ◽  
Uniqueness Theorems ◽  
Half Inverse Problem ◽  
Sturm Liouville ◽  
Derivatives Of

AbstractIn this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem.

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 Partial inverse problems for quadratic differential pencils on a graph with a loop

2020 ◽  
Vol 28 (3) ◽  
pp. 449-463 ◽  
Author(s):  
Natalia P. Bondarenko ◽  
Chung-Tsun Shieh
Keyword(s):  
Inverse Problems ◽  
A Priori ◽  
Spectral Characteristics ◽  
The Other ◽  
Boundary Edge ◽  
Partial Inverse ◽  
Quadratic Pencil ◽  
Sturm Liouville ◽  
Differential Pencils ◽  
Liouville Operators

AbstractIn this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

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