Inverse Problem for Sturm-Liouville Operators with Singularity and Discontinuity Conditions Inside an Interval

2014 ◽  
Vol 6 (1) ◽  
pp. 95-117
Author(s):  
Guldu, R.Kh
Keyword(s):  
Inverse Problem ◽  
Sturm Liouville ◽  
Discontinuity Conditions ◽  
Liouville Operators

scholarly journals Inverse problem for Sturm-Liouville operators on a curve

2019 ◽  
Vol 50 (3) ◽  
pp. 349-359
Author(s):  
Andrey Aleksandrovich Golubkov ◽  
Yulia Vladimirovna Kuryshova
Keyword(s):  
Inverse Problem ◽  
Asymptotic Behavior ◽  
Transfer Matrix ◽  
Liouville Equation ◽  
Inverse Spectral Problem ◽  
Rectifiable Curve ◽  
Uniqueness Of The Solution ◽  
Sturm Liouville ◽  
Discontinuity Conditions ◽  
Liouville Operators

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

scholarly journals Inverse Spectral Problem for Integrodifferential Sturm-Liouville Operators with Discontinuity Conditions

2018 ◽  
Vol 64 (3) ◽  
pp. 427-458 ◽  
Author(s):  
S A Buterin
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Global Solvability ◽  
Convolution Integral ◽  
Main Equation ◽  
Convolution Integral Operator ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Discontinuity Conditions ◽  
Liouville Operators

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

2020 ◽  
Vol 44 (1) ◽  
pp. 124-139
Author(s):  
Ran Zhang ◽  
Natalia Pavlovna Bondarenko ◽  
Chuan Fu Yang
Keyword(s):  
Inverse Problem ◽  
Sturm Liouville ◽  
Liouville Operators

scholarly journals Recovery of the matrix quadratic differential pencil from the spectral data

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Natalia Bondarenko
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Spectral Data ◽  
Quadratic Differential ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
The Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Liouville Operators

AbstractWe consider a pencil of matrix Sturm–Liouville operators on a finite interval. We study the properties of its spectral characteristics and inverse problems that consist in the recovering of the pencil by the spectral data, that is, eigenvalues and so-called weight matrices. This inverse problem is reduced to a linear equation in a Banach space by the method of spectral mappings. A constructive algorithm for the solution of the inverse problem is provided.

scholarly journals An inverse spectral problem for non-selfadjoint Sturm-Liouville operators with nonseparated boundary conditions

2012 ◽  
Vol 43 (2) ◽  
pp. 289-299 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Uniqueness Theorem ◽  
Inverse Spectral Problem ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Nonseparated Boundary Conditions ◽  
Sturm Liouville ◽  
Liouville Operators

Non-selfadjoint Sturm-Liouville operators on a finite interval with nonseparated boundary conditions are studied. We establish properties of the spectral characteristics and investigate an inverse problem of recovering the operators from their spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing the solution.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation

2020 ◽  
Vol 28 (5) ◽  
pp. 651-658
Author(s):  
Shavkat Alimov ◽  
Ravshan Ashurov
Keyword(s):  
Inverse Problem ◽  
Fractional Derivative ◽  
Adjoint Operator ◽  
Fourier Method ◽  
Fixed Time ◽  
Sturm Liouville ◽  
Fractional Differential ◽  
Time Fractional Derivative ◽  
Subdiffusion Equation ◽  
Liouville Operators

AbstractAn inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of {\|Au(t)\|}, where {u(t)} is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and many others.

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