Recovery of Inhomogeneity from Output Boundary Data

Kravchenko, Vladislav V.; Khmelnytskaya, Kira V.; Cetinkaya, F. Ayca

doi:10.3390/math10224349

Cited by 8 publications

(2 citation statements)

References 19 publications

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“…In those papers, the system of linear algebraic equations was obtained with the aid of the Gelfand-Levitan integral equation. In [29], another approach, based on the consideration of the eigenfunctions normalized at the opposite endpoints, was developed, and this idea was used in [13,14,34] and is used in the present work when solving the two-spectrum inverse problems on the edges.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction techniques for quantum trees

Avdonin,

Khmelnytskaya,

Kravchenko

2024

Math Methods in App Sciences

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The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

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Section: Introductionmentioning

confidence: 99%

Reconstruction techniques for quantum trees

Avdonin,

Khmelnytskaya,

Kravchenko

2024

Math Methods in App Sciences

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“…In the paper [15], the authors consider the Sturm-Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. They recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter.…”

mentioning

confidence: 99%

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

Sitnik

2023

Mathematics

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Sinc method in spectrum completion and inverse Sturm–Liouville problems

Kravchenko,

Murcia‐Lozano

2024

Math Methods in App Sciences

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Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

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Recovery of Inhomogeneity from Output Boundary Data

Cited by 8 publications

References 19 publications

Reconstruction techniques for quantum trees

Reconstruction techniques for quantum trees

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

Sinc method in spectrum completion and inverse Sturm–Liouville problems

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