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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2015, Volume 8, Issue 2, Pages 69–80 (Mi vyuru264)  

This article is cited in 1 scientific paper (total in 1 paper)

Mathematical Modelling

On perturbation method for the first kind equations: regularization and application

I. R. Muftahova, D. N. Sidorovabc, N. A. Sidorovc

a Irkutsk State Technical University, Irkutsk, Russian Federation
b Melentiev Energy Systems Institute of Seberian Branch of Russian Academy of Sciences, Irkutsk, Russian Federation
c Irkutsk State University, Irkutsk, Russian Federation

Abstract: One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations $Ax=f$ with bounded operator $A.$ We assume that we know the operator $\tilde{A}$ and source function $\tilde{f}$ only such as $||\tilde{A} - A||\leq \delta,$ $||\tilde{f}-f||< \delta$, The regularizing equation $\tilde{A}x + B(\alpha)x = \tilde{f}$ possesses the unique solution. Here $\alpha \in S$, $S$ is assumed to be an open space in $\mathbb{R}^n$, $0 \in \overline{S}$, $\alpha= \alpha(\delta)$. As result of proposed theory, we suggest a novel algorithm providing accurate results even in the presence of a large amount of noise.

Keywords: operator and integral equations of the first kind; stable differentiation; perturbation method, regularization parameter.

DOI: https://doi.org/10.14529/mmp150206

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Bibliographic databases:

UDC: 517.983
MSC: 47A52
Received: 11.03.2015
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Citation: I. R. Muftahov, D. N. Sidorov, N. A. Sidorov, “On perturbation method for the first kind equations: regularization and application”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:2 (2015), 69–80

Citation in format AMSBIB
\Bibitem{MufSidSid15}
\by I.~R.~Muftahov, D.~N.~Sidorov, N.~A.~Sidorov
\paper On perturbation method for the first kind equations: regularization and application
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2015
\vol 8
\issue 2
\pages 69--80
\mathnet{http://mi.mathnet.ru/vyuru264}
\crossref{https://doi.org/10.14529/mmp150206}
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\elib{https://elibrary.ru/item.asp?id=23442154}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. A. Sidorov, “Classic solutions of boundary value problems for partial differential equations with operator of finite index in the main part of equation”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 27 (2019), 55–70  mathnet  crossref
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