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

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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UDC: 517.983
MSC: 47A52
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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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000422200600006} \elib{https://elibrary.ru/item.asp?id=23442154} 

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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
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