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 Izv. Vyssh. Uchebn. Zaved. Mat., 2009, Number 4, Pages 56–60 (Mi ivm1321)

Brief communications

The necessary and sufficient conditions for the qualified convergence of difference methods for approximate solution of the ill-posed Cauchy problem in a Banach space

V. V. Klyuchev

Mari State University

Abstract: We study properties of the finite-difference methods for approximate solution of the ill-posed Cauchy problem for a homogeneous equation of the first order with a sectorial operator in a Banach space. We obtain the necessary and sufficient conditions for the qualified (with respect to the step of grid) uniform (on a segment) convergence of approximations to the exact solution of the problem. These conditions represent a priori data about the segment, where a solution exists, or about the sourcewise representation of a certain value of the desired solution.

Keywords: Cauchy problem, ill-posed problem, finite-difference approximation methods, sectorial condition, Banach space, sourcewise representation.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:4, 45–48

Bibliographic databases:

UDC: 517.983

Citation: V. V. Klyuchev, “The necessary and sufficient conditions for the qualified convergence of difference methods for approximate solution of the ill-posed Cauchy problem in a Banach space”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 4, 56–60; Russian Math. (Iz. VUZ), 53:4 (2009), 45–48

Citation in format AMSBIB
\Bibitem{Kly09} \by V.~V.~Klyuchev \paper The necessary and sufficient conditions for the qualified convergence of difference methods for approximate solution of the ill-posed Cauchy problem in a~Banach space \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2009 \issue 4 \pages 56--60 \mathnet{http://mi.mathnet.ru/ivm1321} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2581475} \transl \jour Russian Math. (Iz. VUZ) \yr 2009 \vol 53 \issue 4 \pages 45--48 \crossref{https://doi.org/10.3103/S1066369X09040082}