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

We study properties of 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)...

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Bibliographic Details
Published in:	Russian mathematics Vol. 53; no. 4; pp. 45 - 48
Main Author:	Klyuchev, V. V.
Format:	Journal Article
Language:	English
Published:	Heidelberg Allerton Press, Inc 01.04.2009
Subjects:	Mathematics Mathematics and Statistics sectorial condition finite-difference approximation methods sourcewise representation Banach space ill-posed problem Cauchy problem
ISSN:	1066-369X, 1934-810X
Online Access:	Get full text
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Summary:	We study properties of 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 an 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.
ISSN:	1066-369X 1934-810X
DOI:	10.3103/S1066369X09040082