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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Regularization method for an ill-posed Cauchy problem for elliptic equations

Abderafik Benrabah
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• University of 8 Mai 1945, P.O. Box 401, Guelma 24000, Applied Mathematics Laboratory, University Badji Mokhtar, P.O. Box 12, Annaba 23000, Algeria
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• University of 8 Mai 1945, P.O. Box 401, Guelma 24000, Applied Mathematics Laboratory, University Badji Mokhtar, P.O. Box 12, Annaba 23000, Algeria
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/ Faouzia Rebbani
Published Online: 2016-07-30 | DOI: https://doi.org/10.1515/jiip-2015-0075

Abstract

The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by ${u}^{\prime \prime }\left(t\right)-Au\left(t\right)=0$, $0, $u\left(0\right)=\phi$, ${u}^{\prime }\left(0\right)=0$, where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard [23]. We shall give a new regularization method for this problem when the operator A is replaced by ${A}_{\alpha }=A{\left(I+\alpha A\right)}^{-1}$ and $u\left(0\right)=\phi$ is replaced by a nonlocal condition. We show the convergence of this method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.

MSC 2010: 35R30; 47A52; 34K07; 35A35; 34B10

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Revised: 2016-03-21

Accepted: 2016-06-19

Published Online: 2016-07-30

Published in Print: 2017-06-01

The work described in this paper was supported by the Ministry of Higher Education and Scientific Research of Algeria (CNEPRU Project B01120090003).

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 3, Pages 311–329, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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