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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 947379, 11 pages
http://dx.doi.org/10.1155/2013/947379
Research Article

Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

1Department of Mathematics, University 8 Mai 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
2Applied Mathematics Laboratory, University Badji Mokhtar Annaba, P.O. Box 12, 23000 Annaba, Algeria

Received 10 April 2013; Accepted 28 September 2013

Academic Editor: Alberto Fiorenza

Copyright © 2013 Nadjib Boussetila et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.