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

Abderafik Benrabah 1 , Nadjib Boussetila 2  and Faouzia Rebbani 3
  • 1 University of 8 Mai 1945, P.O. Box 401, Guelma, Algeria
  • 2 University of 8 Mai 1945, P.O. Box 401, Guelma, Algeria
  • 3 Applied Mathematics Laboratory, University of Badji Mokhtar, P.O. Box 12, Annaba, Algeria
Abderafik Benrabah
  • Corresponding author
  • 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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, Nadjib Boussetila
  • 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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and Faouzia Rebbani

Abstract

The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u′′(t)-Au(t)=0, 0<t<T, u(0)=φ, u(0)=0, where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard []. We shall give a new regularization method for this problem when the operator A is replaced by Aα=A(I+αA)-1 and u(0)=φ 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.

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