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


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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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.