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J. Hadamard, “Lectures on the Cauchy Problem in Linear Partial Differential Equations,” Yale University Press, New Haven, 1923.

has been cited by the following article:

• TITLE: Data Recovering Problem Using a New KMF Algorithm for Annular Domain

  AUTHORS: Chakir Tajani, Jaafar Abouchabaka, Otman Abdoun

  KEYWORDS: Inverse Problem; Annular Domain; Laplace Equation; Iterative Method; Freefem

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.2 No.2, June 21, 2012

  ABSTRACT: This paper is interested at the Cauchy problem for Laplace’s equation, which is to recover both Dirichlet and Neumann conditions on the inaccessible part of the boundary (inner part) of an annular domain from the over specified conditions on the accessible one (outer part). This work is an extension of the proposed algorithm for a unit circle [1] to annular domain, where, we describe an alternating formulation of the KMF algorithm proposed by Kozlov, Mazya and Fomin, and its relationship with the standard formulation. The new KMF algorithm ameliorates the accuracy of the solution and reduces the number of iterations required to achieve convergence. In the last part, the discussion of the error estimation of solution is presented and some numerical tests, using the software Freefem are given to show the efficiency of the proposed method.