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Data Recovering Problem Using a New KMF Algorithm for Annular Domain

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DOI: 10.4236/ajcm.2012.22012    3,172 Downloads   6,234 Views   Citations

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

C. Tajani, J. Abouchabaka and O. Abdoun, "Data Recovering Problem Using a New KMF Algorithm for Annular Domain," American Journal of Computational Mathematics, Vol. 2 No. 2, 2012, pp. 88-94. doi: 10.4236/ajcm.2012.22012.

References

[1] C. Tajani and J. Abouchabaka, “An Alternating KMF Algorithm to Solve the Cauchy Problem for Laplace’s Equation,” International Journal of Computer Applications, Vol. 38, No. 8, 2012, pp. 30-36.
[2] X. Yang, M. Choulli and J. Cheng, “An Iterative BEM for the Inverse Problem of Detecting Corrosion in a Pipe,” Numerical Mathematics: A Journal of Chinese Universities, Vol. 14, No. 3, 2005, pp. 252-266.
[3] J. Leblond, M. Mahjoub and J. R. Partington, “Analytic Extensions and Cauchy-Type Inverse Problems on Annular Domains: Stability Results,” Journal of Inverse and III-posed Problems, Vol. 14, No. 2, 2006, pp. 189-204. doi:10.1515/156939406777571049
[4] M. Jaoua, J. Leblond and M. Mahjoub, “Robust Numerical Algorithms Based on Analytic Approximation for the Solution of Inverse Problems in Annular Domains,” IMA Journal of Applied Mathematics, Vol. 74, No. 4, 2009, pp. 481-506.
[5] J. Hadamard, “Lectures on the Cauchy Problem in Linear Partial Differential Equations,” Yale University Press, New Haven, 1923.
[6] V. Isakov, “Inverse Problems for Partial Differential Equations,” Applied Mathematical Sciences, Springer, New York, 1998.
[7] L. E. Payne, “Improperly Posed Problems in Partial Differential Equations,” SIAM, Philadelphia, 1975.
[8] M. M. Lavrentiev, “Some Improperly Posed Problems of Mathematical,” Springer-Verlag, Berlin, 1967.
[9] R. Lattes and J. L. Lions, “Mèthode de Quasi-reversibilité et Applications, ” Dunod, Paris, 1967.
[10] A. Cimetière, F. Delvare, M. Jaoua and F. Pons, “Solution of the Cauchy Problem Using Iterated Tikhonov Regularization,” Inverse Problems, Vol. 17, No. 3, 2001, pp. 553-570. doi:10.1088/0266-5611/17/3/313
[11] D. Lesnic, L. Elliott and D. B. Ingham, “An Iterative Boundary Element Method for Solving Numerically the Cauchy Problem for the Laplace Equation,” Engineering Analysis with Boundary Elements, Vol. 20, No. 2, 1997, pp. 123-133. doi:10.1016/S0955-7997(97)00056-8
[12] V. A. Kozlov, V. G. Maz’ya and D. V. Fomin, “An Iterative Method for Solving the Cauchy Problem for Elliptic Equation,” Computational Mathematics and Mathematical Physics, Vol. 31, No. 1, 1991, pp. 45-52.
[13] M. Jourhmane and A. Nachaoui, “An Alternating Method for an Inverse Cauchy Problem,” Numerical Algorithms, Vol. 21, No. 1-4, 1999, pp. 247-260. doi:10.1023/A:1019134102565
[14] C. Tajani and J. Abouchabaka, “Missing Boundary Data Reconstruction by an Alternating Iterative Method,” International Journal of Advances in Engineering and Technology, Vol. 2, No. 1, 2012, pp. 578-586.
[15] M. Jourhmane, D. Lesnic and N. S. Mera, “Relaxation Procedures for an Iterative Algorithm for Solving the Cauchy Problem for the Laplace Equation,” Engineering Analysis with Boundary Elements, Vol. 28, No. 6, 2004, pp. 655-665. doi:10.1016/j.enganabound.2003.07.002
[16] C. Tajani, J. Abouchabaka and O. Abdoun, “Numerical Simulation of an Inverse Problem: Testing the Influence Data,” Proceedings of the 4th International Conference on Approximation Methods and Numerical Modeling in Environment and Natural, Saidia, 23-26 May 2011, pp. 29-32.

  
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