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Relaxation of Alternating Iterative Algorithms for the Cauchy Problem Associated with the Modified Helmholtz Equation

B. Tomas Johansson1, Liviu Marin2
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. E-mail: b. t.
Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O. Box 1-863, 010141 Bucharest, Romania. E-mails:;

Computers, Materials & Continua 2009, 13(2), 153-190.


We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of Kozlov, Maz'ya and Fomin(1991) applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.


Helmholtz Equation, Inverse Problem, Cauchy Problem, Alternating Iterative Algorithms, Relaxation Procedure, Boundary Element Method (BEM).

Cite This Article

B. T. . Johansson and L. . Marin, "Relaxation of alternating iterative algorithms for the cauchy problem associated with the modified helmholtz equation," Computers, Materials & Continua, vol. 13, no.2, pp. 153–190, 2009.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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