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Two regularization methods for a class of inverse boundary value problems of elliptic type

Abdallah Bouzitouna1, Nadjib Boussetila2* and Faouzia Rebbani1

Author Affiliations

1 Applied Mathematics Laboratory, University Badji Mokhtar Annaba, P.O. Box 12, Annaba, 23000, Algeria

2 Department of Mathematics, 8 Mai 1945 Guelma University, P.O. Box 401, Guelma, 24000, Algeria

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Boundary Value Problems 2013, 2013:178  doi:10.1186/1687-2770-2013-178

Published: 2 August 2013

Abstract

This paper deals with the problem of determining an unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1">View MathML</a> in the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M4">View MathML</a>, with the aid of an extra measurement at an internal point. It is well known that such a problem is severely ill-posed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the ill-posed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the Kozlov-Maz’ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

MSC: 35R25, 65J20, 35J25.

Keywords:
ill-posed problems; elliptic problems; cut-off spectral regularization; iterative regularization