On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation c(x)∂t2u−Δu=0 in a bounded smooth domain in ℝd from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data...

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Main Authors: Christian Daveau, Diane Manuel Douady, Abdessatar Khelifi
Format: Article
Language:English
Published: Hindawi Limited 2010-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2010/561395
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spelling doaj-fc6fce1840fe4d23b0a71bdfce892a7a2020-11-24T22:20:41ZengHindawi LimitedJournal of Applied Mathematics1110-757X1687-00422010-01-01201010.1155/2010/561395561395On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary MeasurementsChristian Daveau0Diane Manuel Douady1Abdessatar Khelifi2Département de Mathématiques, CNRS AGM UMR 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, FranceDépartement de Mathématiques, CNRS AGM UMR 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, FranceDépartement de Mathématiques, Université des Sciences de Carthage, 7021 Bizerte, TunisiaThis paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation c(x)∂t2u−Δu=0 in a bounded smooth domain in ℝd from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset Γ of the boundary determines explicitly the coefficient c provided that c is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient c from the knowledge of the difference between the local Dirichlet-to-Neumann maps.http://dx.doi.org/10.1155/2010/561395
collection DOAJ
language English
format Article
sources DOAJ
author Christian Daveau
Diane Manuel Douady
Abdessatar Khelifi
spellingShingle Christian Daveau
Diane Manuel Douady
Abdessatar Khelifi
On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
Journal of Applied Mathematics
author_facet Christian Daveau
Diane Manuel Douady
Abdessatar Khelifi
author_sort Christian Daveau
title On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
title_short On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
title_full On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
title_fullStr On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
title_full_unstemmed On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements
title_sort on a hyperbolic coefficient inverse problem via partial dynamic boundary measurements
publisher Hindawi Limited
series Journal of Applied Mathematics
issn 1110-757X
1687-0042
publishDate 2010-01-01
description This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation c(x)∂t2u−Δu=0 in a bounded smooth domain in ℝd from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset Γ of the boundary determines explicitly the coefficient c provided that c is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient c from the knowledge of the difference between the local Dirichlet-to-Neumann maps.
url http://dx.doi.org/10.1155/2010/561395
work_keys_str_mv AT christiandaveau onahyperboliccoefficientinverseproblemviapartialdynamicboundarymeasurements
AT dianemanueldouady onahyperboliccoefficientinverseproblemviapartialdynamicboundarymeasurements
AT abdessatarkhelifi onahyperboliccoefficientinverseproblemviapartialdynamicboundarymeasurements
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