Recovery of the heat coefficient by two measurements

Amin Boumenir; Vu Kim Tuan

doi:10.3934/ipi.2011.5.775

Article Contents

2011, Volume 5, Issue 4: 775-791. Doi: 10.3934/ipi.2011.5.775

This issue Previous Article Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map Next Article Uniqueness in inverse transmission scattering problems for multilayered obstacles

Recovery of the heat coefficient by two measurements

Amin Boumenir^1, and
Vu Kim Tuan^2,

1.
Department of Mathematics, Kuwait University, Safat -13060
2.
Department of Mathematics, University of West Georgia, Carrollton, GA 30118

Received: March 2010

Revised: August 2011

Published: November 2011

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Abstract

We prove that it takes at most two measurements on the boundary to recover the heat coefficient of a one dimensional heat equation if its lower bound is known. Otherwise a finite number of measurements is needed. We also provide a new constructive algorithm for its recovery. Using asymptotics of eigenfunctions of the associated Sturm-Liouville problem we show that a hot spot initial condition generates all, except maybe a finite number of boundary spectral data. Then a counting argument based on the method of false position helps search for the number of missing boundary spectral data which is then unraveled by a finite number of measurements. Finally, we show how the boundary spectral data is converted into spectral data, and the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators yields the reconstruction of the heat coefficient uniquely.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 34K29.

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