Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation

AbstractWe prove a global uniqueness theorem of reconstruction of a matrix-potential {a(x,t)} of one-dimensional wave equation {\square u+au=0}, {x>0,t>0}, {\square=\partial_{t}^{2}-\partial_{x}^{2}} with zero Cauchy data for {t=0} and given Cauchy data for {x=0}, {u(0,t)=0}, {u_{x}(0,t)=g(t)}. Here {u,a,f}, and g are {n\times n} smooth real matrices, {\det(f(0))\neq 0}, and the matrix {\partial_{t}a} is known.

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A one-dimensional inverse problem for the wave equation

One-Dimensional Inverse Problems of Mathematical Physics - American Mathematical Society Translations: Series 2 ◽

10.1090/trans2/130/04 ◽

1986 ◽

pp. 27-58

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

One Dimensional

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Inversion of the Initial Value for a Time-Fractional Diffusion-Wave Equation by Boundary Data

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2018-0194 ◽

2020 ◽

Vol 20 (1) ◽

pp. 109-120 ◽

Cited By ~ 1

Author(s):

Suzhen Jiang ◽

Kaifang Liao ◽

Ting Wei

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

Fractional Diffusion ◽

Initial Value ◽

Diffusion Wave ◽

One Dimensional ◽

Finite Dimensional Approximation ◽

Finite Dimensional ◽

Dimensional Approximation ◽

Continuation Technique

AbstractIn this study, we consider an inverse problem of recovering the initial value for a multi-dimensional time-fractional diffusion-wave equation. By using some additional boundary measured data, the uniqueness of the inverse initial value problem is proven by the Laplace transformation and the analytic continuation technique. The inverse problem is formulated to solve a Tikhonov-type optimization problem by using a finite-dimensional approximation. We test four numerical examples in one-dimensional and two-dimensional cases for verifying the effectiveness of the proposed algorithm.

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Stability result for two coefficients in a coupled hyperbolic-parabolic system

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2015-0017 ◽

2017 ◽

Vol 25 (3) ◽

Cited By ~ 3

Author(s):

Patricia Gaitan ◽

Hadjer Ouzzane

Keyword(s):

Inverse Problem ◽

Parabolic System ◽

Stability Result ◽

Fixed Time ◽

Spatial Domain ◽

Carleman Estimates ◽

Observation Data ◽

Lipschitz Stability

AbstractThis work is concerned with the study of the inverse problem of determining two coefficients in a hyperbolic-parabolic system using the following observation data: an interior measurement of only one component and data of two components at a fixed time over the whole spatial domain. A Lipschitz stability result is proved using Carleman estimates.

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