Recovering time-dependent singular coefficients of the wave-equation-One Dimensional Case
Résumé
We consider the homogeneous wave equation in the rectangle (0, T )×(0, b), that is, in the one-dimensional space situation.
The conductivity depends on the two variables t, x of time and space, and represents an unknown moving inclusion inside
the background which has constant conductivity. The waves satisfy the homogeneous Dirichlet condition at x = b and
sufficiently smooth but unknown initial conditions at t = 0. We prove that the inclusion is determined by
the Dirichlet-to-Neumann mapping defined on the interface x = 0.
In fact, we show how the inclusion can be reconstructed from the detection of the singularities of the flux of special waves
knowing the singularities of their trace on the interface.
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