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A stability estimate for a solution to the problem of determination of two coefficients of a hyperbolic equation
D. I. Glushkovaa,
V. G. Romanovb a Novosibirsk State University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
We consider the problem of determination of two coefficients
$\sigma(x)$ and
$q(x)$ in a hyperbolic equation. Here
$\sigma(x)$ is the coefficient of the first derivative with respect to
$t$ and
$q(x)$ is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk
$D$. Oscillations are excited by the impulse function
$\delta(t)\delta(x\cdot\nu)$ supported on the straight line
$t=0$,
$x\cdot\nu=0$. Here
$\nu$ is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside
$D$ is measured at the points of the boundary of
$D$ together with the normal derivative on some time interval of a fixed length
$T$ for two different values of the parameter
$\nu$. We prove that, for a sufficiently large
$T$, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.
Keywords:
inverse problem, hyperbolic equation, stability, uniqueness.
UDC:
517.958 Received: 23.12.2002
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