On an inverse problem for 1-dimensional wave equation. (English) Zbl 0682.35106

Zhang, Guanquan

Sci. China, Ser. A 32, No. 3, 257-274 (1989).

By analyzing the propagation of discontinuities the author transformed the inverse problem for one dimensional acoustic wave equation (to determine the propagation velocity from impulse response) to a particular initial value problem of a semilinear system of P.D.E.. He established some a priori estimates for such a system. From these he shows that the stability behavior of its solutions is essentially dependent on the total variation of logarithm of propagation velocity and with the help of Picard iteration he proved that the propagation velocity can always be recovered from the impulse response. It should be pointed out here that the validity of these results need the first derivative of the logarithm of propagation velocity is in \(L^ 1\).

Reviewer: H.Ding

MSC:

35R30	Inverse problems for PDEs
35L05	Wave equation
35G25	Initial value problems for nonlinear higher-order PDEs
35B35	Stability in context of PDEs

Keywords:

transformation; Picard iteration; impulse response