Abstract:
An inverse problem of determining the
two-dimensional kernel of integro-differential wave equation in
medium of weak horizontal properties is considered. Herein the
initial data are equal to zero. The boundary condition of Neyman
type is given at the boundary of semi-plane is an impulse
function. As an additional information the semi-plane line mode is
given. It is assumed that the unknown kernel has the form of
$K(t)=K_0(t)+\varepsilon x K_1(t)+\dots$, where $\varepsilon$ is
a small parameter. In the work, the method of finding $K_0$, $K_1$
with precision correction, having the order $O(\varepsilon^2)$ is
developed. For this, by Fourier transformation the problem is
brought to the sequence of two one-dimensional inverse problems of
determining $K_0$, $K_1$. The first inverse problem for $K_0$ is
reduced to the system of nonlinear integral equations of Volterra
type relative to the unknown functions, and the second being
brought to the system of linear integral equations. Theorems that
characterize the unique solvability of determining unknown
functions for any fixed intercept are proved.
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