A problem of determining

D. K. Durdiev ; Z. R. Bozorov

Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 209-221

Résumé

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.