Far Eastern Mathematical Journal

To content of the issue


A problem of determining the kernel of integrodifferential wave equation with weak horizontal properties


Durdiev D. K., Bozorov Z. R.

2013, issue 2, P. 209-221


Abstract
An inverse problem of determining the two-dimensional kernel of integrodifferential 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 $K(x,t)=K_0(t)+\varepsilon x K_1(t)+…$, 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.

Keywords:
wave equation, inverse problem, delta function, Fourier transformation, integral equation

Download the article (PDF-file)

References

[1] V. G. Romanov, Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984.
[2] V. G. Romanov, Ustoichivost' v obratnykh zadachakh, Nauchnyi mir, M., 2005.
[3] S. I. Kabanikhin, Obratnye i nekorreknye zadachi, Novosibirsk, 2009.
[4] D. K. Durdiev, “Mnogomernaia obratnaia zadacha dlia uravneniia s pamiat'iu”, Sib. matem. zhurn., 35:3 (1994), 574–582.
[5] D. K. Durdiev, “Some multidimensional inverse problems of memory determination in hyperbolic equations”, Zh. Mat. Fiz. Anal. Geom., 3:4 (2007), 411-423.
[6] A. S. Blagoveshchenskii, D. A. Fedorenko, “Uravneniia akustiki v slabo gorizontal'no-neodnorodnoi srede”, Zapiski nauchnykh seminarov POMI, 354 (2008), 81–99.
[7] R. Kurant, Uravneniia s chastnymi proizvodnymi, Mir, M., 1964.
[8] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki. T. 2, Mir, M., 1978.
[9] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsional'nogo analiza, Nauka, M., 1976.

To content of the issue