Restoration of parameters in the boundary conditions for an inhomogeneous cylindrical waveguide

A. O. Vatulyan ; L. V. Vasil'ev ; V. O. Yurov

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 29-37

Résumé

Identification of different characteristics of solid bodies according to the acoustic sounding data has been increasingly attracting the attention of researchers in recent years. In the present paper, we investigate a new inverse problem on determining two parameters (bedding values) entering into the boundary conditions for the boundary-value problem. The boundary problem describes the waves propagation in a hollow inhomogeneous cylindrical waveguide located in a medium. We have performed the solution of this problem previously, we have studied the structure of the dispersion set and obtained the several formulae. These formulae correlate with spectral parameters and bedding values. We have treated the auxiliary Cauchy problems which automatically satisfy boundary conditions on the cylinder’s internal boundary. Solution of boundary problem is found in the form of a linear combination of auxiliary problems. Boundary conditions at the outer boundary are satisfied. For the existence of a nontrivial solution, it is required that the determinant of the emergent system of algebraic equations is zero. Reconstruction of bedding values have been carried out from information on two points of the dispersion set; at that, the approach to solving the inverse problem did not require the explicit representation of the dispersion set. The solution of the inverse problem does not always satisfy a priori information on the non-negativity of the bedding values. In order to obtain a unique reconstruction of the parameters, a unicity theorem is formulated. At the initial stage, the theorem allows to filter out pairs of points of the dispersion set for which there is no solution or it is not unique. Computational experiments show the prevalence of the situation when the dispersion curves can be carried out uniquely through two given points of the dispersion set. Within the framework of the work, an effective method of selecting a pair of parameters with a small error in the input data is to consider the third point of the dispersion set. It is revealed that the reconstruction method presented allows to restore the required parameters with a high enough accuracy.

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