Abstract: 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.
For citation: Vatulyan A. O., Vasil'ev L. V., Yurov V. O. Restoration of
Parameters in the Boundary Conditions for an Inhomogeneous
Cylindrical Waveguide. Vladikavkazskij matematicheskij zhurnal
[Vladikavkaz Math. J.], vol. 20, no. 2, pp.29-37.
1. Yurko V. A. Vvedenie v teoriyu obratnyh spektral'nyh zadach
[Introduction to the Theory of Inverse Spectral Problems], Moscow,
Fizmatlit, 2006, 384 p. (in Russian).
2. Gladwell G. M. L., Movahhedy M. Reconstruction of Mass-Spring
System From Spectral Data I: Theory. Inverse Problems in
Engineering, 1995, vol. 1, no. 2, pp. 179-189. DOI:
3. Akhtyamov A. M. Teoriya identifikacii kraevyh uslovij i eyo
prilozheniya [Theory of Identification of Boundary Conditions and
its Applications], Moscow, Fizmatlit, 2009, 272 p. (in Russian).
4. Akhtyamov A. M., Muftahov A. V., Akhtyamova A. A. On the
Determination of the Fixation and Loading of one of the Ends of a
Rod According to the Natural Frequencies of its Oscillations. Vestn.
Udmurtsk. un-ta. Matem. Mekh. Komp'yut. nauki [Herald of the Udmurt
University. Mathematics. Mechanics. Computers of Science], 2013.
vol. 3, pp. 114-129 (in Russian).
5. Aitbaeva A. A., Ahtyamov A. M. Identification of Fixedness and
Loading of an End of the Euler–Bernoulli Beam by the Natural
Frequencies of its Vibrations. Journal of Applied and Industrial
Mathematics, 2017, vol. 11, no. 1, pp. 1-7. DOI:
6. Vatulyan A. O., Vasilev L. V. Determination of the Parameters of
an in\-Homogeneous Beam Elastic Fixation. Ekologicheskij vestnik
nauchnyh centrov CHEHS [Ecological Bulletin of Research Centers of
the Black Sea Economic Cooperation], 2015, no. 3, pp. 14-20 (in
7. Morassi A., Dilena M. On Point Mass Identification in Rods and
Beams from Minimal Frequency Measurements. Inverse Problems in
Engineering, 2002, vol. 10, no. 3, pp. 183-201. DOI:
8. Grinchenko V. T., Meleshko V. V. Garmonicheskie kolebaniya i
volny v uprugih telah [Harmonic Oscillations and Waves in Elastic
Bodies], Kiev, "Nauk. dumka", 1981, 284 p. (in Russian).
9. Shardakov I. N., Shestakov A. P., Cvetkov R. V. Mathematical
Modeling of Wave Processes in Main Pipelines. XVII Zimnyaya shkola
po mekhanike sploshnyh sred. Perm'. 18-22 Fevralya 2013g. Tezisy
dokladov [XVII Winter School on the Mechanics of Continuous Media.
Perm, February 18-22, 2013. Abstracts of Reports], IMSS Uro RAN,
2013, 252 p. (in Russian).
10. Vatul’yan A. O., Yurov V. O. On the Dispersion Relations for an
Inhomogeneous Waveguide With Attenuation. Mech. Solids, 2016, vol.
51, no. 5, p. 576-582. DOI: 10.3103/S0025654416050101
11. Abzalimov R. R., Akhtyamov A. M. Diagnostika i vibrozashchita
truboprovodnykh sistem i khranilishch [Diagnosis and Vibration
Protection of Pipeline Systems and Storags], Ufa State Petroleum
Technological University, 2006, 118 p. (in Russian).
12. VatulТyan A. O., Yurov V. O. On the Properties of a Dispersion
Set for an Inhomogeneous Cylindrical Waveguide, Vladikavkazskij
matematicheskij zhurnal [Vladikavkaz Math. J.],† 2018, vol. 20, no.
1, pp. 50-60 (in Russian). DOI 10.23671/VNC.2018.1.11397.
13. Aybinder A. B. Raschet magistral'nykh i promyslovykh
truboprovodov na prochnost' i ustoychivost': spravochnoe posobie
[Calculation of Main and Field Pipelines for Strength and Stability:
Reference Book], Moscow, Nedra, 1991, 287 p. (in Russian).
14. Kalitkin N. N. Chislennye metody: uchebnoe posobie [Numerical
Methods. Tutorial], SPb., BHV-SPb,† 2011, 592 p. (in Russian).