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7vector case; three and six for 2-tensor case. The recovery algorithms are based on the approximate inverse
method [2, 3].
This research was partially supported by RFBR and DFG according to the research project 19-51-12008.
References
1. Sharafutdinov V. A. Slice-by-slice reconstruction algorithm for vector tomography with incomplete data. Inverse
Problems. 2007. V. 23, No 6. P. 2603�2627.
2. Louis A. K., Maass P. A mollifier method for linear operator equations of the first kind. Inverse Problems. 1990.
V. 6, No 3. P. 427�440.
3. Derevtsov E. Yu., Louis A. K., Maltseva S. V., Polyakova A. P., Svetov I. E. Numerical solvers based on the method of
approximate inverse for 2D vector and 2-tensor tomography problems. 2017. Inverse Problems. V 33, No 12, 124001.
An inverse problem for a system of nonlinear parabolic equations
E. V. Tabarintseva
South Ural State University, Chelyabinsk
Email: eltab@rambler.ru
DOI 10.24412/cl-35065-2021-1-02-16
An inverse problem for a system of nonlinear parabolic equations is considered in the present paper.
Namely, it is required to restore the initial condition by a given time-average value of the solution to the
system of the nonlinear parabolic equations. An exact in the order error estimate of the optimal method for
solving the inverse problem through the error estimate for the corresponding linear problem is obtained. A
stable approximate solution to the unstable nonlinear problem under study is constructed by means of the
projection regularization method which consists of using the representation of the approximate solution as a
partial sum of the Fourier series and the auxiliary boundary conditions method. An exact in the order estimate
for the error of the projection regularization method is obtained on one of the standard correctness classes. As
a consequence, it is proved the optimality of the projection regularization method. As an example of a
nonlinear system of parabolic equations, which has important practical applications, a spatially distributed
model of blood coagulation is considered. Numerical examples are given to confirm the theoretical results.
This work was supported by Act 211 Government of the Russian Federation, contract � 02.A03.21.0011.
References
1. Ivanov V. K., Vasin V.V., Tanana V.P. The Theory of Linear Ill-Posed Problems and Its Applications. VCP, 2002.
2. Denisov A. M. Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary
condition // Comput. Math. Math. Phys. 2014. V. 54 (10), P. 1513-1521.
3. Lobanov A. I., Starogilova T.K., Guria G.T. Numerical investigation of pattern formation in blood coagulation //
Matematicheskoe modelirovanie. 1997. V. 9 (3), P. 83-95.
Determination of the variable density of the rod from the natural frequencies of longitudinal vibrations
I. M. Utyashev
Mavlyutov Institute of Mechanics UFRC RAS, Ufa
Email: utyashevim@mail.ru
DOI 10.24412/cl-35065-2021-1-02-17
Rods of various configurations are elements of many structures and machines. Therefore, the acoustic and
vibration diagnostics of such parts has been widely developed [1-3]. The paper considers the problem of de-
termining the variable density of the rod from the natural frequencies of longitudinal vibrations. It is assumed
that the density changes along the axis and is described by a power function. This approach allows one to de-
termine the law of density variation from a finite set of eigenvalues. The results of the study can find applica-
tions for finding hidden defects in steel and composite rods, which arise during the production process or due
to corrosion.
References
1. Lokshin A.A., Sagomonyan E.A. On the motion of a rod of variable density // Moscow University Bulletin. Ser.1.
Mathematics, mechanics, 1987. No. 3. P. 93�95.
2. Ponomarev S.D., Biderman V.L., Makushin V.M. Strength calculations in mechanical engineering. M.: Mashgiz.
1959.
3. Kravchuk AS, Kravchuk AI, Tarasyuk IA Longitudinal vibrations of layered and structurally inhomogeneous
composite rods. Vestnik VolGU. Series 1: Mathematics. Physics. 2016. No. 3 (34). P. 41-52.
Method for improving the reconstruction quality in pulsed x-ray tomography
I. P. Yarovenko1,2, I. V. Prokhorov1,2
1Institute of Applied Mathematics FEB RAS
2Far-Eastern Federal University, Vladivostok
Email: prokhorov@iam.dvo.ru
DOI 10.24412/cl-35065-2021-1-02-19
Recently, successful development of short-pulse X-ray sources along with appearance of high timing reso-
lution detectors give wider opportunities to construct new schemes of tomographic imaging [1]. The paper is
devoted to a method of improving quality of imaging based on serial irradiation of medium with x-ray pulses of
different durations. We investigate an inverse problem for non-stationary radiation transfer equation that
consists of determining an attenuation coefficient. The time-integral intensity at the domain boundary is con-
sidered to be known. The vanishing rate of the scattered component was estimated with the probe pulse dura-
tion. The estimation is used to extrapolate the transfer equation solution and to construct an approximate
method for medium imaging. Extrapolation methods are not unfrequently used to improve reconstruction
quality of various imaging modalities. However, algorithms applied to resulting image or its spectrum are pre-
vailing. Our approach differs from the previous methods in that we apply an extrapolation method to projec-
tion data with approximation function taking into account the nature and structural features of the measured
signal. We carried out numerical experiments using a well-known digital phantom [2]. The results obtained
show the effectiveness of the algorithm proposed to suppress a scattering effect in X-ray tomography.
The reported study was funded by RFBR (project number 20-01-00173) and the Ministry of Education and Science of
the Russian Federation (Additional Agreement no. 075�02�2020�1482�1).
References
1. Fetisov G.V. X-ray diffraction methods for structural diagnostics of materials: progress and achievements//
Physics-Uspekhi. 2020. V. 63. P. 2�32.
2. Steiding C., Kolditz D., Kalender W.A. A quality assurance framework for the fully auto-mated and objective
evaluation of image quality in cone-beam computed tomography // Medical Physics. 2014. V.41, 031901.
