Abstract: Integral operators represent the heart of tomography approaches, consisting in nondestructive mode of obtaining information, which is accumulated along lines of integration. A list of integral transforms, describing initial data for integral geometry and tomography problems, is very vast [1]. Weighted ray transforms of tensor fields [2], [3] stand out among the others as they are the solutions of differential equations of transport type. We prove the uniqueness theorems of boundary-value and initial boundary-value problems for the derived equations. Thus, the integral geometry and tensor tomography problems can be treated as inverse problems of right-hand part reconstruction by known certain families of weighted ray transforms of tensor fields. Next variety of integral operators applies for investigations of integral geometry and refraction tensor tomography problems. A generalization of back projection operator provides the operators of angular moments for weighted ray transforms of tensor fields. Differential operators of tensor analysis, applying to tensor fields in conjunction with the operators of weighted ray transforms and angular moments, present favorable tools for investigation of integral geometry and tomography problems [2]. Some part of them can be considered as conservation laws in framework of iterative methods for approximate solutions of refraction tensor tomography problems. The reported study was funded partially by the Program of fundamental researches of SB RAS No. I.1.5 (project 0314-2016-0011), RFBR and DFG according to the research project No. 19-51-12008. References [1] V. A. Sharafutdinov. Integral geometry of tensor elds. Utrecht: VSP, The Netherlands, 1994. [2] E.Yu. Derevtsov, I. E. Svetov. Tomography of tensor elds in the plane. Eurasian J. Math. Comp. Applications, V. 3 (2), 2015, pp. 24{68. [3] E.Yu. Derevtsov. On a generalization of attenuated ray transform in tomography. Sib. J. of Pure and Applied Math., V. 18 (4), 2018, pp. 29{41 (in Russian).

Cite: Derevtsov E.Y.
Differential equations and integral operators at investigations of integral geometry and tomography problems
In compilation The International Conference "Harmonic Analysis and PDE" in honor of Vladimir Maz'ya Holon, May 26‐31, 2019. 2019.

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