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|Title:||Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields|
|Author(s):||Derevtsov, Evgeny Yu.|
Volkov, Yuriy S.
|Editor(s):||Sergeyev, Yaroslav D.|
Kvasov, Dmitri E.
|Title:||Numerical Computations: Theory and Algorithms|
|Year of Publication:||2020|
|Free key words:||Tensor tomography|
Attenuated ray transform
|DDC notations:||510 Mathematics|
|Publikation type:||Conference Paper|
|Abstract:||Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components ξj of direction vector ξ participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.|
|DOI of the first publication:||10.1007/978-3-030-40616-5_8|
|URL of the first publication:||https://link.springer.com/chapter/10.1007/978-3-030-40616-5_8|
|Link to this record:||urn:nbn:de:bsz:291--ds-394231|
|Date of registration:||30-Mar-2023|
|Notes:||Third Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms (NUMTA 2019), Le Castella – Isola Capo Rizzuto (Crotone), Italy, June 15–21, 2019|
|Faculty:||MI - Fakultät für Mathematik und Informatik|
|Department:||MI - Mathematik|
|Professorship:||MI - Prof. Dr. Thomas Schuster|
|Collections:||SciDok - Der Wissenschaftsserver der Universität des Saarlandes|
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