Inverse problem for the system of viscoelasticity in anisotropic media with tetragonal form of elasticity modulus
D. K. DurdievInstitute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State UniversityZavqiddin BozorovInstitute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State UniversityA.A. BoltaevInstitute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University; Vladikavkaz Scientific Center of the Russian Academy of Sciences
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For the reduced canonical system of integro-differential equations of viscoelasticity, direct and inverse problems of determining the velocity field of elastic waves and the relaxation matrix are considered. The problems are replaced by a closed system of Volterra integral equations of the second kind with respect to the Fourier transform in the variables $x_{1}$ and $x_{2}$ for the solution of the direct problem and unknowns of the inverse problem. Further, the method of contraction mappings in the space of continuous functions with a weighted norm is applied to this system. Thus, we prove global existence and uniqueness theorems for solutions of the problems.
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