Geodesic
Inverse problem for viscoelastic system in a vertically layered medium
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 4, pp. 30-47 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables $x_1$ and $x_2$ of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables $x_1$, $x_2$, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem. 
@article{VMJ_2022_24_4_a2,
     author = {A. A. Boltaev and D. K. Durdiev},
     title = {Inverse problem for viscoelastic system in a vertically layered medium},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {30--47},
     year = {2022},
     volume = {24},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2022_24_4_a2/}
}
TY  - JOUR
AU  - A. A. Boltaev
AU  - D. K. Durdiev
TI  - Inverse problem for viscoelastic system in a vertically layered medium
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2022
SP  - 30
EP  - 47
VL  - 24
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VMJ_2022_24_4_a2/
LA  - en
ID  - VMJ_2022_24_4_a2
ER  - 
%0 Journal Article
%A A. A. Boltaev
%A D. K. Durdiev
%T Inverse problem for viscoelastic system in a vertically layered medium
%J Vladikavkazskij matematičeskij žurnal
%D 2022
%P 30-47
%V 24
%N 4
%U http://geodesic.mathdoc.fr/item/VMJ_2022_24_4_a2/
%G en
%F VMJ_2022_24_4_a2
A. A. Boltaev; D. K. Durdiev. Inverse problem for viscoelastic system in a vertically layered medium. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 4, pp. 30-47. http://geodesic.mathdoc.fr/item/VMJ_2022_24_4_a2/

[1] Mura T., Micromechanics of Defects in Solids, Second Revised Edition, Northwestern University, USA, IL, Evanston, 1987

[2] Galin L. A., Contact Problems of the Theory of Elasticity and Viscoelasticity, Nauka, M., 1980 (in Russian) | MR

[3] Kilbas A. A, Integral Equations: Course of Lectures, Belarusian State University, Minsk, 2005 (in Russian)

[4] Durdimurod D., Shishkina E. and Sitnik S., “The Explicit Formula for Solution of Anomalous Diffusion Equation in the Multi-Dimensional Space”, Lobachevskii Journal of Mathematics, 42:6 (2021), 1264–1273 | DOI | MR

[5] Godunov S. K., Equations of Mathematical Physics, Nauka, Ch. Ed. Physical-Mat. Lit, M., 1979 (in Russian) | MR

[6] Romanov V. G., Inverse Problems of Mathematical Physics, Utrecht, The Netherlands, 1987

[7] Lorenzi A., “An Identification Problem Related to a Nonlinear Hyperbolic Integro-Differential Equation”, Nonlinear Analysis, Theory, Methods and Applications, 22:1 (1994), 21–44 | DOI | MR

[8] Janno J. and Von Wolfersdorf L., “Inverse Problems for Identification of Memory Kernels in Viscoelasticity”, Mathematical Methods in the Applied Sciences, 20:4 (1997), 291–314 | 3.0.CO;2-W class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[9] Romanov V. G., “Stability Estimates for the Solution in the Problem of Determining the Kernel of the Viscoelasticity Equation”, Journal of Applied and Industrial Mathematics, 6:3 (2012), 360–370 | DOI | MR

[10] Totieva Zh. D. and Durdiev D. Q., “The Problem of Determining the Multidimensional Kernel of Viscoelasticity Equation”, Vladikavkaz Mathematical Journal, 17:4 (2015), 18–43 | DOI | MR

[11] Durdiev D. K., “Some Multidimensional Inverse Problems of Memory Determination in Hyperbolic Equations”, Journal of Mathematical Physics, Analysis, Geometry, 3:4 (2007), 411–423 | MR

[12] Durdiev D. K. and Safarov Z. S., “Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain”, Mathematical Notes, 97:6 (2015), 867–877 | DOI | MR

[13] Romanov V. G., “On the Determination of the Coefficients in the Viscoelasticity Equations”, Siberian Mathematical Journal, 55:3 (2014), 503–510 | DOI | MR

[14] Romanov V. G., “The Problem of Determining the Kernel of the Viscoelasticity Equation”, Doklady Akademii Nauk, 446:1 (2012), 18–20 (in Russian) | MR

[15] Durdiev D. K. and Rakhmonov A. A., “Inverse Problem for the System Integro-Differential Equation SH Waves in a Visco-Elastic Porous Medium: Global Solvability”, Theoretical and Mathematical Physics, 195:3 (2018), 923–937 | DOI | MR

[16] Durdiev D. K. and Rakhmonov A. A., “The Problem of Determining Two-Dimensional Kernel in a System of Integro-Differential Equations of a Viscoelastic Porous Medium”, Journal of Applied and Industrial Mathematics, 14:2 (2020), 281–295 | DOI | MR

[17] Durdiev D. K. and Rahmonov A. A., “A 2D Kernel Determination Problem in a Viscoelastic Porous Medium with a Weakly Horizontally Inhomogeneity”, Mathematical Methods in the Applied Sciences, 43:15 (2020), 8776–8796 | DOI | MR

[18] Durdiev D. K. and Totieva Z. D., “The Problem of Determining the One-Dimensional Matrix Kernel of the System of Viscoelasticity Equations”, Mathematical Methods in the Applied Sciences, 41:17 (2018), 8019–8032 | DOI | MR

[19] Totieva Z. D. and Durdiev D. K., “The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation”, Mathematical Notes, 103:1–2 (2018), 118–132 | DOI | MR

[20] Safarov J. SH. and Durdiev D. K., “Inverse Problem for an Integro-Differential Equation of Acoustics”, Differential Equations, 54:1 (2018), 134–142 | DOI | MR

[21] Durdiev D. K. and Totieva Z. D., “The Problem of Determining the One-Dimensional Kernel of Viscoelasticity Equation with a Source of Explosive Type”, Journal of Inverse and Ill-Posed Problems, 28:1 (2020), 43–52 | DOI | MR

[22] Durdiev U. D., “An Inverse Problem for the System of Viscoelasticity Equations in Homogeneous Anisotropic Media”, Journal of Applied and Industrial Mathematics, 13:4 (2019), 623–628 | DOI | MR

[23] Durdiev D. K. and Turdiev Kh. Kh., “Inverse Problem for a First-Order Hyperbolic System with Memory”, Differential Equations, 56:12 (2020), 1634–1643 | DOI | MR

[24] Durdiev D. K. and Turdiev Kh. Kh., “The Problem of Finding the Kernels in the System of Integro-Differential Maxwell's Equations”, Journal of Applied and Industrial Mathematics, 15:2 (2021), 190–211 | DOI | MR

[25] Kolmogorov A. N. and Fomin S. V., Elements of Function Theory and Functional Analysis, Nauka, M., 1989 (in Russian) | MR