The problem of determining the matrix kernel of the anisotropic viscoelasticity equations system

Zh. D. Totieva

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 2, pp. 58-66

Résumé

We consider the problem of determining the matrix kernel $K(t)=\mathrm{diag}(K_1, K_2, K_3)(t)$, $ t>0,$ occurring in the system of integro-differential viscoelasticity equations for anisotropic medium. The direct initial boundary value problem is to determine the displacement vector function $u(x,t)=(u_1,u_2,u_3)(x,t),$ $x=(x_1,x_2,x_3) \in R^3,$ $x_3>0$. It is assumed that the coefficients of the system (density and elastic modulus) depend only on the spatial variable $x_3>0$. The source of perturbation of elastic waves is concentrated on the boundary of $x_3=0$ and represents the Dirac Delta function (Neumann boundary condition of a special kind). The inverse problem is reduced to the previously studied problems of determining scalar kernels $K_i(t)$, $ i=1,2,3$. As an additional condition, the value of the Fourier transform in $x_2$ of the function $u(x,t)$ is given on the surface $x_3=0$. Theorems of global unique solvability and stability of the solution of the inverse problem are given. The idea of proving global solvability is to apply the contraction mapping principle to a system of nonlinear Volterra integral equations of the second kind in a weighted Banach space.

Bibliographie

[1] Kristensen R. M., Introduction to the Theory of Viscoelasticity, Mir, M., 1974, 340 pp. (in Russian)

[2] Rabotnov Yu. N., Elements of Hereditary Mechanics of Solids, Nauka, M., 1977, 384 pp. (in Russian)

[3] Rabotnov Yu. N., Mechanics of a Deformable Solid, Nauka, M., 1988, 712 pp. (in Russian)

[4] Pipkin A. C., Lectures on Viscoelasticity Theory, Springer, Berlin, 1986, 199 pp.

[5] Khohlov A. V., “The Qualitative Analysis of Theoretic Curves Generated by Linear Viscoelasticity Constitutive Equation”, Sience and Education, 2016, no. 5, 187–245

[6] Durdiev D. K., Totieva Zh. D., “The Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation”, Journal of Applied and Industrial Mathematics, 16:2 (2013), 72–82 (in Russian) | MR | Zbl

[7] Durdiev D. Q., Totieva Zh. D., “The Problem of Determining the Multidimensional Kernel of Viscoelasticity Equation”, Vladikavkaz Math. J., 17:4 (2015), 18–43 (in Russian) | DOI | MR

[8] Durdiev D. K., Totieva Zh. D., “The Problem of Determining the One-Dimensional Kernel of the Electroviscoelasticity Equation”, Siberian Mathematical J., 58:3 (2017), 427–444 | DOI | DOI | MR | Zbl

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

[10] Safarov Zh. Sh., Durdiev D. K., “Inverse Problem for Integro-Differential Equation of Acoustics”, Differential Equations, 54:1 (2018), 136–147 (in Russian) | DOI

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

[12] Durdiev D. K., Durdiev U. D., “The problem of kernel determination from viscoelasticity system integro-differential equations for homogeneous anisotropic media”, Nanosystems: Physics, Chemistry, Mathematics, 7:3 (2016), 405–409 | DOI | Zbl

[13] Durdiev D. K., Totieva Zh. D., “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equations”, Math. Meth. Appl. Sci., 17:17 (2018), 8019–8032 | DOI | MR | Zbl

[14] Tuaeva Zh. D., “Multidimensional Mathematical Model of Seismic Memory”, Research on Differential Equations and Mathematical Modeling, VNC RAN, Vladikavkaz, 2008, 297–306 (in Russian)