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 Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 2, Pages 58–66 (Mi vmj693)

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

Zh. D. Totievaab

a North Ossetian State University, 44-46 Vatutin St., Vladikavkaz 362025, Russia
b Southern Mathematical Institute VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia

Abstract: 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.

Key words: inverse problem, stability, delta function, elastic moduli, coefficients, matrix kernel.

DOI: https://doi.org/10.23671/VNC.2019.2.32117

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UDC: 517.958
MSC: 35L20, 35R30, 35Q99

Citation: Zh. D. Totieva, “The problem of determining the matrix kernel of the anisotropic viscoelasticity equations system”, Vladikavkaz. Mat. Zh., 21:2 (2019), 58–66

Citation in format AMSBIB
\Bibitem{Tot19} \by Zh.~D.~Totieva \paper The problem of determining the matrix kernel of the anisotropic viscoelasticity equations system \jour Vladikavkaz. Mat. Zh. \yr 2019 \vol 21 \issue 2 \pages 58--66 \mathnet{http://mi.mathnet.ru/vmj693} \crossref{https://doi.org/10.23671/VNC.2019.2.32117}