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DOI: 10.23671/VNC.2019.2.32117

The Problem of Determining the Matrix Kernel of the Anisotropic Viscoelasticity Equations System

Totieva, Zh. D.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 2.
We consider the problem of determining the matrix kernel \(K(t)=diag(K_1, K_2, K_3)(t)\), \( t>0,\) 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.
Keywords: inverse problem, stability, delta function, elastic moduli, coefficients, matrix kernel
Language: Russian Download the full text  
For citation: Totieva, Zh. D. The Problem of Determining the Matrix Kernel of the Anisotropic Viscoelasticity Equations System, Vladikavkaz Math. J., 2019, vol. 21, no. 2, pp. 58-66 (in Russian). DOI 10.23671/VNC.2019.2.32117
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