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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.
Abstract:
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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