Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain

Abstract

The one-dimensional integro-differential equation arising in the theory of viscoelasticity with constant density and Lam´e coefficients is considered. The direct problem is to determine the displacement function from the initial boundary-value problem for this equation, provided that the initial conditions are zero. The spatial domain is the closed interval [0, l], and the boundary condition is given by the stress function in the form of a concentrated perturbation source at the left endpoint of this interval and as zero at the right endpoint. For the direct problem, we study the inverse problem of determining the kernel appearing in the integral term of the equation. To find it, we introduce an additional condition for the displacement function at x = 0. The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved.

The problem of kernel determination from viscoelasticity system integro-differential equations for homogeneous anisotropic media

“…Among the problems that are closer to the present work can be identified [3][4][5][6][7][8][9]. In papers [3,4], the unique solvability and stability of the solution for the inverse problem for the identification of a memory kernel from Maxwell's system integro-differential equations for a homogeneous anisotropic media are studied.…”

“…By Fourier's method, this problem is reduced to solving the Volterra integral equations with respect to the unknown functions of the time-dependent variable. In papers [6,7] (see also references therein) the problem of determining the multidimensional kernel in viscoelasticity equation for an inhomogeneous isotropic medium is investigated. In [8,9], the problem of the one-dimensional kernel reconstruction from viscoelasticity equation in the bounded and unbounded domains has been studied.…”

“…In papers [6,7] (see also references therein) the problem of determining the multidimensional kernel in viscoelasticity equation for an inhomogeneous isotropic medium is investigated. In [8,9], the problem of the one-dimensional kernel reconstruction from viscoelasticity equation in the bounded and unbounded domains has been studied. The theorems for the global unique solvability of these problems in the class of continuous functions with weighted norms were proved.…”

“…The theorems for the global unique solvability of these problems in the class of continuous functions with weighted norms were proved. The basic feature inherent in [3,4,[6][7][8][9] and this paper is to use a boundary-localized and/or a fixed point of the spatial domain source, for the initiation of the physical process of wave propagation. Finally, we recall that the papers [10][11][12][13] are concerned with the problems of kernel determination from integro-differential equations with an integral of the convolution type.…”

The problem of determining the one‐dimensional matrix kernel of the system of viscoelasticity equations

“…The subject matter of previous studies 1-10 is related most closely to that of the present paper. These papers deal with problems of determining the kernel depending only on the temporal variable (one-dimensional inverse problem) for the cases of distributed [1][2][3][4] and concentrated [5][6][7][8][9][10] wave excitation sources. These problems can be reduced to the solution of integral equations of Volterra type for the unknown functions.…”