The integro-differential system of viscoelasticity equations with a source of explosive type is considered. It is assumed that the coefficients of the equations depend only on one spatial variable. The problem of determining the kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to one inverse problem for scalar hyperbolic equations. This inverse problem is replaced by an equivalent system of integral equations for unknown functions. The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied. The theorem of global unique solvability is proved and the stability estimate of solution to the inverse problem is obtained.

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The inverse problem of determining 2D spatial part of integral member kernel in integro-differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial-boundary value problem for the half-space x > 0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain ( , t) ∈ R 2 , x > 0. Local existence and uniqueness theorem for the solution to the inverse problem is obtained.The theory and application of hyperbolic integro-differential equations play an important role in the mathematical modeling of many fields: physical, biological phenomens, and engineering sciences, in which it is necessary to take into consideration the effect of real problems. In many cases, PDEs of the electrodynamics and elasticity with integral convolution terms are reduced to one integro-differential equation; the main part of which is second-order hyperbolic operator. For the last 30 years, there has been much work related to problems of identification of memory kernel in these equations. Here, we mention some of them 1-8 that are close to this work and the references therein for more details. In Durdiev and Durdiev and Totieva, 3-5 the local in time existence and the uniqueness results for of some multidimensional inverse problems for the second-order hyperbolic integro-differential equations in the class of functions having certain smoothness in the time variable and analyticity with respect to the spatial variables were obtained. Problems of determining the spatial part of the multidimensional kernel were investigated in the works of Durdiev and Safarov, Lorenzi and Romanov, In this paper, using the method of the work of Romanov, 10 we focus on the inverse problem of recovering a spatial part of a kernel in the integral term of hyperbolic integro-differential equation. The presented results give a convenient approach for numerical solving of the inverse problem.Math Meth Appl Sci. 2019; :7440-7451.