The problem of determining kernels in a two-dimensional system of viscoelasticity equations

For a two-dimensional system of integro-differential equations of viscoelasticity in an isotropic medium, the direct and inverse problems of determining the stress vector and particle velocity, as well as the diagonal hereditarity matrix, are studied. First, the system of two-dimensional viscoelasticity equations was transformed into a system of first-order linear equations. The thus composed system of first-order integro-differential equations with the help of its special matrix was reduced to a normal form with respect to time and one of the spatial variables. Then, using the Fourier transform with respect to another spatial variable and integrating over the characteristics of the equations based on the initial and boundary conditions, it was replaced by a system of Volterra integral equations of the second kind, equivalent to the original problem. An existence and uniqueness theorem for the solution of the direct problem is given. To solve the inverse problem using the integral equations of the direct problem and additional conditions, a closed system of integral equations for unknown functions and some of their linear combinations is constructed. Further, the contraction mapping method (Banach principle) is applied to this system in the class of continuous functions with an exponential weighted norm. Thus, we prove the global existence and uniqueness theorem for the solutions of the stated problems. The proof of the theorems is constructive, i.e. with the help of the obtained integral equations, for example, by the method of successive approximations, a solution to the problems can be constructed.