Inverse Problem of Determining a Kernel of the Viscoelasticity Equation with Distributed Data in a Limited Domain
Abstract
Inverse problem of determining the kernel of the integral term of viscoelasticity with distributed data in a limited domain is considered. First, the direct problem is studied. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using the generalized Gronwall inequality, we obtain a priori estimates of the solution and its derivatives through the unknown kernel, which will be used in studying the inverse problem. The problem of determining the memory kernel in oscillatory process comes down to nonlinear integral Volterra equation of the first kind of convolution type, which transforms to the Volterra equation of the second kind. The method of contraction mappings is used to prove the global unique solvability of the problem posed in the space of continuous functions with weighted norms.