Finite Velocity of Propagation of Perturbations in the Cauchy Problem for the Wave Integro-Differential Equation with the Fractional-Exponential Memory Function

Abstract

The paper is devoted to the study of the Cauchy problem for the Volterra integro-differential equation, the main part of which is a wave equation perturbed by an integral operator in the form of a Volterra convolution (a wave integro-differential equation). The kernel function of the integral operator (the memory function) is a sum of fractional-exponential functions (Rabotnov functions) with positive coefficients. The influence of the integral operator on the perturbation propagation velocity in the Cauchy problem for the wave integro-differential equation is investigated. This wave integro-differential equation describes wave propagation in viscoelastic media, heat propagation in media with memory (the Gurtin–Pipkin equation), and has a number of other important applications.