On the Properties of the Fundamental Solution of a One-Dimensional Wave Integro-Differential Operator With a Fractional-Exponential Memory Function

The properties of the fundamental solution of the linear Volterra integro-differential operator, which is a one-dimensional wave linear differential operator with partial derivatives, perturbed the Volterra integral operator of convolution, are investigated. The kernel function of the integral operator is the sum of fractional exponential functions (Rabotnov functions) with positive coefficients. For linear Volterra integro-differential operators with second-order partial derivatives, the concept of hyperbolicity with respect to a cone is introduced. It is established that hyperbolicity with respect to a cone is equivalent to the localization of the support of the fundamental solution of a second-order linear Volterra integro-differential operator in the conjugate cone. Hyperbolicity relative to the cone is established for one-dimensionalwave integrodifferential operator with a fractional-exponential memory function.