Stochastic Zener model with complex order fractional derivatives

Abstract

We analyze the dissipation inequality for the constitutive equation of a complex order fractional Zener model and obtain appropriate thermodynamical restrictions for the wave-type model equation in terms of its Laplace transform. These constraints obtained on the model parameters are less restrictive than the ones known in the previous literature. The main results of this paper are related to explicitly solving this equation in spaces of tempered distributions, proving the existence and uniqueness of the solution and discussing its regularity properties. The second set of results is related to the analysis of including random perturbations such as white noise in the model resulting in stochastic wave propagation models. We analyze various stochastic body forces, random initial excitations, and random initial velocities as input data followed by deriving the stochastic solution and calculating its most important statistical characteristics. All these results significantly extend our previous results related to a real order fractional Zener model. MSC[2020]: 26A33, 35L05, 35R11, 35R60, 60G10, 60G15, 74D05, 74J05, 82D30