The Equation of Motion for a Three-Dimensional Fractional Model of Viscoelasticity

Abstract

Over the past 4 decades, the differential equations of fractional order and the Abel integral equations have been considered to be the very important and useful tools for studying the hereditary and memorable properties of various processes. In some cases they lead us to the more adequate models as compared to the models based on an entire order of derivatives. In the main, the focus was on the study of models based on the fractional diffusion-wave differential equations. Such equations have been successfully used for the modeling of various phenomena, for example, for studying the behavior of viscoelastic materials and for studying the diffusion processes in porous regions that appear in fractal geometry, as well as for white noise process modeling, and so on.     This article addresses the equation of motion for a three-dimensional fractional model of viscoelastic behavior, when the constitutive relationships contain the derivatives of fractional order, in Riemann- Liouville sense. The hyperbolicity of the obtained equations of motion is shown.