Generalized viscoelastic models: Their fractional equations with solutions

Recently fractional calculus (FC) has encountered much success in the description of complex dynamics. In particular Fe has proved to be a valuable tool to handle viscoelastic aspects. In this paper we construct fractional theological constitutive equations on the basis of well known mechanical models, especially the Maxwell, the Kelvin-Voigt, the Zener and the Poynting-Thomson model. To this end we introduce a fractional element, in addition to the standard purely elastic and purely viscous elements. As we proceed to show, many of the fractional differential equations which we obtain by this construction method admit closed form, analytical solutions in terms of Fox X-functions of the Mittag-Leffler type.