Fractional differential equations and viscoelastic damping

Physical systems are mostly modelled by differential equations together with initial conditions and boundary conditions. For systems showing some kind of `memory\´ local differential equations are not appropiate. Particularly in the description of damping behaviour of vis-coelastic media, fractional differential equations are suggested instead by many authors. Many of these approaches use ad hoc definitions of the fractional derivative given by Riemann and Liouville. Usually this leads to the introduction of a time to where the system `is switched on\´. As a consequence the description of the system is not homogeneous in time whereas the system itself is clearly expected to be. Another consequence is the fact that the influence of the whole history of the system for times smaller than to has to be described by a finite set of initial conditions. We have recently shown that this is not possible [11]. This paper sketches a functional analytic approach using pseudodifferential equations which avoids these problems and especially adresses the physically important question of `causality\´. It leads to an impulse response separated into an exponentially damped oscillatory part and a "slow" relaxation. Moreover, sufficient criteria on the coefficients of the equation are given which guarantee the causality of the system. Though the spirit of the method roots in rather deep operator theory its results are easily understood. To demonstrate the use of these tools as an effective few-parameter model we compare in the second part its predictions for the frequency response of viscoelastic rods with measurement. The chosen reference example PTFE (Teflon) shows that different from classical models our model describes the behaviour in a wide frequency range within the accuracy of the measurement. Even dispersion effects are included.