On physical interpretations of fractional integration and differentiation

Is there a relation between fractional calculus and fractal geometry? Can a fractional order system be represented by a causal dynamical model? These are the questions recently debated in the scientific community. The author intends to answer to these questions. In the first part of the paper, some recently suggested models are reviewed and no convincing evidence is found for any dynamical model of a fractional order system having been built with the help of fractals. Linear filters with constant lumped parameters have a very limited use as approximations of fractional order systems. The model suggested in the paper is a state-space representation with parameters as functions of the independent variable. Regularization of fractional differentiation is considered and asymptotic error estimates, as well as simulation results, are presented.