Scope:
The Fractional Calculus (FC) is a generalisation of the traditional calculus that leads to similar concepts and tools, but with a much wider applicability. By allowing derivative and integral operations of arbitrary order, it is to traditional calculus what the real number line is to the set of integers. For almost 300 years it was seen as an interesting, but abstract, mathematical concept. We now know that the non-integer order operators can describe dynamical behaviour of materials and processes over vast time and frequency scales with very concise and computable models.
Of particular interest to the signal processing community, as well as to the control systems and many others, is the fact the fractional systems have both short and long term memory. While the first corresponds to the “distribution of time constants” associated with the distribution of poles and zeroes in the complex plane, the second corresponds to infinitely many interlaced close to each other poles and zeros that in the limit originate a branch cut line. This translates to a lack specific time scale and, therefore, no new resonance or other instability effects. This incorporates the power law behaviour found in natural systems that show the greatest robustness to variation of environmental parameters. On the other hand, the intrinsic causality characteristic of the fractional derivative forces us to include time ordering into the setup of differential equations.
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics and notably control theory and signal and image processing. In these last three fields, some important considerations such as modeling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability, and robustness are now linked to long-range dependence phenomena. Considering the number of sessions now organized in intentional conferences, of special issues reserved by journal editors, and of articles in conferences and journals, fractional differentiation and its applications is an important issue for the international scientific community.