Fractional differentiation and its applications

Fractional differentiation, also called non-integer differentiation, is a concept that dates back to the beginning of differential calculus when it came to the attention of Leibniz and L’Hospital (1695) who exchanged letters about the half-order derivative. Since then many famous mathematicians and physicists have studied fractional integrals and derivatives mainly from a theoretical point of view, the main ideas being related to the names of Abel, Liouville, Riemann, Grunwald and Letnikov. By the beginning of the twentieth century only a few applications had been proposed, namely by O. Heaviside (1880) and A. Gemant (1936) who developed fractional models for electrical and mechanical engineering applications. In spite of these contributions this area of research remained almost unknown for many applied scientists until twenty or thirty years ago, when a considerable attention started to be paid to systems governed by fractional differential equations (now commonly called fractional systems). All over the world a spreading scientific community has indeed brought to light that many real physical systems (systems with long memory or hereditary behavior) are well characterized by fractional differential equations. Fractional differentiation and fractional systems play now a very important role in various fields such as biology, bio-physics, control theory, economics, electrical engineering, electronics, electromagnetism, electrochemistry, image and signal processing, mechanics, mechatronics, physics, rheology, material modeling and thermal engineering. In this context, the first IFAC workshop on Fractional Differentiation and is Applications, FDA’04, was held in Bordeaux, France, in 2004. This workshop aimed at bringing together experts in the field of fractional differentiation and its applications and all interested researchers, from universities and industries, to look at the state of the art and current research lines in theory, methodology, applications and tools.
This book integrates three parts gathering a selection of articles presented during FDA’04. Its attempts to give to the reader a presentation of current research and the latest industrial applications of fractional differentiation. The first part is dedicated to mathematical tools and geometrical and physical aspects. The second part presents applications in the domains of econophysics, mechanics, material modeling, thermal systems, electronics and electrical systems. Finally, the third part presents applications in systems analysis, implementation and simulation, system identification and system control.