The concept of fractional derivative (Fractional derivative (FD)) was introduced
after 1695 as a simply academic generalization of integer derivative. An FD
generalizes the order of differentiation from positive integers Set of natural numbers
(N) to real Set of real numbers (R), or even to complex Set of complex numbers (C)
numbers. A detailed presentation of the old historical steps of fractional calculus
(Fractional calculus (FC)) with references is presented in the papers [45, 50].
In the last two decades, it was established that a series of phenomena can be
studied in terms of FC. It was established that the rheologic properties [34] of some
polymers can be expressed with the aid of fractional differential models [1, 2, 10,
15, 20, 26, 28, 29, 36]. Fractional phenomena were established as the damping
phenomena in the high-density polyurethane foams [42], nuclear reactor dynamics
[35], thermoelasticity [33], mechanical vibrations [8], or biological tissues [5, 19].
An analysis of the integer and fractional entropy is performed in [44].
This review of the possible applications of FC in the real world justifies the
necessity of its extensive study. The aim of this book is to introduce a series of
problems and methods insufficiently discussed in the field of FC.
A series of examples based on symbolic computation, written in Maple© and
Mathematica©, are presented. The reader can find other useful applications for the
case of integer order systems in the book of Inna Shingareva and C. LizárragaCelaya [41], which can be extended to the case of fractional calculus, or the book of
problems [17].
This book is organized in six chapters.