Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (ISSN Book 198)

This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'. This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models. In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research. A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications. Key Features * A unique survey of many applications of fractional calculus * Presents basic theory * Includes a unified presentation of selected classical results, which are important for applications * Provides many examples * Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory * The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches * Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'.
This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models.
In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research.
A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications.


Key Features
* A unique survey of many applications of fractional calculus
* Presents basic theory
* Includes a unified presentation of selected classical results, which are important for applications
* Provides many examples
* Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory
* The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches
* Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives Content: List of figures Pages xv-xvi Preface Pages xvii-xxi Acknowledgements Pages xxiii-xxiv Chapter 1 Special functions of the fractional calculus Pages 1-39 Chapter 2 Fractional derivatives and integrals Pages 41-119 Chapter 3 Existence and uniqueness theorems Pages 121-136 Chapter 4 The laplace transform method Pages 137-147 Chapter 5 Fractional Green's Function Pages 149-158 Chapter 6 Other methods for solution of fractional order equations Pages 159-198 Chapter 7 Numerical evaluation of fractional derivatives Pages 199-221 Chapter 8 Numerical solution of fractional differential equations Pages 223-242 Chapter 9 Fractional-order systems and controllers Pages 243-260 Chapter 10 Survey of applications of the fractional calculus Pages 261-307 Appendix: Tables of fractional derivatives Pages 309-311 Bibliography Pages 313-335 Index Pages 337-340 This book is a ready reference to fractional calculus for researchers and engineers needing applications tools for modeling, numerical methods, and plot visualization (2D and 3D). Convenient tables of fractional derivatives and integrals will be provided. This book is entirely devoted to applications of fractional derivatives and fractional differential equations. A solid combination of selected classical and recent results, numerous easy-to-follow examples and figures, and the general style of presentation should help those who need fractional-order models in their research. In this chapter some basic theory of the special functions which are used in the other chapters is given.