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.
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 is by no means the first (or the last) book on the subject
of fractional calculus, but indeed it is one that would
undoubtedly attract the attention (and successfully serve the needs) of
mathematical, physical, and engineering scientists looking
for applications of fractional calculus. I, therefore, recommend this well-written
book to all users of fractional calculus."
H. M. Srivastava, Zentralblatt MATH
CONTENTS
Preface.
Acknowledgments. Special Functions Of Preface. Acknowledgements. Special Functions of
the Fractional Calculus . Gamma Function. Mittag-Leffler Function. Wright Function. Fractional
Derivatives and Integrals . The Name of the Game. Grünwald-Letnikov Fractional
Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential
Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional
Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of
Fractional Derivatives. Mellin Transforms of Fractional Derivatives. Existence and
Uniqueness Theorems . Linear Fractional Differential Equations. Fractional
Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of
Solution. Dependence of a Solution on Initial Conditions. The Laplace Transform Method .
Standard Fractional Differential Equations. Sequential Fractional Differential Equations. Fractional
Green's Function . Definition and Some Properties. One-Term Equation. Two-Term
Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity
Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General
Case: n-term Equation. Other Methods for the Solution of Fractional-order
Equations . The Mellin Transform Method. Power Series Method. Babenko's Symbolic
Calculus Method. Method of Orthogonal Polynomials. Numerical Evaluation of Fractional
Derivatives . Approximation of Fractional Derivatives. The "Short-Memory"
Principle. Order of Approximation. Computation of Coefficients. Higher-order
Approximations. Numerical Solution of Fractional Differential Equations . Initial
Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions.
The "Short-Memory" Principle in Initial Value Problems for Fractional
Differential Equations. Fractional-Order Systems and Controllers. Fractional-Order
Systems and Fractional-Order Controllers . Example. On Viscoelasticity. Bode's
Analysis of Feedback Amplifiers. Fractional Capacitor Theory. Electrical Circuits.
Electroanalytical Chemistry. Electrode-Electrolyte Interface. Fractional Multipoles.
Biology. Fractional Diffusion Equations. Control Theory. Fitting of Experimental Data. The
"Fractional-Order" Physics? Bibliography. Tables of Fractional Derivatives .
Index.
340 pages