Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

Ignor Podlubny and Kenneth V. Thimann (Eds.)9780125588409, 0125588402

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

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Table of contents :
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