Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications
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Table of contents
- Cover
- Contentsvii
- Prefacexvii
- Acknowledgementsxxiii
- Chapter 1. Special Functions of the Fractional Calculus1
- 1.1 Gamma Function1
- 1.2 Mittag-Leffler Function16
- 1.3 Wright Function37
- Chapter 2. Fractional Derivatives and Integrals41
- 2.1 The Name of the Game41
- 2.2 Grünwald Letnikov Fractional Derivatives43
- 2.3 Riemann Liouville Fractional Derivatives62
- 2.4 Some Other Approaches77
- 2.5 Sequential Fractional Derivatives86
- 2.6 Left and Right Fractional Derivatives88
- 2.7 Properties of Fractional Derivatives90
- 2.8 Laplace Transforms of Fractional Derivatives103
- 2.9 Fourier Transforms of Fractional Derivatives109
- 2.10 Mellin Transforms of Fractional Derivatives112
- Chapter 3. Existence and Uniqueness Theorems121
- 3.1 Linear Fractional Differential Equations122
- 3.2 Fractional Differential Equation of a General Form126
- 3.3 Existence and Uniqueness Theorem as a Method of Solution131
- 3.4 Dependence of a Solution on Initial Conditions133
- Chapter 4. The Laplace Transform Method137
- 4.1 Standard Fractional Differential Equations138
- 4.2 Sequential Fractional Differential Equations144
- Chapter 5. Fractional Green's Function149
- 5.1 Definition and Some Properties150
- 5.2 One-term Equation153
- 5.3 Two-term Equation154
- 5.4 Three-term Equation155
- 5.5 Four-term Equation156
- 5.6 General Case: n-term Equation157
- Chapter 6. Other Methods for the Solution of Fractional-order Equations159
- 6.1 The Mellin Transform Method159
- 6.2 Power Series Method161
- 6.3 Babenko's Symbolic Calculus Method168
- 6.4 Method of Orthogonal Polynomials173
- Chapter 7. Numerical Evaluation of Fractional Derivatives199
- 7.1 Riemann Liouville and Grünwald–Letnikov Definitions of the Fractionalorder Derivative199
- 7.2 Approximation of Fractional Derivatives200
- 7.3 The "Short-Memory" Principle203
- 7.4 Order of Approximation204
- 7.5 Computation of coefficients208
- 7.6 Higher-order approximations209
- 7.7 Calculation of Heat Load Intensity Change in Blast Furnace Walls210
- 7.8 Finite-part Integrals and Fractional Derivatives219
- Chapter 8. Numerical Solution of Fractional Differential Equations223
- 8.1 Initial Conditions: Which Problem to Solve?223
- 8.2 Numerical Solution224
- 8.3 Examples of Numerical Solutions224
- 8.4 The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations242
- Chapter 9. Fractional-order Systems and Controllers243
- 9.1 Fractional-order Systems and Fractional-order Controllers244
- 9.2 Example251
- 9.3 On Fractional-order System Identification257
- 9.4 Conclusion259
- Chapter 10. Survey of Applications of the Fractional Calculus261
- 10.1 Abel's Integral Equation261
- 10.2 Viscoelasticity268
- 10.3 Bode's Analysis of Feedback Amplifiers277
- 10.4 Fractional Capacitor Theory278
- 10.5 Electrical Circuits279
- 10.6 Electroanalytical Chemistry290
- 10.7 Electrode-Electrolyte Interface291
- 10.8 Fractional Multipoles293
- 10.9 Biology294
- 10.10 Fractional Diffusion Equations296
- 10.11 Control Theory298
- 10.12 Fitting of Experimental Data299
- 10.13 "Fractional-order" Physics?305
- Appendix: Tables of Fractional Derivatives309
- Bibliography313
- Index337
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 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