FRACTIONAL DIFFERENTIAL EQUATIONS An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications by Igor Podlubny Technical University of Kosice, Slovak Republic ACADEMIC PRESS San Diego Boston New York London Sydney Tokyo Toronto
Contents Preface Acknowledgements xvii xxiii 1 Special Functions of the Fractional Calculus 1 1.1 Gamma Function 1 1.1.1 Definition of the Gamma Function 1 1.1.2 Some Properties of the Gamma Function... 2 1.1.3 Limit Representation of the Gamma Function. 4 1.1.4 Beta Function 6 1.1.5 Contour Integral Representation 10 1.1.6 Contour Integral Representation of 1/F(z)... 12 1.2 Mittag-Leffler Function 16 1.2.1 Definition and Relation to Some Other Functions 17 1.2.2 The Laplace Transform of the Mittag-Leffler Function in Two Parameters 20 1.2.3 Derivatives of the Mittag-Leffler Function.... 21 1.2.4 Differential Equations for the Mittag-Leffler Function 23 1.2.5 Summation Formulas 23 1.2.6 Integration of the Mittag-Leffler Function... 24 1.2.7 Asymptotic Expansions 29 1.3 Wright Function 37 1.3.1 Definition 37 1.3.2 Integral Representation 37 1.3.3 Relation to Other Functions 38 2 Fractional Derivatives and Integrals 41 2.1 The Name of the Game 41 2.2 Grünwald-Letnikov Fractional Derivatives 43 vii
i CONTENTS 2.2.1 Unification of Integer-order Derivatives and Integrals 43 2.2.2 Integrals of Arbitrary Order 48 2.2.3 Derivatives of Arbitrary Order 52 2.2.4 Fractional Derivative of (t - a) ß 55 2.2.5 Composition with Integer-order Derivatives... 57 2.2.6 Composition with Fractional Derivatives... 59 2.3 Riemann-Liouville Fractional Derivatives 62 2.3.1 Unification of Integer-order Derivatives and Integrals 63 2.3.2 Integrals of Arbitrary Order 65 2.3.3 Derivatives of Arbitrary Order 68 2.3.4 Fractional Derivative of (t - a) ß 72 2.3.5 Composition with Integer-order Derivatives... 73 2.3.6 Composition with Fractional Derivatives... 74 2.3.7 Link to the Grünwald-Letnikov Approach.... 75 2.4 Sonic Other Approaches 77 2.4.1 Caputo's Fractional Derivative 78 2.4.2 Generalized Functions Approach 81 2.5 Sequential Fractional Derivatives 86 2.6 Left and Right Fractional Derivatives 88 2.7 Properties of Fractional Derivatives 90 2.7.1 Linearity 90 2.7.2 The Leibniz Rule for Fractional Derivatives... 91 2.7.3 Fractional Derivative of a Composite Function. 97 2.7.4 Riemann- Liouville Fractional Differentiation of an Integral Depending on a Parameter... 98 2.7.5 Behaviour near the Lower Terminal 99 2.7.6 Behaviour far from the Lower Terminal 101 2.8 Laplace Transforms of Fractional Derivatives 103 2.8.1 Basic Facts on the Laplace Transform 103 2.8.2 Laplace Transform of the Riemann-Liouville Fractional Derivative 104 2.8.3 Laplace Transform of the Caputo Derivative.. 106 2.8.4 Laplace Transform of the Grünwald-Letnikov Fractional Derivative 106 2.8.5 Laplace Transform of the Miller-Ross Sequential Fractional Derivative 108 2.9 Fourier Transforms of Fractional Derivatives '. 109 2.9.1 Basic Facts on the Fourier Transform 109
CONTENTS ix 2.Ü.2 Fourier Transform of Fractional Integrals... 110 2.9.3 Fourier Transform of Fractional Derivatives... 111 2.10 Mollin Ti'ansforms of Fractional Derivatives 112 2.10.1 Basic Facts on the Mollin Transform 112 2.10.2 Meilin Transform of the Riemann-Liouville Fractional Integral 115 2.10.3 Mellin Transform of the Riemann-Liouville Fractional Derivative 115 2.10.4 Mellin Transform of the Caputo Fractional Derivative 116 2.10.5 Mellin Transform of the Miller-Ross Fractional Derivative 117 3 Existence and Uniqueness Theorems 121 3.1 Linear Fractional Differential Equations 122 3.2 Fractional Differential Equation of a General Form... 126 3.3 Existence and Uniqueness Theorem as a Method of Solution 131 3.4 Dependencc of a Solution on Initial Conditions 133 4 The Laplace Transform Method 137 4.1 Standard Fractional Differential Equations 138 4.1.1 Ordinary Linear Fractional Differential Equations 138 4.1.2 Partial Linear Fractional Differential Equations 140 4.2 Sequential Fractional Differential Equations 144 4.2.1 Ordinary Linear Fractional Differential Equations 144 4.2.2 Partial Linear Fractional Differential Equations 146 5 Fractional Green's Function 149 5.1 Definition and Some Properties 150 5.1.1 Definition 150 5.1.2 Properties 150 5.2 One-term Equation 153 5.3 Two-term Equation 154 5.4 Three-term Equation 155 5.5 Four-term Equation 156
x CONTENTS 5.6 General Case: n-term Equation 157 6 Other Methods for the Solution of Fractional-order Equations 159 6.1 The Meilin Transform Method 159 6.2 Power Series Method 161 6.2.1 One-term Equation 162 6.2.2 Equation with Non-constant Coefficients... 166 6.2.3 Two-term Non-linear Equation 167 6.3 Babenko's Symbolic Calculus Method 168 6.3.1 The Idea of the Method 169 6.3.2 Application in Heat and Mass Transfer 170 6.3.3 Link to the Laplace Transform Method 172 6.4 Method of Orthogonal Polynomials 173 6.4.1 The Idea of the Method 174 6.4.2 General Scheme of the Method 179 6.4.3 Riesz Fractional Potential 181 6.4.4 Left Riemann-Liouville Fractional Integrals and Derivatives 186 6.4.5 Other Spectral Relationships For the Left Riemann-Liouville Fractional Integrals 188 6.4.6 Spectral Relationships For the Right Riemann-Liouville Fractional Integrals 189 6.4.7 Solution of Arutyunyan's Equation in Creep Theory 191 6.4.8 Solution of Abel's Equation. 192 6.4.9 Finite-part Integrals 192 6.4.10 Jacobi Polynomials Orthogonal with Non-integrable Weight Function 195 7 Numerical Evaluation of Fractional Derivatives 199 7.1 Riemann-Liouville and Grünwald-Letnikov Definitions of the Fractional-order Derivative 199 7.2 Approximation of Fractional Derivatives 200 7.2.1 Fractional Difference Approach 200 7.2.2 The Use of Quadrature Formulas 200 7.3 The "Short-Memory" Principle 203 7.4 Order of Approximation 204 7.5 Computation of coefficients 208 7.6 Higher-order approximations 209
CONTENTS XI 7.7 Calculation of Heat Load Intensity Change in Blast Furnace Walls 210 7.7.1 Introduction to the Problem 211 7.7.2 Fractional-order Differentiation and Integration 211 7.7.3 Calculation of the Heat Flüx by Fractional Order Derivatives - Method A 212 7.7.4 Calculation of the Heat Flux Based on the Simulation of the Thermal Field of the Furnace Wall - Method B 215 7.7.5 Comparison of the Methods 218 7.8 Finite-part Integrals and Fractional Derivatives 219 7.8.1 Evaluation of Finite-part Integrals Using Fractional Derivatives 220 7.8.2 Evaluation of Fractional Derivatives Using Finite-part Integrals 220 Numerical Solution of Fractional Differential Equations 223 8.1 Initial Conditions: Which Problem to Solve? 223 8.2 Numerical Solution 224 8.3 Examples of Numerical Solutions 224 8.3.1 Relaxation-oscillation Equation 224 8.3.2 Equation with Constant Coefficients: Motion of an Immersed Plate 225 8.3.3 Equation with Non-constant Coefficients: Solution of a Gas in a Fluid 231 8.3.4 Non-Linear Problem: Cooling of a Semi-infinite Body by Radiation.. 235 8.4 The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations 242 Fractional-order Systems and Controllers 243 9.1 Fractional-order Systems and Fractional-order Controllers 244 9.1.1 Fractional-order Control System 244 9.1.2 Fractional-order Transfer Functions 245 9.1.3 New Function of the Mittag-Leffler Type... 246 9.1.4 General Formula 247
Xll CONTENTS 9.1.5 The Unit-impulse and Unit-step Response... 248 9.1.6 Some Special Cases 248 9.1.7 PJ A _D"-controller 249 9.1.8 Open-loop System Response 250 9.1.9 Closed-loop System Response 250 9.2 Example 251 9.2.1 Fractional-order Controlled System 252 9.2.2 Integer-order Approximation 252 9.2.3 Integer-order PZ)-controller 253 9.2.4 Fractional-order Controller 256 9.3 On Fractional-order System Identification 257 9.4 Conclusion 259 10 Survey of Applications of the Fractional Calculus 261 10.1 Abel's Integral Equation 261 10.1.1 General Remarks 262 10.1.2 Some Equations Reducible to Abel's Equation. 263 10.2 Viscoelasticity 268 10.2.1 Integer-order Models 268 10.2.2 Fractional-order Models 271 10.2.3 Approaches Related to the Fractional Calculus. 275 10.3 Bode's Analysis of Feedback Amplifiers 277 10.4 Fractional Capacitor Theory 278 10.5 Electrical Circuits 279 10.5.1 Tree Fractance 279 10.5.2 Chain Fractance 280 10.5.3 Electrical Analogue Model of a Porous Dyke.. 282 10.5.4 Westerlund's Generalized Voltage Divider... 282 10.5.5 Fractional-order Chua-Hartley System 286 10.6 Electroanalytical Chemistry 290 10.7 Electrode-Electrolyte Interface 291 10.8 Fractional Multipoles 293 10.9 Biology 294 10.9.1 Electric Conductance of Biological Systems... 294 10.9.2 Fractional-order Model of Neurons 295 10.10 Fractional Diffusion Equations 296 10.11 Control Theory 298 10.12 Fitting of Experimental Data 299 10.12.1 Disadvantages of Classical Regression Models.. 299 10.12.2 Fractional Derivative Approach 300
CONTENTS xiii 10.12.3 Example: Wires at Nizna Slana Mines 301 10.13 "Fractional-order" Physics? 305 Appendix: Tables of Fractional Derivatives 309 Bibliography 313 Index 337