FRACTIONAL DIFFERENTIAL EQUATIONS

Transcription

1 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

2 Contents Preface Acknowledgements xvii xxiii 1 Special Functions of the Fractional Calculus Gamma Function Definition of the Gamma Function Some Properties of the Gamma Function Limit Representation of the Gamma Function Beta Function Contour Integral Representation Contour Integral Representation of 1/F(z) Mittag-Leffler Function Definition and Relation to Some Other Functions The Laplace Transform of the Mittag-Leffler Function in Two Parameters Derivatives of the Mittag-Leffler Function Differential Equations for the Mittag-Leffler Function Summation Formulas Integration of the Mittag-Leffler Function Asymptotic Expansions Wright Function Definition Integral Representation Relation to Other Functions 38 2 Fractional Derivatives and Integrals The Name of the Game Grünwald-Letnikov Fractional Derivatives 43 vii

3 i CONTENTS Unification of Integer-order Derivatives and Integrals Integrals of Arbitrary Order Derivatives of Arbitrary Order Fractional Derivative of (t - a) ß Composition with Integer-order Derivatives Composition with Fractional Derivatives Riemann-Liouville Fractional Derivatives Unification of Integer-order Derivatives and Integrals Integrals of Arbitrary Order Derivatives of Arbitrary Order Fractional Derivative of (t - a) ß Composition with Integer-order Derivatives Composition with Fractional Derivatives Link to the Grünwald-Letnikov Approach Sonic Other Approaches Caputo's Fractional Derivative Generalized Functions Approach Sequential Fractional Derivatives Left and Right Fractional Derivatives Properties of Fractional Derivatives Linearity The Leibniz Rule for Fractional Derivatives Fractional Derivative of a Composite Function Riemann- Liouville Fractional Differentiation of an Integral Depending on a Parameter Behaviour near the Lower Terminal Behaviour far from the Lower Terminal Laplace Transforms of Fractional Derivatives Basic Facts on the Laplace Transform Laplace Transform of the Riemann-Liouville Fractional Derivative Laplace Transform of the Caputo Derivative Laplace Transform of the Grünwald-Letnikov Fractional Derivative Laplace Transform of the Miller-Ross Sequential Fractional Derivative Fourier Transforms of Fractional Derivatives ' Basic Facts on the Fourier Transform 109

4 CONTENTS ix 2.Ü.2 Fourier Transform of Fractional Integrals Fourier Transform of Fractional Derivatives Mollin Ti'ansforms of Fractional Derivatives Basic Facts on the Mollin Transform Meilin Transform of the Riemann-Liouville Fractional Integral Mellin Transform of the Riemann-Liouville Fractional Derivative Mellin Transform of the Caputo Fractional Derivative Mellin Transform of the Miller-Ross Fractional Derivative Existence and Uniqueness Theorems Linear Fractional Differential Equations Fractional Differential Equation of a General Form Existence and Uniqueness Theorem as a Method of Solution Dependencc of a Solution on Initial Conditions The Laplace Transform Method Standard Fractional Differential Equations Ordinary Linear Fractional Differential Equations Partial Linear Fractional Differential Equations Sequential Fractional Differential Equations Ordinary Linear Fractional Differential Equations Partial Linear Fractional Differential Equations Fractional Green's Function Definition and Some Properties Definition Properties One-term Equation Two-term Equation Three-term Equation Four-term Equation 156

5 x CONTENTS 5.6 General Case: n-term Equation Other Methods for the Solution of Fractional-order Equations The Meilin Transform Method Power Series Method One-term Equation Equation with Non-constant Coefficients Two-term Non-linear Equation Babenko's Symbolic Calculus Method The Idea of the Method Application in Heat and Mass Transfer Link to the Laplace Transform Method Method of Orthogonal Polynomials The Idea of the Method General Scheme of the Method Riesz Fractional Potential Left Riemann-Liouville Fractional Integrals and Derivatives Other Spectral Relationships For the Left Riemann-Liouville Fractional Integrals Spectral Relationships For the Right Riemann-Liouville Fractional Integrals Solution of Arutyunyan's Equation in Creep Theory Solution of Abel's Equation Finite-part Integrals Jacobi Polynomials Orthogonal with Non-integrable Weight Function Numerical Evaluation of Fractional Derivatives Riemann-Liouville and Grünwald-Letnikov Definitions of the Fractional-order Derivative Approximation of Fractional Derivatives Fractional Difference Approach The Use of Quadrature Formulas The "Short-Memory" Principle Order of Approximation Computation of coefficients Higher-order approximations 209

6 CONTENTS XI 7.7 Calculation of Heat Load Intensity Change in Blast Furnace Walls Introduction to the Problem Fractional-order Differentiation and Integration Calculation of the Heat Flüx by Fractional Order Derivatives - Method A Calculation of the Heat Flux Based on the Simulation of the Thermal Field of the Furnace Wall - Method B Comparison of the Methods Finite-part Integrals and Fractional Derivatives Evaluation of Finite-part Integrals Using Fractional Derivatives Evaluation of Fractional Derivatives Using Finite-part Integrals 220 Numerical Solution of Fractional Differential Equations Initial Conditions: Which Problem to Solve? Numerical Solution Examples of Numerical Solutions Relaxation-oscillation Equation Equation with Constant Coefficients: Motion of an Immersed Plate Equation with Non-constant Coefficients: Solution of a Gas in a Fluid Non-Linear Problem: Cooling of a Semi-infinite Body by Radiation The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations 242 Fractional-order Systems and Controllers Fractional-order Systems and Fractional-order Controllers Fractional-order Control System Fractional-order Transfer Functions New Function of the Mittag-Leffler Type General Formula 247

7 Xll CONTENTS The Unit-impulse and Unit-step Response Some Special Cases PJ A _D"-controller Open-loop System Response Closed-loop System Response Example Fractional-order Controlled System Integer-order Approximation Integer-order PZ)-controller Fractional-order Controller On Fractional-order System Identification Conclusion Survey of Applications of the Fractional Calculus Abel's Integral Equation General Remarks Some Equations Reducible to Abel's Equation Viscoelasticity Integer-order Models Fractional-order Models Approaches Related to the Fractional Calculus Bode's Analysis of Feedback Amplifiers Fractional Capacitor Theory Electrical Circuits Tree Fractance Chain Fractance Electrical Analogue Model of a Porous Dyke Westerlund's Generalized Voltage Divider Fractional-order Chua-Hartley System Electroanalytical Chemistry Electrode-Electrolyte Interface Fractional Multipoles Biology Electric Conductance of Biological Systems Fractional-order Model of Neurons Fractional Diffusion Equations Control Theory Fitting of Experimental Data Disadvantages of Classical Regression Models Fractional Derivative Approach 300

8 CONTENTS xiii Example: Wires at Nizna Slana Mines "Fractional-order" Physics? 305 Appendix: Tables of Fractional Derivatives 309 Bibliography 313 Index 337