Fractional Calculus: High-Precision Algorithms and Numerical Implementations

Preface
Contents
1 Introduction to Fractional Calculus
	Dingyü Xue 慮搠 Lu Bai
		1.1 Historic Review of Fractional Calculus
		1.2 Fractional Calculus Phenomena and Modeling Examples in Nature
		1.3 Historic Review of Fractional Calculus Computations
			1.3.1 Numerical Computing in Fractional Calculus
			1.3.2 Numerical Computing in Fractional-Order Ordinary Differential Equations
			1.3.3 Numerical Computing in Fractional-Order Partial Differential Equations
		1.4 Tools in Fractional Calculus and Fractional-Order Control
		1.5 Structures in the Book
			1.5.1 Main Contents
			1.5.2 Reading Suggestions
		References-8pt
2 Commonly Used Special Functions: Definitions  and Computing
	Dingyü Xue 慮搠 Lu Bai
		2.1 Error and Complementary Error Functions
		2.2 Gamma Functions
			2.2.1 Definition and Properties of Gamma Functions
			2.2.2 Complex Gamma Functions
			2.2.3 Other Forms of Gamma Functions
			2.2.4 Incomplete Gamma Functions
		2.3 Beta Functions
			2.3.1 Definition and Properties of Beta Functions
			2.3.2 Complex Beta Functions
			2.3.3 Incomplete Beta Functions
		2.4 Dawson Functions
		2.5 Hypergeometric Functions
		2.6 Mittag-Leffler Functions
			2.6.1 One-Parameter Mittag-Leffler Functions
			2.6.2 Two-Parameter Mittag-Leffler Functions
			2.6.3 Multi-Parameter Mittag-Leffler Functions
			2.6.4 The Relationship Between Mittag-Leffler  and Hypergeometric Functions
			2.6.5 Derivatives of Mittag-Leffler Functions
			2.6.6 Numerical Evaluation of Mittag-Leffler Functions and Their Derivatives
		2.7 Exercises
		References-8pt
3 Definitions and Numerical Evaluations of Fractional Calculus
	Dingyü Xue 慮搠 Lu Bai
		3.1 Fractional-Order Integral Formula
			3.1.1 Cauchy Integral Formula
			3.1.2 Derivative and Integral Formulas for Commonly Used Functions
		3.2 Definition and Numerical Evaluation of Grünwald–Letnikov Integrals and Derivatives
			3.2.1 Formulations in High-Order Integer-Order Derivatives
			3.2.2 Definition of Grünwald–Letnikov Fractional-Order Derivatives
			3.2.3 Numerical Evaluation of Grünwald–Letnikov Fractional-Order Derivatives and Integrals
			3.2.4 Podlubny\'s Matrix Algorithm
			3.2.5 Exploring Short-Time Memory Effects
		3.3 Definition and Evaluation of Riemann–Liouville Derivatives and Integrals
			3.3.1 High-Order Integer-Order Integral Formulas
			3.3.2 Definitions of Riemann–Liouville Fractional-Order Derivatives and Integrals
			3.3.3 Riemann–Liouville Derivative and Integral Formulas for Commonly Used Functions
			3.3.4 Initial Time Translation Properties
			3.3.5 Numerical Evaluation of Riemann–Liouville Derivatives and Integrals
			3.3.6 Symbolic Computing in Riemann–Liouville Derivatives
		3.4 Caputo Fractional Calculus Definition
			3.4.1 Definition of Caputo Derivatives and Integrals
			3.4.2 Commonly Used Caputo Derivative Formulas
			3.4.3 Symbolic Computing in Caputo Calculus
		3.5 The Relationship Among Different Fractional Calculus Definitions
			3.5.1 The Relationship Between Grünwald–Letnikov and Riemann–Liouville Definitions
			3.5.2 The Relationship Between Caputo and Riemann–Liouville Definitions
			3.5.3 Numerical Evaluations of Caputo Derivatives and Integrals
		3.6 Properties and Geometrical Interpretations of Fractional Calculus
			3.6.1 Properties of Fractional Calculus
			3.6.2 Geometrical Interpretations of Fractional Integrals
		3.7 Exercises
		References-8pt
4 High-Precision Numerical Algorithms and Implementation in Fractional Calculus
	Dingyü Xue 慮搠 Lu Bai
		4.1 Generating Function Construction for Arbitrary Integer Orders
		4.2 Trials on High-Precision Algorithms for Grünwald–Letnikov Derivatives
			4.2.1 An FFT-Based Algorithm
			4.2.2 A Recursive Formula for Generating Function Coefficients
		4.3 High-Precision Algorithm and Implementation for Grünwald–Letnikov Definition
			4.3.1 Decomposition and Compensation for Nonzero Initial Value Functions
			4.3.2 High-Precision Algorithm and Its Implementation
			4.3.3 Testing and Assessment of the Algorithms
			4.3.4 Revisit to the Matrix Algorithm
		4.4 High-Precision Algorithm for Caputo Derivatives
			4.4.1 The Algorithm and Its Implementation
			4.4.2 Testing and Assessment of the Algorithm
			4.4.3 Solutions of a Benchmark Problem
		4.5 Computing of Higher Fractional-Order Derivatives
			4.5.1 High-Precision Algorithms for Higher Integer-Order Derivatives
			4.5.2 Computing of Higher Fractional-Order Derivatives
		4.6 Exercises
		References-8pt
5 Approximations of Fractional-Order Operators and Systems
	Dingyü Xue 慮搠 Lu Bai
		5.1 Representation and Analysis of Linear Integer-Order Models
			5.1.1 Mathematical Model Input and Manipulations
			5.1.2 Time and Frequency Domain Responses
			5.1.3 Modeling and Analysis of Linear Fractional-Order Systems
		5.2 Some Approximation Methods with Continued Fractions
			5.2.1 Continued Fraction Approximation
			5.2.2 Carlson Approximation
			5.2.3 Matsuda–Fujii Approximation
			5.2.4 The Relationship between Fitting Quality and Filter Parameters
		5.3 Oustaloup Filter Approximations
			5.3.1 Oustaloup Filter
			5.3.2 An Improved Oustaloup Filter
		5.4 Integer-Order Approximation of FOTFs
			5.4.1 High-Order Approximation of FOTFs
			5.4.2 Reduction of Fractional-Order Models
		5.5 Approximation of Irrational Fractional-Order Transfer Functions
			5.5.1 Approximation of Implicit Irrational Models
			5.5.2 Frequency Response Fitting Methods
			5.5.3 Charef Approximation
			5.5.4 Optimum Charef Filter Design for Complicated Irrational Models
		5.6 Discrete Filter Approximations
			5.6.1 FIR Filter Approximation
			5.6.2 IIR Filter Approximation
			5.6.3 Discrete Filters for Step and Impulse Response Invariants
		5.7 Exercises
		References-8pt
6 Analytical and Numerical Solutions of Linear Fractional-Order Differential Equations
	Dingyü Xue 慮搠 Lu Bai
		6.1 Introduction to Linear Fractional-Order Differential Equations
			6.1.1 The General Form of Linear Fractional-Order Differential Equations
			6.1.2 Initial Value Problems of Fractional-Order Derivatives Under Different Definitions
			6.1.3 An Important Laplace Transform Formula
		6.2 Analytical Solutions of Some Linear FODEs
			6.2.1 One-Term FODEs
			6.2.2 Two-Term FODEs
			6.2.3 Three-Term FODEs
			6.2.4 General n-Term FODEs
		6.3 Analytical Solutions of Linear Commensurate-Order FODEs
			6.3.1 The General Form of Linear Commensurate-Order Differential Equations
			6.3.2 Some Commonly Used Laplace Transform Formulas for Linear FODEs
			6.3.3 Analytical Solutions of Linear Commensurate-Order Differential Equations
		6.4 A Closed-Form Algorithm for Linear FODEs with Zero Initial Conditions
			6.4.1 A Closed-Form Algorithm
			6.4.2 Impulse Responses of Linear FODEs
			6.4.3 Validating Numerical FODE Solutions
			6.4.4 A Matrix-Based Algorithm
			6.4.5 A High-Precision Closed-Form Algorithm
		6.5 Numerical Solutions of Caputo Equations with Nonzero Initial Conditions
			6.5.1 Mathematical Descriptions of Caputo Equations
			6.5.2 Taylor Axillary Functions
			6.5.3 High-Precision Algorithm for Caputo Equations
		1. Equivalent Initial Condition Problem
		2. High-Precision Algorithm for the FODEs
		6.6 Solutions of Linear Fractional-Order State Space Models
			6.6.1 State Space Descriptions of Linear FODEs
			6.6.2 State Transition Matrix
			6.6.3 Commensurate-Order State Space Models
		6.7 Numerical Solutions of Irrational Differential Equations
			6.7.1 Descriptions of Irrational Transfer Functions
			6.7.2 Solutions Based on Numerical Laplace Inverse Transform
			6.7.3 Time Response Computing of Closed-Loop Irrational Systems
			6.7.4 Time Responses Under Arbitrary Input Signals
		6.8 Stability Assessment of Linear Fractional-Order Systems
			6.8.1 Stability Assessment of Linear Commensurate-Order Systems
			6.8.2 Stability Assessment of Non-Commensurate-Order Systems
			6.8.3 Stability Assessment of Irrational Systems
		6.9 Exercises
		References-8pt
7 Numerical Solutions  of Nonlinear FODEs
	Dingyü Xue 慮搠 Lu Bai
		7.1 Descriptions of FODEs
			7.1.1 General form of FODEs
			7.1.2 Commensurate-Order State Space Models
			7.1.3 Extended State Space Models
		7.2 Numerical Solutions of Nonlinear Caputo Equations
			7.2.1 Numerical Solutions of Scalar Commensurate-Order Equations
			7.2.2 Solutions of Commensurate-Order Caputo Equations
			7.2.3 Numerical Solutions of Extended FOSS Models
			7.2.4 An Algebraic Equation-Based FODE Solver
		7.3 High-Precision Algorithm for Caputo Equations
			7.3.1 Predictor Equation
			7.3.2 Corrector Solution Method
		7.4 Exercises
		References-8pt
8 Block Diagram-Based Solutions of FODEs
	Dingyü Xue 慮搠 Lu Bai
		8.1 Introduction of FOTF Toolbox and Blockset
			8.1.1 Input and Connections of Fractional-Order Transfer Functions
			8.1.2 Fractional-Order State Space Models
			8.1.3 Analysis Functions for Linear Fractional-Order Systems
			8.1.4 The FOTF Blockset
		8.2 Block Diagram-Based Solutions of FODEs with Zero Initial Conditions
			8.2.1 Simulink Modeling Rules
			8.2.2 Simulink Environment Settings
		1. Solver Parameters Setting
		2. Input and Output Format Setting
			8.2.3 Simulink Modeling and Solutions for FODEs
			8.2.4 Validations of Numerical Solutions for Nonlinear FODEs
		8.3 Block Diagram-Based Solutions of Caputo Equations  with Nonzero Initial Conditions
			8.3.1 Modeling and Solutions of Explicit Caputo Equations
			8.3.2 Simulink Modeling of FOSS Models
			8.3.3 Handling of State Space Models with Order Higher  Than 1
		1. Low-Level Modeling Method
		2. State Augmentation Method
		8.4 Simulation of Fractional-Order Feedback Control Systems with Simulink
			8.4.1 Fractional-Order Transfer Function Block
			8.4.2 Fractional-Order PID Controllers and Closed-Loop System Simulation
			8.4.3 Simulation of Multivariable Systems
		8.5 Exercises
		References-8pt
9 Numerical Solutions of Special Fractional-Order Differential Equations
	Dingyü Xue 慮搠 Lu Bai
		9.1 Implicit Fractional-Order Differential Equations
			9.1.1 A High-Precision Matrix Algorithm for Caputo Equations
			9.1.2 Block Diagram-Based Solutions of Implicit FODEs
			9.1.3 Stiff ODE-Based Method
			9.1.4 The Fitting with the Implicit Block
		9.2 Solutions of Fractional-Order Delay Differential Equations
			9.2.1 Design of Benchmark Problems
			9.2.2 Modeling of Nonzero History Functions
			9.2.3 Solutions of FODDEs
		9.3 Boundary Value Problems in FODEs
			9.3.1 Mathematical Forms of BVPs
			9.3.2 Optimization and Algebraic Equation Modeling in Shooting Methods
			9.3.3 Fast Restart Mode in Simulink
			9.3.4 Solutions of BVPs
		9.4 Numerical Solutions of Time-Fractional Partial Differential Equations
		9.5 Exercises
		References-8pt
A Supplementary  Information
		Appendix A: Benchmark Problems for Fractional-order Differential Equations
		A.1  Mathematical Descriptions and Proofs in Benchmark Problems
			A.1.1 Initial Value Problems in FODEs
			A.1.2 Boundary Value Problems for FODEs
			A.1.3 Fractional-Order Delay Differential Equations
		A.2  A Simulink Group for Benchmark Problems
		A.3 Exercises
		References
		Appendix B: Inverse Laplace Transforms for Fractional-order and Irrational Functions
		B.1  Commonly Used Special Functions in Fractional Calculus
		B.2 Inverse Laplace Transform Table
		References
		Appendix C: FOTF Toolbox Functions  and Models
		C.1  Fundamental Computing Functions
			C.1.1 Special Functions and Other Supporting Functions
			C.1.2 Numerical Computing in Fractional-Order Derivatives
			C.1.3 Filter Design
			C.1.4 Solutions of Linear FODEs
			C.1.5  Solutions of Nonlinear FODEs
		C.2  Object-Oriented Program Design
			C.2.1 FOTF Class for Fractional-Order Transfer Functions
			C.2.2 FOSS Class for Fractional-Order State Space Models
		C.3  Simulink Models
			C.3.1 FOTF Blockset in Simulink
			C.3.2 Important Reusable Fractional-Order Models
		References
		Index