Fractional Calculus : High-Precision Algorithms and Numerical Implementations

• Autore: Xue Dingyü
• Pubblicazione: Singapore : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (413 pages)
• Altri autori: BaiLu
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Fractional Calculus
• ISBN: 981-9920-70-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910855383603321
• Lo trovi qui: Univ. Federico II