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Index
Cover
Title Page
Copyright
Table of Contents
Preface
Acknowledgments
About the Companion Website
Chapter 1: Introduction
1.1 Background
1.2 The Fractional Integral and Derivative
1.3 The Traditional Trigonometry
1.4 Previous Efforts
1.5 Expectations of a Generalized Trigonometry and Hyperboletry
Chapter 2: The Fractional Exponential Function via the Fundamental Fractional Differential Equation
2.1 The Fundamental Fractional Differential Equation
2.2 The Generalized Impulse Response Function
2.3 Relationship of the F-function to the Mittag-Leffler Function
2.4 Properties of the F-Function
2.5 Behavior of the F-Function as the Parameter a Varies
2.6 Example
Chapter 3: The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus
3.1 Introduction
3.2 Functions for the Fractional Calculus
3.3 The R-Function: A Generalized Function
3.4 Properties of the Rq,v(a, t)-Function
3.5 Relationship of the R-Function to the Elementary Functions
3.6 R-Function Identities
3.7 Relationship of the R-Function to the Fractional Calculus Functions
3.8 Example: Cooling Manifold
3.9 Further Generalized Functions: The G-Function and the H-Function
3.10 Preliminaries to the Fractional Trigonometry Development
3.11 Eigen Character of the R-Function
3.12 Fractional Differintegral of the Timescaled R-Function
3.13 R-Function Relationships
3.14 Roots of Complex Numbers
3.15 Indexed Forms of the R-Function
3.16 Term-by-Term Operations
3.17 Discussion
Chapter 4: R-Function Relationships
4.1 R-Function Basics
4.2 Relationships for Rm,0 in Terms of R1,0
4.3 Relationships for R1/m,0 in Terms of R1,0
4.4 Relationships for the Rational Form Rm/p,0 in Terms of R1/p,0
4.5 Relationships for R1/p,0 in Terms of Rm/p,0
4.6 Relating Rm/p,0 to the Exponential Function R1,0(b, t) = ebt
4.7 Inverse Relationships – Relationships for R1,0 in Terms of Rm,k
4.8 Inverse Relationships – Relationships for R1,0 in Terms of R1/m,0
4.9 Inverse Relationships – Relationships for eat = R1,0(a, t) in Terms of Rm/p,0
4.10 Discussion
Chapter 5: The Fractional Hyperboletry
5.1 The Fractional R1-Hyperbolic Functions
5.2 R1-Hyperbolic Function Relationship
5.3 Fractional Calculus Operations on the R1-Hyperbolic Functions
5.4 Laplace Transforms of the R1-Hyperbolic Functions
5.5 Complexity-Based Hyperbolic Functions
5.6 Fractional Hyperbolic Differential Equations
5.7 Example
5.8 Discussions
Chapter 6: The R1-Fractional Trigonometry
6.1 R1-Trigonometric Functions
6.2 R1-Trigonometric Function Interrelationship
6.3 Relationships to R1-Hyperbolic Functions
6.4 Fractional Calculus Operations on the R1-Trigonometric Functions
6.5 Laplace Transforms of the R1-Trigonometric Functions
6.6 Complexity-Based R1-Trigonometric Functions
6.7 Fractional Differential Equations
Chapter 7: The R2-Fractional Trigonometry
7.1 R2-Trigonometric Functions: Based on Real and Imaginary Parts
7.2 R2-Trigonometric Functions: Based on Parity
7.3 Laplace Transforms of the R2-Trigonometric Functions
7.4 R2-Trigonometric Function Relationships
7.5 Fractional Calculus Operations on the R2-Trigonometric Functions
7.6 Inferred Fractional Differential Equations
Chapter 8: The R3-Trigonometric Functions
8.1 The R3-Trigonometric Functions: Based on Complexity
8.2 The R3-Trigonometric Functions: Based on Parity
8.3 Laplace Transforms of the R3-Trigonometric Functions
8.4 R3-Trigonometric Function Relationships
8.5 Fractional Calculus Operations on the R3-Trigonometric Functions
Chapter 9: The Fractional Meta-Trigonometry
9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity
9.2 The Meta-Fractional Trigonometric Functions: Based on Parity
9.3 Commutative Properties of the Complexity and Parity Operations
9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions
9.5 R-Function Representation of the Fractional Meta-Trigonometric Functions
9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions
9.7 Special Topics in Fractional Differintegration
9.8 Meta-Trigonometric Function Relationships
9.9 Fractional Poles: Structure of the Laplace Transforms
9.10 Comments and Issues Relative to the Meta-Trigonometric Functions
9.11 Backward Compatibility to Earlier Fractional Trigonometries
9.12 Discussion
Chapter 10: The Ratio and Reciprocal Functions
10.1 Fractional Complexity Functions
10.2 The Parity Reciprocal Functions
10.3 The Parity Ratio Functions
10.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions
10.5 Relationships
10.6 Discussion
Chapter 11: Further Generalized Fractional Trigonometries
11.1 The G-Function-Based Trigonometry
11.2 Laplace Transforms for the G-Trigonometric Functions
11.3 The H-Function-Based Trigonometry
11.4 Laplace Transforms for the H-Trigonometric Functions
Introduction to Applications
Chapter 12: The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry
12.1 Fractional Differential Equations
12.2 Fundamental Fractional Differential Equations of the First Kind
12.3 Fundamental Fractional Differential Equations of the Second Kind
12.4 Preliminaries – Laplace Transforms
12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots
12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots
12.7 Fractional Differential Equations Containing Repeated Roots
12.8 Fractional Differential Equations of Non-Commensurate Order
12.9 Indexed Fractional Differential Equations: Multiple Solutions
12.10 Discussion
Chapter 13: Fractional Trigonometric Systems
13.1 The R-Function as a Linear System
13.2 R-System Time Responses
13.3 R-Function-Based Frequency Responses
13.4 Meta-Trigonometric Function-Based Frequency Responses
13.5 Fractional Meta-Trigonometry
13.6 Elementary Fractional Transfer Functions
13.7 Stability Theorem
13.8 Stability of Elementary Fractional Transfer Functions
13.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions
13.10 Discussion
Chapter 14: Numerical Issues and Approximations in the Fractional Trigonometry
14.1 R-Function Convergence
14.2 The Meta-Trigonometric Function Convergence
14.3 Uniform Convergence
14.4 Numerical Issues in the Fractional Trigonometry
14.5 The R2Cos- and R2Sin-Function Asymptotic Behavior
14.6 R-Function Approximations
14.7 The Near-Order Effect
14.8 High-Precision Software
Chapter 15: The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry
15.1 The Fractional Spiral Functions
15.2 Analysis of Spirals
15.3 Relation to the Classical Spirals
15.4 Discussion
Chapter 16: Fractional Oscillators
16.1 The Space of Linear Fractional Oscillators
16.2 Coupled Fractional Oscillators
Chapter 17: Shell Morphology and Growth
17.1 Nautilus pompilius
17.2 Shell 5
17.3 Shell 6
17.4 Shell 7
17.5 Shell 8
17.6 Shell 9
17.7 Shell 10
17.8 Ammonite
17.9 Discussion
Chapter 18: Mathematical Classification of the Spiral and Ring Galaxy Morphologies
18.1 Introduction
18.2 Background – Fractional Spirals for Galactic Classification
18.3 Classification Process
18.4 Mathematical Classification of Selected Galaxies
18.5 Analysis
18.6 Discussion
18.7 Appendix: Carbon Star
Chapter 19: Hurricanes, Tornados, and Whirlpools
19.1 Hurricane Cloud Patterns
19.2 Tornado Classification
19.3 Low-Pressure Cloud Pattern
19.4 Whirlpool
19.5 Order in Physical Systems
Chapter 20: A Look Forward
20.1 Properties of the R-Function
20.2 Inverse Functions
20.3 The Generalized Fractional Trigonometries
20.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments
20.5 Applications: Fractional Field Equations
20.6 Fractional Spiral and Nonspiral Properties
20.7 Numerical Improvements for Evaluation to Larger Values of atq
20.8 Epilog
Appendix A: Related Works
A.1 Introduction
A.2 Miller and Ross
A.3 West, Bologna, and Grigolini
A.4 Mittag-Leffler-Based Fractional Trigonometric Functions
A.5 Relationship to Current Work
Appendix B: Computer Code
B.1 Introduction
B.2 Matlab® R-Function
B.3 Matlab® R-Function Evaluation Program
B.4 Matlab® Meta-Cosine Function
B.5 Matlab® Cosine Evaluation Program
B.6 Maple® 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine
Appendix C: Tornado Simulation
Appendix D: Special Topics in Fractional Differintegration
D.1 Introduction
D.2 Fractional Integration of the Segmented tp-Function
D.3 Fractional Differentiation of the Segmented tp-Function
D.4 Fractional Integration of Segmented Fractional Trigonometric Functions
D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions
Appendix E: Alternate Forms
E.1 Introduction
E.2 Reduced Variable Summation Forms
E.3 Natural Quency Simplification
References
Index
End User License Agreement
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