The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science

Addresses the rapidly growing ­field of fractional calculus and provides simpli­fied solutions for linear commensurate–order fractional differential equations

­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is the result of the authors work in fractional calculus, and more particularly, in functions for the solutions of fractional di­fferential equations, which is fostered in the behavior of generalized exponential functions. The authors discuss how fractional trigonometry plays a role analogous to the classical trigonometry for the fractional calculus by providing solutions to linear fractional di­fferential equations. The book begins with an introductory chapter that o­ffers insight into the fundamentals of fractional calculus, and topical coverage is then organized in two main parts. Part One develops the definitions and theories of fractional exponentials and fractional trigonometry. Part Two provides insight into various areas of potential application within the sciences. The fractional exponential function via the fundamental fractional differential equation, the generalized exponential function, and R–function relationships are discussed in addition to the fractional hyperboletry, the R1–fractional trigonometry, the R2–fractional trigonometry, and the R3–trigonometric functions. ­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science also:

Presents fractional trigonometry as a tool for scientists and engineers and discusses how to apply fractional–order methods to the current toolbox of mathematical modelers Employs a mathematically clear presentation in an e­ ort to make the topic broadly accessible  Includes solutions to linear fractional di­fferential equations and generously features graphical forms of functions to help readers visualize the presented concepts Provides e­ffective and efficient methods to describe complex structures

­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is an ideal reference for academic researchers, research engineers, research scientists, mathematicians, physicists, biologists, and chemists who need to apply new fractional calculus methods to a variety of disciplines. The book is also appropriate as a textbook for graduate– and PhD–level courses in fractional calculus.

Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry.

Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional–order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann s complementary–function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University.

Preface

Acknowledgements

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

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

3. The Generalized Fractional Exponential Function

3.1 Introduction

3.2 Functions for the Fractional Calculus

3. 3 The R–Function: A Generalized Function

3.4 Properties of the 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 A Further Generalized Function: The G–Function and H–function

3.10 Preliminaries to the Fractional Trigonometry Develpoment

3.11 Eigen Character of the R–Function

3.12 Fractional Differintegral of the Time–Scaled 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

4. R–Function Relationships

4.1 R–Function Basics

4.2 Relationships for  in Terms of

4.3 Relationships for  in Terms of

4.4 Relationships for the Rational Form in Terms of

4.5 Relationships for  in Terms of

4.6 Relating  to the Exponential Function

4.7 Inverse Relationships Relationships for  in Terms of

4.8 Inverse Relationships Relationships for  in Terms of

4.9 Inverse Relationships Relationships for  in Terms of

4.10 Discussion

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 Discussion

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

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

7.5.9 Summary of Fractional Calculus Operations on the R2 Trigonometric Functions

7.6 Inferred Fractional Differential Equations

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

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 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

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

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

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 Equation 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

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 Functions Based Frequency Responses

13.5 Fractional Meta–Trigonometry

13.6 Elementary Fractional Transfer Functions

13.7 Stability Theorem (Matignon)

13.8 Stability of Elementary Fractional Transfer Functions

13.9 Insights into the Behavior of the Fractional Meta–Trigonometric Functions

13.10 Discussion

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

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 Classical Spirals

15.4 Discussion

16. Fractional Oscillators

16.1 The Space of Linear Fractional Oscillators

16.2 Coupled Fractional Oscillators

17. Shell Morphology and Growth

17.1 Nautilus Pompilius

17.2 –17.7 Various Shells

17.8 Ammonite

17.9 Discussion

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

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

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 Properties and Non–Spiral Properties

20.7 Numerical Improvements for Evaluation to Larger Values of atq

20.8 Epilog

Appendix A Related Works

Appendix B Computer Code

Appendix C Tornado Simulation

Appendix D Special Topics in Fractional Differintegration

D.1 Fractional Integration of the Segmented Function

D.2 Fractional Differentiation of the Segmented Function

D.3 Fractional Integration of Segmented Fractional Trigonometric Functions

D.4 Fractional Differentiation of Segmented Fractional Trigonometric Functions

Appendix E Alternate Forms

E.1 Reduced Variable Summation Forms

E.2 Natural Quency Simplification

Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry.

Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional–order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann s complementary–function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University.