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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Addresses the rapidly growing -field of fractional calculus and provides simpli-fied 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 di-fferential 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 di-fferential equations. The book begins with an introductory chapter that o-ffers 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 di-fferential equations and generously features graphical forms of functions to help readers visualize the presented concepts * Provides e-ffective 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.
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Table of Contents
Cover
Title Page
Copyright
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
R
q
,
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
R
m
,0
in Terms of
R
1,0
4.3 Relationships for
R
1/
m
,0
in Terms of
R
1,0
4.4 Relationships for the Rational Form
R
m
/
p
,0
in Terms of
R
1/
p
,0
4.5 Relationships for
R
1/
p
,0
in Terms of
R
m
/
p
,0
4.6 Relating
R
m
/
p
,0
to the Exponential Function
R
1,0
(
b
,
t
) =
e
bt
4.7 Inverse Relationships – Relationships for
R
1,0
in Terms of
R
m
,
k
4.8 Inverse Relationships – Relationships for
R
1,0
in Terms of
R
1/
m
,0
4.9 Inverse Relationships – Relationships for
e
at
=
R
1,0
(
a
,
t
) in Terms of
4.10 Discussion
Chapter 5: The Fractional Hyperboletry
5.1 The Fractional
R
1
-Hyperbolic Functions
5.2
R
1
-Hyperbolic Function Relationship
5.3 Fractional Calculus Operations on the
R
1
-Hyperbolic Functions
5.4 Laplace Transforms of the
R
1
-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
R
1
-Trigonometric Functions
6.2
R
1
-Trigonometric Function Interrelationship
6.3 Relationships to
R
1
-Hyperbolic Functions
6.4 Fractional Calculus Operations on the
R
1
-Trigonometric Functions
6.5 Laplace Transforms of the
R
1
-Trigonometric Functions
6.6 Complexity-Based
R
1
-Trigonometric Functions
6.7 Fractional Differential Equations
Chapter 7: The R2-Fractional Trigonometry
7.1
R
2
-Trigonometric Functions: Based on Real and Imaginary Parts
7.2
R
2
-Trigonometric Functions: Based on Parity
7.3 Laplace Transforms of the
R
2
-Trigonometric Functions
7.4
R
2
-Trigonometric Function Relationships
7.5 Fractional Calculus Operations on the
R
2
-Trigonometric Functions
7.6 Inferred Fractional Differential Equations
Chapter 8: The R3-Trigonometric Functions
8.1 The
R
3
-Trigonometric Functions: Based on Complexity
8.2 The
R
3
-Trigonometric Functions: Based on Parity
8.3 Laplace Transforms of the
R
3
-Trigonometric Functions
8.4
R
3
-Trigonometric Function Relationships
8.5 Fractional Calculus Operations on the
R
3
-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
R
2
Cos
- and
R
2
Sin
-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
at
q
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
t
p
-Function
D.3 Fractional Differentiation of the Segmented
t
p
-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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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Overview of the development of the fractional trigonometry and its applications.
Chapter 2: The Fractional Exponential Function via the Fundamental Fractional Differential Equation
Figure 2.1 The -function versus time as varies from 0.25 to 2.0 in 0.25 increments.
Figure 2.2 Both sheets of the Laplace transform of the
F
-function in the
s
-plane.
Figure 2.3 The
w
-plane for , with .
Figure 2.4 Step responses corresponding to various pole locations in the
w
-plane, for .
Figure 2.5 Supercapacitor circuit example.
Figure 2.6 The
w
-plane stability diagram for supercapacitor.
Figure 2.7 Supercapacitor impulse response.
Chapter 3: The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus
Figure 3.1 Effect of
q
on
R
q
,0
(1,
t
),
v
= 0.0, ,
q
= 0.25–2.5 in steps of 0.25. (a)
a
= 1.0. (b)
a
= −1.0. Source: Adapted from Lorenzo and Hartley, NASA [69].
Figure 3.2 Effect of
v
on ,
v
= −0.50 to 0.50 in steps of 0.25,
a
= −1.0. (a)
q
= 0.25. (b)
q
= 0.75.
Figure 3.3 Effect of
a
on (a) ,
q
= 0.25, and (b) ,
q
= 0.75,
v
= 0,
a
= −1.5 to 0.50 in steps of 0.5.
Figure 3.4 Generalized exponential constant versus
q
.
Figure 3.5 Cooling manifold.
Figure 3.6 versus
t
for
q
= 0.75,
a
= −1.0,
v
= 0, and
r
= 0.25–2.0 in steps of 0.25.
Figure 3.7
versus
t
for
q
= 1.0,
a
= −1.0,
v
= 0, and
r
= 0.25–2.0 in steps of 0.25.
Figure 3.8 versus
t
for
q
= 1.5,
a
= −1.0,
v
= 0, and
r
= 0.25–2.0 in steps of 0.25.
Figure 3.9
R
-Function . Real and imaginary parts for
q
= 0.2–1.0 in steps of 0.2.
Chapter 5: The Fractional Hyperboletry
Figure 5.1 versus
t
-Time, for
q
= 0.2–2.0 in steps of 0.2,
a
= 1,
v
= 0, and
k =
0.
Figure 5.2
versus
t
-Time, for
q
= 0.2–2.0 in steps of 0.2,
a
= 1,
v
= 0, and
k
= 0.
Figure 5.3 versus
t
-Time, for
q
= 0.2–3.0 in steps of 0.2,
a
= 1,
v
= 0, and
k
= 0.
Figure 5.4 Effect of the
a
parameter on versus
t
-Time, for (a)
q
= 0.25, (b)
q
= 0.50, (c)
q
= 0.75,
a
= 0.2–2.0 in steps of 0.2,
v
= 0, and
k =
0.
Figure 5.5 Effect of the
a
parameter on
versus
t
-Time, for (a)
q
= 0.25, (b)
q
= 0.50, (c)
q
= 0.75,
a
= 0.2–2.0 in steps of 0.2,
v
= 0, and
k
= 0.
Figure 5.6 Effect of the
a
parameter on versus
t
-Time, for (a)
q
= 0.25, (b)
q
= 0.50, (c)
q
= 0.75, (d)
q
= 2.00,
a
= 0.2–2.0 in steps of 0.2,
v
= 0, and
k
= 0.
Figure 5.7 Effect of
v
on (a) , (b) , (c) versus
t
-Time, for
v = q =
0
.
2–2.0 in steps of 0.2,
a
= 1, and
k =
0.
Figure 5.8 Effect of
v
on versus
t
-Time, for
v
=
q
− 1,
q =
1.0–2.0 in steps of 0.2,
a
= 1, and
k =
0.
Figure 5.9 Effect of
v
on
versus
t
-Time, for
v
=
q −
1,
q =
1.0–2.0 in steps of 0.2,
a
= 1, and
k =
0.
Figure 5.10 Effect of
v
on versus
t
-Time, for
v
=
q
− 1,
q =
1.0–2.0 in steps of 0.2,
a
= 1, and
k =
0.
Figure 5.11 and versus
t
-Time, for
k
= 0 to 6.
Figure 5.12 Expanding foam reactor.
Chapter 6: The R1-Fractional Trigonometry
Figure 6.1 (a) versus
t
-Time, for
q
= 0.2–1.2 in steps of 0.1,
v
= 0,
a
= 1. (b) versus
t
-Time, for
q
= 0.2–1.2 in steps of 0.1,
v
= 0,
a
= 1. (a, b)
Figure 6.2 versus
t
-Time, for
q
= 0.2–1.2 in steps of 0.2,
v
= 0,
a
= 1.
Figure 6.3 Effect of
a
on for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.25,
v
= 0 and
k
= 0.
Figure 6.4 Effect of
a
on
for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.75,
v
= 0 and
k
= 0.
Figure 6.5 Effect of
a
on
for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.25,
v
= 0 and
k
= 0.
Figure 6.6 Effect of
a
on
for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.75,
v
= 0 and
k
= 0.
Figure 6.7 Effect of
a
on
for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.25,
v
= 0 and
k
= 0.
Figure 6.8 Effect of
a
on for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.75,
v
= 0 and
k
= 0.
Figure 6.9 Effect of
v
on for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.25,
a
= 1.0 and
k
= 0.
Figure 6.10 Effect of
v
on
for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.75,
a
= 1.0 and
k
= 0.
Figure 6.11 Effect of
v
on
for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.25,
a
= 1.0 and
k
= 0.
Figure 6.12 Effect of
v
on
for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.75,
a
= 1.0 and
k
= 0.
Figure 6.13 Effect of
v
on
for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.25,
a
= 1.0 and
k
= 0.
Figure 6.14 Effect of
v
on for
v
= −0.20–0.20 in steps of 0.10, with
q
= 0.75,
a
= 1.0 and
k
= 0.
Figure 6.15 Phase plane versus for
q
= 1.1 and
q
= 0.25–1.75 in steps of 0.25,
a
= 1.0,
v
= 0.0.
Figure 6.16 Conjugate product ,
q
= 0.2–1.4 in steps of 0.2,
k
= 0.
Figure 6.17 Conjugate sum ,
q
= 0.2–1.2 in steps of 0.2,
k
= 0,
a
= 1.0.
Chapter 7: The R2-Fractional Trigonometry
Figure 7.1 versus
t
-Time for
q
= 0.2–2.0 in steps of 0.1, with
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.2 versus
t
-Time for
q
= 0.2–2.0 in steps of 0.1, with
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.3 versus
t
-Time for
q
= 0.2–1.9 in steps of 0.1, with
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.4 Effect of
a
for versus
t
-Time for
a
= 0.2–1.1 in steps of 0.1, with
q
= 0.25,
k
= 0,
v
= 0.0.
Figure 7.5 Effect of
a
for versus
t
-Time for
a
= 0.2–1.1 in steps of 0.1, with
q
= 0.75,
k
= 0,
v
= 0.0.
Figure 7.6 Effect of
a
for versus
t
-Time for
a
= 0.2–1.1 in steps of 0.1, with
q
= 0.25,
a
= 1.0,
k
= 0,
v
= 0.0.
Figure 7.7 Effect of
a
for versus
t
-Time for
a
= 0.2–1.1 in steps of 0.1, with
q
= 0.75,
a
= 1.0,
k
= 0,
v
= 0.0.
Figure 7.8 Effect of
a
for versus
t
-Time for
a
= 0.7–1.0 in steps of 0.1, with
q
= 0.25,
a
= 1.0,
k
= 0,
v
= 0.0.
Figure 7.9 Effect of
a
for versus
t
-Time for
a
= 0.7–1.0 in steps of 0.1, with
q
= 0.75,
a
= 1.0,
k
= 0,
v
= 0.0.
Figure 7.10 Effect of
v
for versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.25,
a
= 1.0,
k
= 0.
Figure 7.11 Effect of
v
for
versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.75,
a
= 1.0,
k
= 0.
Figure 7.12 Effect of
v
for
versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.25,
a
= 1.0,
k
= 0.
Figure 7.13 Effect of
v
for
versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.75,
a
= 1.0,
k
= 0.
Figure 7.14 Effect of
v
for
versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.25,
a
= 1.0,
k
= 0.
Figure 7.15 Effect of
v
for versus
t
-Time for
v
= −0.6 to 0.6 in steps of 0.6, with
q
= 0.75,
a
= 1.0,
k
= 0.
Figure 7.16 Effect of
k
for versus
t
-Time for
k
= 0–6 in steps of 1, with
q
= 6/7,
a
= 1.0,
k
= 0,
v
= 0.
Figure 7.17 Phase plane plot versus for
q
= 0.4–2.0 in steps of 0.2,
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.18 and versus t-Time for
q
= 0.2–1.0 in steps of 0.2 and
q
= 0.5, with
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.19 and versus t-Time for
q
= 0.2–1.0 in steps of 0.2 and
q
= 0.5, with
a
= 1.0,
v
= 0,
k
= 0.
Figure 7.20 Phase plane plot versus for
q
= 0.2–1.0 in steps of 0.2, and
q
= 0.5,
a
= 1.0,
v
= 0,
k
= 0. Arrows indicate increasing
t
.
Figure 7.21 Phase plane plot versus for
q
= 0.2–1.0 in steps of 0.2, and
q
= 0.5,
a
= 1.0,
v
= 0,
k
= 0. Arrows indicate increasing
t
.
Chapter 8: The R3-Trigonometric Functions
Figure 8.1 and versus
t
-Time for
a
= 1,
k
= 0,
v
= 0,
q
= 0.2–1.0 in steps of 0.2.
Figure 8.2 and versus
t
-Time for
a
= 1,
k
= 0,
v
= 0,
q
= 1.0–1.5 in steps of 0.1.
Figure 8.3 Effect of
a
, , and versus
t
-Time for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.25,
k
= 0,
v
= 0.
Figure 8.4 Effect of
a
, , and versus
t
-Time for
a
= 0.25–1.0 in steps of 0.25, with
q
= 0.75,
k
= 0,
v
= 0.
Figure 8.5 Effect of
v
, , and versus
t
-Time for
v
= −0.6–0.6 in steps of 0.3, with
q
= 0.25,
k
= 0,
a
= 1.
Figure 8.6 Effect of
v
, , and versus
t
-Time for
v
= −0.6–0.6 in steps of 0.3, with
q
= 0.75,
k
= 0,
a
= 1.
Figure 8.7 Effect of
k
, for , with
k
= 0–6 in steps of 1,
q
= 5/7,
a
= 1.0,
v
= 0.
Figure 8.8 Phase plane versus for
q
= 1.0–2.0 in steps of 0.2, with
a
= 1.0,
v
= 0, and
t
= 0–7.0.
Figure 8.9 versus
t
-Time for
a
= 1,
v
= 0,
k
= 0, and
q
= 0.1–0.5 in steps of 0.1.
Figure 8.10
versus
t
-Time for
a
= 1,
v
= 0,
k
= 0, and
q
= 0.1–0.5 in steps of 0.1.
Figure 8.11
versus
t
-Time for
a
= 1,
v
= 0,
k
= 0, and
q
= 0.1–0.5 in steps of 0.1.
Figure 8.12 versus
t
-Time for
a
= 1,
v
= 0,
k
= 0, and
q
= 0.1–0.5 in steps of 0.1.
Figure 8.13 Phase plane versus for
q
= 0.5–1.5 in steps of 0.1,
a
= 1.0,
k
= 0,
v
= 0,
t
= 0–10.
Figure 8.14 Phase plane versus for
q
= 0.5–1.5 in steps of 0.1,
a
= 1.0,
k
= 0,
v
= 0,
t
= 0–12.
Chapter 9: The Fractional Meta-Trigonometry
Figure 9.1 Graphical display of for in steps of .
Figure 9.2 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.05,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.3 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.00,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.4 Effect of
on
with
= 1.0–3.0 in steps of 0.2, with
q
= 0.75,
a
= 1,
,
v
= 0,
k
= 0.
Figure 9.5 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 0.25,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.6 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.05,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.7 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.00,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.8 Effect of
on
with
= 1.0–3.0 in steps of 0.2, with
q
= 0.75,
a
= 1,
,
v
= 0,
k
= 0.
Figure 9.9 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 0.50,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.10 Effect of on with (a) = 1.0–2.0 and (b) = 2.0–3.0 in steps of 0.2, with
q
= 1.05,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.11 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.00,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.12 Effect of
on
with
= 1.0–3.0 in steps of 0.2, with
q
= 0.75,
a
= 1,
,
v
= 0,
k
= 0.
Figure 9.13 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 0.50,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.14 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 1.05,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.15 Effect of
on
with
= 1.0–3.0 in steps of 0.2, with
q
= 1.05,
a
= 1,
,
v
= 0,
k
= 0.
Figure 9.16 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 0.75,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.17 Effect of on with = 1.0–3.0 in steps of 0.2, with
q
= 0.50,
a
= 1, ,
v
= 0,
k
= 0.
Figure 9.18 Phase plane versus for = 1–3 in steps of 0.2, with
a
= 1.0,
q
= 1.00, ,
k
= 0,
v
= 0. Arrows indicate increasing function.
Figure 9.19 Phase plane
versus
for
= 1–3 in steps of 0.2, with
a
= 1.0,
q
= 1.05,
k
= 0,
,
v
= 0. Arrows indicate increasing function parameter.
Figure 9.20 Phase plane versus for = 1–3 in steps of 0.2, with
a
= 1.0,
q
= 0.95,
k
= 0, ,
v
= 0.
Figure 9.21 Phase plane versus for = 1–3 in steps of 0.2, with
a
= 1.0,
q
= 0.50,
k
= 0, ,
v
= 0. Arrows indicate increasing function parameter.
Figure 9.22 Phase plane versus for = 1–3 in steps of 0.2, with
v
= 1.0,
a
= 1.0,
q
= 1.00,
k
= 0, . Arrows indicate increasing function parameter.
Figure 9.23 Phase plane versus for = 1–3 in steps of 0.2, with
v
= −1.0,
a
= 1.0,
q
= 1.00,
k
= 0, .
Figure 9.24 Phase plane versus for = 1–1.5 in steps of 0.1, with
q
= 1.15,
v
= 0,
a
= 1.0,
k
= 0, .
Figure 9.25 Phase plane versus for = 1–3 in steps of 0.25, with
q
= 0.95,
v
= 0,
a
= 1.0,
k
= 0, .
Figure 9.26 Phase plane versus for = 1–2 in steps of 0.2, with
a
= −1.0,
q
= 0.85,
v
= 0,
k
= 0, ,
t
= 0–12.
Figure 9.27 Phase plane versus and versus for = 0.2, with
a
= 1.0,
q
= 1.10,
v
= −1,
k
= 0, ,
t
= 0–18.
Figure 9.28 Phase plane
versus
and for
versus
= 0, with
a
= 1.0,
q
= 1.05,
v
= 0.2,
k
= 0,
,
t
= 0–20.
Figure 9.29 Phase plane
versus
and for
versus
,
= 0.1, with
a
= 1.0,
q
= 1.05,
v
= 0,
k
= 0,
,
t
= 0–20.
Figure 9.30 Phase plane
versus
and for
versus
= 0.1, with
a
= 1.0,
q
= 1.05,
v
= 0.2,
k
= 0,
,
t
= 0–20.
Figure 9.31 Phase plane versus and for versus , = 0, with
a
= 1.0,
q
= 1.05,
v
= 0.05,
k
= 0, ,
t
= 0–10.
Figure 9.32 The effect of
q
for ,
q
= 0.1–1.0 in steps of 0.1, with
a
= 1,
v
= 0,
k
= 0, = 0.5, = 0.5.
Figure 9.33 The effect of
q
for
,
q
= 0.1–1.0 in steps of 0.1, with
a
= 1,
v
= 0,
k
= 0,
= 0.5,
= 0.5.
Figure 9.34 The effect of
q
for
,
q
= 0.1–1.0 in steps of 0.1, with
a
= 1,
v
= 0,
k
= 0,
= 0.5,
= 0.5.
Figure 9.35 The effect of
q
for ,
q
= 0.1–1.0 in steps of 0.1, with
a
= 1,
v
= 0,
k
= 0, = 0.5, = 0.5.
Figure 9.36 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0.
Figure 9.37 The effect of
and
,
, for
, with
= 0.0–2.0 in steps of 0.2, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0.
Figure 9.38 The effect of
and
,
, for
, with
= 0.0–2.0 in steps of 0.2, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0.
Figure 9.39 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0.
Figure 9.40 The effect of
a
for , with
a
= 0.25–2.0 in steps of 0.25, and with
q
= 0.5, = 0.5, = 0.5,
v
= 0,
k
= 0.
Figure 9.41 The effect of
a
for , with
a
= 0.25–2.0 in steps of 0.25, and with
q
= 0.5, = 0.5, = 0.5,
v
= 0,
k
= 0.
Figure 9.42 The effect of
v
for , with
v
= −1.0–1.0 in steps of 0.25, and with
q
= 0.5,
a
= 1, = 0.5, = 0.5,
k
= 0.
Figure 9.43 The effect of
v
for , with
v
= −1.0–1.0 in steps of 0.25, and with
q
= 0.5,
a
= 1, = 0.5, = 0.5,
k
= 0.
Figure 9.44 Effect of
k
, for
k
= 0–4, with
q
= 3/5,
a
= 1, = 0.5, = 0.5,
v
= 0.
Figure 9.45 Phase plane showing the effect of and , , for versus with = 0.0–1.0 in steps of 0.1, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0,
t
= 0–9.
Figure 9.46 Phase plane showing the effect of and , , for versus with = 1.0–2.0 in steps of 0.1, and with
q
= 0.5,
a
= 1,
v
= 0,
k
= 0,
t
= 0–9.
Figure 9.47 Phase plane showing the effect of for versus with = 1.0–2.0 in steps of 0.1, and with
q
= 0.5,
a
= 1,
v
= 0, = 1,
k
= 0.
Figure 9.48 . Phase plane showing the effect of for versus with = 1.0–2.0 in steps of 0.1, and with
q
= 0.5,
a
= 1,
v
= 0, = 3,
k
= 0.
Figure 9.49 Phase plane for versus with = 0.16, = 1.25, and with
q
= 0.80,
a
= 0.8,
v
= 0,
k
= 0,
t
= 0–40.
Figure 9.50 Phase plane for versus with = 0.5, and with
q
= 1.09,
a
= 1.0,
v
= 0,
k
= 0,
t
= 0–19.
Figure 9.51 Phase plane for versus and versus with = 0.5–0.8, = 0.6, and with
q
= 1.04,
v
= −0.3,
a
= 1.0,
k
= 0,
t
= 0–4.8.
Figure 9.52 Phase plane for versus and versus with = 0.5–0.8, = 0.6, and with
q
= 1.04,
v
= −0.3,
a
= 1.0,
k
= 0,
t
= 0–4.8.
Figure 9.53 Taxonomy of the fractional meta-trigonometric functions.
Chapter 10: The Ratio and Reciprocal Functions
Figure 10.1 Effect of
q
with; versus
t
-Time for , , , and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.2 Effect of
q
with;
versus
t
-Time for
,
,
, and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.3 Effect of
q
with; versus
t
-Time for , , , and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.4 Effect of
q
with;
versus
t
-Time for
,
,
, and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.5 Effect of
q
with;
versus
t
-Time for
,
,
, and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.6 Effect of
q
with; versus
t
-Time for , , , and
q
= 0.25–1.5 in steps of 0.25,
t
= 0–10.
Figure 10.7 Effect of
q
with versus
t
-Time for , ,
a
= 1.0, ,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.8 Effect of
q
with
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.9 Effect of
q
with
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.10 Effect of
q
with
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.11 Effect of
q
with
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.12 Effect of
q
with 1/
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.13 Effect of
q
with
versus
t
-Time for
,
,
a
= 1.0,
,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.14 Effect of
q
with 1/ versus
t
-Time for ,,
a
= 1.0, ,
q
= 0.25–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.15 Effect of
q
with versus
t
-Time for , ,
a
= 1.0, ,
q
= 0.5–1.25 in steps of 0.25,
t
= 0–10.
Figure 10.16 Effect of
q
with versus
t
-Time for , ,
a
= 1.0, ,
q
= 0.5–1.25 in steps of 0.25,
t
= 0–10.
Chapter 12: The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry
Figure 12.1 Multiple solutions,
x
(
t
), for equation (12.69), given by equations (12.70,
k
= 0), (12.71,
k
= 1), and (12.72,
k
= 2).
a
= 1, , , and .
Chapter 13: Fractional Trigonometric Systems
Figure 13.1 System impulse response, , equation (13.8), with
q =
0.25–2.0 in steps of 0.25, .
Figure 13.2 System step response, , equation (13.9), with
q =
0.25–2.0 in steps of 0.25, .
Figure 13.3 System frequency response for equation (13.10) with
v
= 0, that is with
q =
0.25–2.0 in steps of 0.25, . (a) Magnitude ratio in decibel. (b) Phase angle in degrees.
Figure 13.4 System frequency response for , with
q =
0.25–2.0 in steps of 0.25, .
Figure 13.5 System frequency response for , with
q =
0.25–2.0 in steps of 0.25. (a) Magnitude ratio in decibel. (b) Phase angle in degrees.
Figure 13.6 Stability and resonance regions of the elementary transfer function of the second kind in the versus
u
plane. Area 1, stable with no resonant frequency; Area 2, stable with a single resonant frequency; Area 3, stable with two resonant frequencies; Area 4, unstable.
Figure 13.7 Neutrally stable, underdamped stable, and oscillatory unstable transfer functions of the second kind. with
q
= 1/3,
v
= 0,
a
= 1.0, , .
Chapter 14: Numerical Issues and Approximations in the Fractional Trigonometry
Figure 14.1 Size of the
n
th term versus
n
, with
q
= 1.0 and .
Figure 14.2 and the approximation versus
t
-Time. Solid line is and dashed is approximation,
q
= 0.4 and 0.6,
v
= 0.0,
a
= 1.75.
Figure 14.3 Approximation of by .
Figure 14.4 Approximation of by equation (14.19).
Figure 14.5 Approximation of by equation (14.22),
q
= 1.0–2.0 in steps of 0.25. Approximations are dashed and overlap at
q
= 2.0.
Figure 14.6 Approximation of by equation (14.24). With
q
= 1.0–2.0 in steps of 0.25,
a
= −4,
v
= 0. Approximations are dashed and overlap at
q
= 2.0.
Figure 14.7 Approximation of by equation (14.25). With
q
= 2.0–3.0 in steps of 0.25,
a
= −1,
v
= 0. Approximations are dashed and overlap at
q
= 2.0.
Figure 14.8 Approximation of by equation (14.26) with
q
= 0.2–1.0 in steps of 0.2. Note overlap of approximation and function for all
q
at times . Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 14.9 Number of functions versus Order-
q
for
q
denominators .
Figure 14.10 Reciprocal of Number of functions versus Order-
q
for
q
denominators .
Chapter 15: The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry
Figure 15.1 Phase plane plot, versus for
q
= 0.1–1 in steps of 0.1. With
a
= 1,
v
= 0,
t
= 0–7.4.
Figure 15.2 Phase plane plot, versus . Effect of
q
, with
q
= 1.0–2.0 in steps of 0.1,
a
= 1.0,
v
= 0,
t
= 0 –7.4. Arrows indicate
t
increasing.
Figure 15.3 Phase plane plots, versus for
q
= 0.2–3 in steps of 0.2 with
a
= 1,
v
= 0. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer. (a) , (b) , (c)
Figure 15.4 Phase plane plot for versus for
q
= 0.2–2 in steps of 0.2,
v
= 0,
a
= 1,
k
= 0,
t
= 0–10.
Figure 15.5 Phase plane plot for versus for
q
= 2–5 in steps of 0.25,
v
= 0,
a
= 1,
k
= 0,
t
= 0–4.
Figure 15.6 Effect of
v
on versus , phase plane plot with
q
= 1.20,
v
= 0–0.15 in steps of 0.05,
a
= 0.25,
k
= 0,
t
= 0–28.
Figure 15.7 Four views of versus versus
h
for
k
= 0–38.
Figure 15.8 (a–d). Four views of versus versus
h
, for
k
= 0–38. With
a
= 1.0, and
v
= 0.
Figure 15.9 Phase plane plot for versus for
q
= 0.8,
v
= −1,
a
= 1.0, in steps of 0.1, ,
k
= 0 and
t
= 0–20.
Figure 15.10 Phase plane plot for versus and versus (dotted) for
q
= 1.07,
v
= −0.070,
a
= 1.0, , in steps of 0.05,
k
= 0 and
t
= 0–10.
Figure 15.11 Phase plane plot for versus and versus (dashed) for
q
= 0.5,
v
= 1,
a
= 1.0, , ,
k
= 0 and
t
= 0–22.
Figure 15.12 Phase plane plot for versus and versus (dashed) for
q
= 1.2,
v
= −0.8,
a
= 1.0, , ,
k
= 0 and
t
= 0–14.
Figure 15.13 Phase plane plot for versus and versus (dashed) for
q
= 0.5,
v
= 0,
a
= 1.0, , ,
k
= 0 and
t
= 0–22.
Figure 15.14 Phase plane plot for versus and versus (dotted) for
q
= 1.06,
v
= −0.01,
a
= 1.0, , in steps of 0.05,
k
= 0 and
t
= 0–2.5.
Figure 15.15 Coordinate systems for both polar and parametric forms.
Figure 15.16 Parametric spiral interpreted as a three-dimensional space curve.
Figure 15.17 Polar plot showing two simple counterclockwise spirals bounding (a) a general continuous and (b) a general discontinuous counterclockwise spiral.
Figure 15.18 Velocity components.
Figure 15.19 Angular retardation demonstrated for spiral defined by with
q
= 1.1 for test spiral and
q
= 1.0 for reference spiral; for both spirals,
a
= 1,
k
= 0, and
v
= 0, and .
Figure 15.20 Time retardation demonstrated for spiral defined by with
q
= 1.1 for test spiral and
q
= 1.0 for reference spiral; for both spirals,
a
= 1,
k
= 0, and
v
= 0,
.
Figure 15.21 Rectangular plots for some of the classical spirals.
Figure 15.22 Radius versus -angle for spiral defined by ,
q
= 0.2–2.0 by 0.2,
v
= 0,
a
= 1,
k
= 0 with marker data points for
t
= 2.0, 3.0, 4.0, and 5.0.
Figure 15.23 Radius versus angle for spiral defined by ,
q
= 0.1–1.0 by 0.1,
v
= 0,
a
= 1,
k
= 0 with marker data points for
t
= 1.0, 2.0, 3.0, 4.0, and 5.0.
Figure 15.24 Radius versus angle for spiral defined by versus ,
q
= 0.1–1.0 by 0.1,
v
= 0,
a
= 1,
k
= 0 with marker data points for
t
= 1.0, 2.0, 3.0, 4.0, and 5.0.
Figure 15.25 Comparison of an Archimedean spiral 15-25a and a Fractional spiral 15-25b. (a) Archimedean spiral defined by with
B
= 1.25 and
m
= 1.8, . (b) Fractional spiral with versus and versus , dashed line, for
q
= 1.15,
v
= 0,
a
= 1.0, , ,
k
= 0 and
t
= 0–8.
Figure 15.26 Comparison of the Spiral of Theodorus, and a fractional spiral. Solid line is the Theodorus spiral with
k
f
= 1000,
b
= −1–33.35 in steps of 0.05, and circles are a meta-trigonometric spiral with
q
= 1.0239,
v
= −0.22, , ,
k
=0, .
Chapter 16: Fractional Oscillators
Figure 16.1 Block diagram for interpreted as a physical system.
Figure 16.2 A set of linear fractional oscillators based on the complexity function , with from equation (16.9), ,
q
= 0.5,
v
= 0,
a
= 1.0, and
k
= 0.
Figure 16.3 A set of linear fractional oscillators based on the parity function , with from equation (16.14), ,
q
= 0.5,
v
= 0,
a
= 1.0, , and
k
= 0.
Figure 16.4 Effect of variation of
k
, for a set of linear fractional oscillators based on the parity function , with from equation (16.14)
q
= 0.9,
v
= 0,
a
= 1.0, , .
Figure 16.5 Spatial discretization.
Figure 16.6 Coupled fractional oscillators. Individual oscillators are indicated by dashed boxes.
Chapter 17: Shell Morphology and Growth
Figure 17.1 Shell number 2. Morphological models: (a) Logarithmic spiral, (b)
R
1
-trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels.
Figure 17.3 Shell number 4. Morphological models: (a) Logarithmic spiral, (b)
R
1
-trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels.
Figure 17.4 Shell 4,
x, y
, and radius components of
R
1
-trigonometric spiral shown in Figure 17.3b versus fractional spiral rotation in radians.
Figure 17.5 Expanded view spiral radii (in pixels) for logarithmic and fractional spirals versus spiral angle theta (in radians).
Figure 17.6 Shell number 1. Siphuncle morphological models: (a) Image without model, (b)
R
1
-trigonometric spiral of observed locus, (c) panel (b) enlarged, (d) spirals enlarged further with model of hidden locus indicated. See text and Table 17.3 for parameters.
Figure 17.7 Shell number 4, Growth rate in mm/day versus
t
– time (days), ().
Figure 17.8 Shell 5. (a) Normal view of cut shell, (b) top view of spire, (c) spire with spiral fit, (d) enlarged view of spire, (e) enlarged view with spiral fit, (f) view of body interior, (g) body interior with spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.9 Shell 6. (a) Cut shell side view, (b) shell top view, (c) view showing interior fractional spiral fit, (d) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.10 Shell 7. (a) External view of cut, (b) cut section of shell, (c) fractional spiral fit, (d) magnified view of spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.11 Shell 8. (a) External view of cut, (b) magnified view near origin, (c) view of cut section, (d) inner spiral fit, (e) outer spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.12 Shell 9. (a) External view of cut, (b) fractional spiral fit, (c) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.13 Shell 10. (a) Top view, (b) location of cut, (c) side view of cut, (d) fractional spiral fit. (e) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.14 Ammonite fossil. (a) External view of fossil, (b) fractional spiral fit, (c) magnified view, (d) magnified view of fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Chapter 18: Mathematical Classification of the Spiral and Ring Galaxy Morphologies
Figure 18.1 The morphological classification of the galaxies as suggested by Edwin Hubble.
Figure 18.2 Effect of order,
q
. versus and versus ,
q
= 1.0–1.8 in steps of 0.2,
v
= 0.0,
a
= 1.0, .
Figure 18.3 Effect of , versus and versus ,
q
= 1.2,
v
= 0.2,
a
= 1.0, ,
k
= 0.
Figure 18.4 Effect of
v
, versus and versus , .
Figure 18.5 Effect of Order,
q
, with , versus and versus ,
q
= 1.0–1.75 in steps of 0.25,
a
= 1.0, .
Figure 18.6 Effect of , versus and versus , . .
Figure 18.7 Effect of , versus and versus , .
Figure 18.8 Effect of small parameter changes in , versus with
q
= 1.06,
v
= −0.5, ,
a
= 1.0,
k
= 0, .
Figure 18.9 Manipulation of a fractional spiral for image comparison. (a) Face-on view of scaled test spiral. (b) Spiral rotated about
y
0
axis by 40°. (c) Spiral rotated about
x
0
axis by 65°. (d) Spiral rotated about
x
1
axis by 65° and spiral rotated about
y
3
axis by 40°. (e) In-plane rotation of
x
4
,
y
4
spiral by −15°.
Figure 18.10 Sample data display for NGC 1300. This display shows three scaling of the same spiral. Two spirals to enclose the spiral arms along with a median spiral. A nucleus outline is shown displaced from the spiral center.
Figure 18.11 Classification for NGC 4314. See text for fit parameters. Solid curve is primary arm and dashed is secondary arm.
Figure 18.12 Classification for NGC 1365. See text for fit parameters. Dashed curve is primary arm and phantom arm extends primary to origin.
Figure 18.13 Classification for M 95. See text for fit parameters.
Figure 18.14 Classification for NGC 2997. See text for fit parameters.
Figure 18.15 Classification for NGC4622. See text for parameters.
Figure 18.16 Classification for M 66 or NGC 3627. See text for fit parameters.
Figure 18.17 (a) Classification for NGC 4535. See text for fit parameters for first model. (b) Classification for NGC 4535. See text for fit parameters for second model.
Figure 18.18 Classification for NGC 1300. See text for fit parameters.
Figure 18.19 Classification for Hoag's Object. See text for fit parameters.
Figure 18.20 Classification for M 51. See text for fit parameters.
Figure 18.21 Classification for AM 0644-741. See text for fit parameters.
Figure 18.22 Classification for ESO 269-G57. See text for fit parameters.
Figure 18.23 Classification for NGC 1313. See text for fit parameters.
Figure 18.24 Galaxy order parameter
q
versus Hubble class.
Figure 18.25 Galaxy order parameter
q
− 1 versus
v
for galaxies with and .
Figure 18.26 Limits for
v
parameter, for galaxies with , , and .
Figure 18.27 Effect of and , Hard barred spiral. versus and versus , , .
Figure 18.28 Effect of and . Normal spiral. versus and versus , , .
Figure 18.29 Classification for Carbon Star 3068. See text for fit parameters. (a) Original image (b) magnified view, (c) with analytical fit.
Chapter 19: Hurricanes, Tornados, and Whirlpools
Figure 19.1 Hurricane Fran off the Florida coast. (a) Reference image, (b) model. See text for spiral parameters.
Figure 19.2 Hurricane Isabel. (a) Hurricane Isabel reference image, (b) model. See text for spiral parameters.
Figure 19.3 Magnified view of Hurricane Isabel showing eye and fit of extended streams. See text for parameters. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 19.4 Simplified tornado model, 3D time slice.
Figure 19.5 Tornado morphology as function of
k
index, for
q
= 48/49,
v
= 0, and
k
= 0, 1, 2, 3, 4. (a)
k
= 0, (b)
k
= 1, (c)
k
= 2, (d)
k
= 3, (e)
k
= 4.
Figure 19.6 Tornado morphology
k
= 1, for
q
= 98/99.
Figure 19.7 Low-pressure system over Iceland. (a) Reference view, (b) fractional spiral model, and (c) fractional model enlarged. See text for spiral parameters.
Figure 19.8 Whirlpool off the coast of Japan, March 2011. (a) Reference image, (b) fractional spiral model. See text for model parameters.
Chapter 20: A Look Forward
Figure 20.1 Classical connections.
Figure 20.2
R
-Function as a function of time with
q
= 0.25 to 2.5 in steps of 0.25,
v
= 0, and
a
= 1. Circles mark the approximate local minima. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 20.3 Inverse
R
-Function, , with
q
= 0.25 to 2.5 in steps of 0.25,
v
= 0, and
a
= 1. Circles mark the approximate local minima from Figure 20.2. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 20.4 Negative time issue. versus
t
-time for
a
= 1.0, with
q
= 0.5,
v
= 0.5,
k
= 0, ; , and . For , the function is interpreted as .
Figure 20.5 Negative time issue. versus
t
-time for
a
= 1.0, with
q
= 1.0 and
v
= 0, ,
k
= 0, . For , the function is interpreted as . Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Appendix B: Computer Code
Figure B.1
R
-Function evaluation output.
Figure B.2 Evaluation program output.
Appendix C: Tornado Simulation
Figure C.1 Simulation final frame.
List of Tables
Chapter 3: The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus
Table 3.1 Special fractional calculus functions
Table 3.2 Special case coefficients for indexed forms of the
R
-function using equation (3.136) or (3.140)
Chapter 4: R-Function Relationships
Table 4.1
R
-Function relationships
positive parameter
Table 4.2
R
-Function relationships
negative parameter
Chapter 6: The R1-Fractional Trigonometry
Table 6.1 Summary of the principal
R
1
-trigonometric,
R
1
-hyperbolic, and the traditional functions
Chapter 7: The R2-Fractional Trigonometry
Table 7.1 Summary of
R
2
-functions
Table 7.2 Summary of
R
2
-functions
Chapter 8: The R3-Trigonometric Functions
Table 8.1 Summary of
R
3
-functions
Table 8.2 Summary of
R
3
-functions
Chapter 9: The Fractional Meta-Trigonometry
Table 9.1 Special values of the generalized trignobolic functions
Table 9.2 Summary of the meta-trigonometric functions
Table 9.3 Summary of the meta-trigonometric functions
Chapter 15: The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry
Table 15.1 The classical spirals in polar, rectangular, and parametric forms
Chapter 17: Shell Morphology and Growth
Table 17.1 Shell dimensions
Table 17.2 Parameters for morphological fit using
R
1
-trigonometric fractional spirals
Table 17.3 Parameters for morphological fit of shell 1 siphuncle locus using the
R
1
-trigonometric fractional spirals
Chapter 18: Mathematical Classification of the Spiral and Ring Galaxy Morphologies
Table 18.1 Summary of galaxy modeling data and Hubble classifications as found on the NASA NED database
Chapter 19: Hurricanes, Tornados, and Whirlpools
Table 19.1 Fitting parameters for Hurricane Fran
Table 19.2 Parameters for Whirlpool model
Table 19.3 Order and spatial stability associated with various physical processes
