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E-grāmata: Fractional Trigonometry - With Applications to Fractional Differential Equations and Science: With Applications to Fractional Differential Equations and Science [Wiley Online]

(University of Akron, OH), (NASA Glenn Research Center, Cleveland, OH)
  • Formāts: 464 pages
  • Izdošanas datums: 30-Dec-2016
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 1119139449
  • ISBN-13: 9781119139447
Citas grāmatas par šo tēmu:
  • Wiley Online
  • Cena: 137,50 €*
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Fractional Trigonometry - With Applications to Fractional Differential Equations and Science: With Applications to Fractional Differential Equations and SciencePalielināt
  • Formāts: 464 pages
  • Izdošanas datums: 30-Dec-2016
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 1119139449
  • ISBN-13: 9781119139447
Citas grāmatas par šo tēmu:
Wiley Online E-booksMore info about Wiley Online
Rokasgrāmatas mājas lapa: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119139447
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 Riemanns complementary-function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University.
Preface xv
Acknowledgments xix
About the Companion Website xxi
1 Introduction 1(8)
1.1 Background
2(1)
1.2 The Fractional Integral and Derivative
3(3)
1.2.1 Grunwald Definition
3(1)
1.2.2 Riemann-Liouville Definition
4(1)
1.2.3 The Nature of the Fractional-Order Operator
5(1)
1.3 The Traditional Trigonometry
6(2)
1.4 Previous Efforts
8(1)
1.5 Expectations of a Generalized Trigonometry and Hyperboletry
8(1)
2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation 9(10)
2.1 The Fundamental Fractional Differential Equation
9(1)
2.2 The Generalized Impulse Response Function
10(1)
2.3 Relationship of the F-function to the Mittag-Leffler Function
11(1)
2.4 Properties of the F-Function
12(1)
2.5 Behavior of the F-Function as the Parameter a Varies
13(3)
2.6 Example
16(3)
3 The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus 19(28)
3.1 Introduction
19(1)
3.2 Functions for the Fractional Calculus
19(3)
3.2.1 Mittag-Leffler's Function
20(1)
3.2.2 Agarwal's Function
20(1)
3.2.3 Erdelyi's Function
20(1)
3.2.4 Oldham and Spanier's, Hartley's, and Matignon's Function
20(1)
3.2.5 Robotnov's Function
21(1)
3.2.6 Miller and Ross's Function
21(1)
3.2.7 Gorenflo and Mainardi's, and Podlubny's Function
21(1)
3.3 The R-Function: A Generalized Function
22(1)
3.4 Properties of the Rqv(a, t)-Function
23(4)
3.4.1 Differintegration of the R-Function
23(2)
3.4.2 Relationship Between Rq,mq and Rq,o
25(2)
3.4.3 Fractional-Order Impulse Function
27(1)
3.5 Relationship of the R-Function to the Elementary Functions
27(2)
3.5.1 Exponential Function
27(1)
3.5.2 Sine Function
27(1)
3.5.3 Cosine Function
28(1)
3.5.4 Hyperbolic Sine and Cosine
28(1)
3.6 R-Function Identities
29(2)
3.6.1 Trigonometric-Based Identities
29(1)
3.6.2 Further Identities
30(1)
3.7 Relationship of the R-Function to the Fractional Calculus Functions
31(1)
3.7.1 Mittag-Leffler's Function
31(1)
3.7.2 Agarwal's Function
31(1)
3.7.3 Erdelyi's Function
31(1)
3.7.4 Oldham and Spanier's, and Hartley's Function
31(1)
3.7.5 Miller and Ross's Function
32(1)
3.7.6 Robotnov's Function
32(1)
3.7.7 Gorenflo and Mainardi's, and Podlubny's Function
32(1)
3.8 Example: Cooling Manifold
32(2)
3.9 Further Generalized Functions: The G-Function and the H-Function
34(4)
3.9.1 The G-Function
34(2)
3.9.2 The H-Function
36(2)
3.10 Preliminaries to the Fractional Trigonometry Development
38(1)
3.11 Eigen Character of the R-Function
38(1)
3.12 Fractional Differintegral of the TimeScaled R-Function
39(1)
3.13 R-Function Relationships
39(1)
3.14 Roots of Complex Numbers
40(1)
3.15 Indexed Forms of the R-Function
41(3)
3.15.1 R-Function with Complex Argument
41(1)
3.15.2 Indexed Forms of the R-Function
42(37)
3.15.2.1 Complexity Form
42(1)
3.15.2.2 Parity Form
43(1)
3.16 Term-by-Term Operations
44(2)
3.17 Discussion
46(1)
4 R-Function Relationships 47(16)
4.1 R-Function Basics
47(1)
4.2 Relationships for Rm,o in Terms of R1,o
48(2)
4.3 Relationships for R1/m,o in Terms of R1,o
50(1)
4.4 Relationships for the Rational Form Rm/p,o in Terms of R1p,o
51(2)
4.5 Relationships for R1/po in Terms of Rm/p,o
53(1)
4.6 Relating Rm/p,o to the Exponential Function R1,o(b, t) = ebt
54(2)
4.7 Inverse Relationships-Relationships for R1,o in Terms of Rm,k
56(1)
4.8 Inverse Relationships-Relationships for R1,o in Terms of R1/m,o
57(2)
4.9 Inverse Relationships-Relationships for eat = R1,o(a, t) in Terms of Rm/p,o
59(2)
4.10 Discussion
61(2)
5 The Fractional Hyperboletry 63(16)
5.1 The Fractional R1-Hyperbolic Functions
63(9)
5.2 R1-Hyperbolic Function Relationship
72(1)
5.3 Fractional Calculus Operations on the R1-Hyperbolic Functions
72(1)
5.4 Laplace Transforms of the R1-Hyperbolic Functions
73(1)
5.5 Complexity-Based Hyperbolic Functions
73(1)
5.6 Fractional Hyperbolic Differential Equations
74(2)
5.7 Example
76(1)
5.8 Discussions
77(2)
6 The R1-Fractional Trigonometry 79(18)
6.1 R1-Trigonometric Functions
79(9)
6.1.1 R1-Trigonometric Properties
81(7)
6.2 R1-Trigonometric Function Interrelationship
88(1)
6.3 Relationships to R1-Hyperbolic Functions
89(1)
6.4 Fractional Calculus Operations on the R1-Trigonometric Functions
89(1)
6.5 Laplace Transforms of the R1-Trigonometric Functions
90(2)
6.5.1 Laplace Transform of R1 Cosq,v(a, k, t)
90(1)
6.5.2 Laplace Transform of R1Sinq,v(a, k, t)
91(1)
6.6 Complexity-Based R1-Trigonometric Functions
92(2)
6.7 Fractional Differential Equations
94(3)
7 The R2-Fractional Trigonometry 97(32)
7.1 R2-Trigonometric Functions: Based on Real and Imaginary Parts
97(5)
7.2 R2-Trigonometric Functions: Based on Parity
102(9)
7.3 Laplace Transforms of the R2-Trigonometric Functions
111(2)
7.3.1 R2Cosq,v(a, k, t)
111(1)
7.3.2 R2Sinq,v(a, k, t)
112(1)
7.3.3 L{R2Coflq,v(a, k, t)}
113(1)
7.3.4 L{R2Flutq,v(a, k, t))
113(1)
7.3.5 L{R2Covibq,v(a, k, t)}
113(1)
7.3.6 L{R2Vibq,v(a, k, t)}
113(1)
7.4 R2-Trigonometric Function Relationships
113(6)
7.4.1 R2Cosq,v(a, k, t) and R2Sinq,v(a, k, t) Relationships and Fractional Euler Equation
114(2)
7.4.2 R2Rotq,v(a, t) and R2Corq,v (a, t) Relationships
116(1)
7.4.3 R2Coflq,v(a, t) and R2Flutq,v(a, t) Relationships
116(2)
7.4.4 R2Covibq,v(a, t) and R2Vibq,v(a, t) Relationships
118(1)
7.5 Fractional Calculus Operations on the R2-Trigonometric Functions
119(8)
7.5.1 R2Cosq,v(a, k, t)
119(2)
7.5.2 R2Siq,v(a, k, t)
121(1)
7.5.3 R2Corq,v(a, t)
122(1)
7.5.4 R2Rotq,v(a, t)
122(1)
7.5.5 R2Coflutq,v(a, t)
122(1)
7.5.6 R2Flutq,v(a, k, t)
123(1)
7.5.7 R2Covibq,v(a, k, t)
123(1)
7.5.8 R2Vibq,v(a, k, t)
124(1)
7.5.9 Summary of Fractional Calculus Operations on the R2-Trigonometric Functions
124(3)
7.6 Inferred Fractional Differential Equations
127(2)
8 The R3-Trigonometric Functions 129(30)
8.1 The R3-Trigonometric Functions: Based on Complexity
129(5)
8.2 The R3-Trigonometric Functions: Based on Parity
134(6)
8.3 Laplace Transforms of the R3-Trigonometric Functions
140(1)
8.4 R3 -Trigonometric Function Relationships
141(5)
8.4.1 R3Cosq,v (a, t) and R3Sinq,v(a, t) Relationships and Fractional Euler Equation
142(1)
8.4.2 R3Rotq,v(a, t) and R3Corq,v(a, t) Relationships
143(1)
8.4.3 R3Coflq,v(a, t) and R3Flutq,v,(a, t) Relationships
144(1)
8.4.4 R3Covibq,v(a, t) and R3Vibq,v(a, t) Relationships
145(1)
8.5 Fractional Calculus Operations on the R3-Trigonometric Functions
146(13)
8.5.1 R3Cosq,v(a, k, t)
146(2)
8.5.2 R3Sinq,v(a, k, t)
148(1)
8.5.3 R3Corq,v(a, t)
149(1)
8.5.4 R3Rotq,v(a, t)
150(1)
8.5.5 R3Coflutq,v(a, k, t)
150(2)
8.5.6 R3Flutq,v(a, k, t)
152(1)
8.5.7 R3Covibq,v(a, k, t)
153(1)
8.5.8 R3Vibq,v(a, k, t)
154(3)
8.5.9 Summary of Fractional Calculus Operations on the R3-Trigonometric Functions
157(2)
9 The Fractional Meta-Trigonometry 159(58)
9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity
160(6)
9.1.1 Alternate Forms
161(1)
9.1.2 Graphical Presentation-Complexity Functions
161(5)
9.2 The Meta-Fractional Trigonometric Functions: Based on Parity
166(13)
9.3 Commutative Properties of the Complexity and Parity Operations
179(9)
9.3.1 Graphical Presentation- Parity Functions
181(7)
9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions
188(4)
9.5 R-Function Representation of the Fractional Meta-Trigonometric Functions
192(3)
9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions
195(11)
9.6.1 Cosq,v(a, alpha, beta, k, t)
195(2)
9.6.2 Sinq,v(a, alpha, beta, k, t)
197(1)
9.6.3 Corq,v(a, alpha, beta, t)
198(1)
9.6.4 Rotq,v(a, alpha, beta, t)
198(1)
9.6.5 Coflutq,v(a, alpha , beta, k, t)
199(1)
9.6.6 Flutq,v(a, alpha, beta, k, t)
200(2)
9.6.7 Covibq,v(a, alpha, beta k, t)
202(1)
9.6.8 Vikq,v(a, alpha, beta, k, t)
203(1)
9.6.9 Summary of Fractional Calculus Operations on the Meta-Trigonometric Functions
204(2)
9.7 Special Topics in Fractional Differintegration
206(1)
9.8 Meta-Trigonometric Function Relationships
206(8)
9.8.1 Cosq,v(a, alpha, beta, t) and Sinq,v(a, alpha, beta, t) Relationships
206(1)
9.8.2 Corq,v(a, alpha, beta, t) and Rotq,v(a, alpha, beta, t) Relationships
207(1)
9.8.3 Covibq,v(a, alpha, beta, t) and Vibq,v(a, alpha, beta, t) Relationships
208(1)
9.8.4 Coflq,v(a, alpha, beta, t) and Flutq,v(a, alpha,beta, t) Relationships
208(1)
9.8.5 Coflq,v(a, alpha, beta, t) and Vibq,v(a, alpha, beta, t) Relationships
209(2)
9.8.6 Cosq,v(a, alpha, beta, t) and Sinq,v(a, alpha, beta, t) Relationships to Other Functions
211(1)
9.8.7 Meta-Identities Based on the Integer-order Trigonometric Identities
211(35)
9.8.7.1 The cos(-x) = cos(x)-Based Identity for Cosq,v(a, alpha, beta, t)
211(1)
9.8.7.2 The sin(-x) = - sin(x)-Based Identity for Sinq,v(a, alpha, beta, t)
212(1)
9.8.7.3 The Cosv(a, alpha, beta, t)<=> Sinq,v(a, alpha, beta, t) Identity
212(1)
9.8.7.4 The sin(x) = sin(x ± mpi/2)-Based Identity for Sinq,v(a, alpha, beta, t)
213(1)
9.9 Fractional Poles: Structure of the Laplace Transforms
214(1)
9.10 Comments and Issues Relative to the Meta-Trigonometric Functions
214(1)
9.11 Backward Compatibility to Earlier Fractional Trigonometries
215(1)
9.12 Discussion
215(2)
10 The Ratio and Reciprocal Functions 217(12)
10.1 Fractional Complexity Functions
217(2)
10.2 The Parity Reciprocal Functions
219(2)
10.3 The Parity Ratio Functions
221(4)
10.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions
225(1)
10.5 Relationships
226(1)
10.6 Discussion
227(2)
11 Further Generalized Fractional Trigonometries 229(12)
11.1 The G-Function-Based Trigonometry
229(1)
11.2 Laplace Transforms for the G-Trigonometric Functions
230(4)
11.3 The H-Function-Based Trigonometry
234(1)
11.4 Laplace Transforms for the H-Trigonometric Functions
235(6)
Introduction to Applications 241(2)
12 The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry 243(16)
12.1 Fractional Differential Equations
243(2)
12.2 Fundamental Fractional Differential Equations of the First Kind
245(1)
12.3 Fundamental Fractional Differential Equations of the Second Kind
246(1)
12.4 Preliminaries-Laplace Transforms
246(4)
12.4.1 Fractional Cosine Function
246(2)
12.4.2 Fractional Sine Function
248(1)
12.4.3 Higher-Order Numerator Dynamics
248(1)
12.4.3.1 Fractional Cosine Function
248(1)
12.4.3.2 Fractional Sine Function
248(1)
12.4.4 Parity Functions-The Flutter Function
249(1)
12.4.5 Additional Transform Pairs
250(1)
12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots
250(2)
12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots
252(1)
12.6.1 Repeated Real Fractional Roots
252(1)
12.6.2 Repeated Complex Fractional Roots
253(1)
12.7 Fractional Differential Equations Containing Repeated Roots
253(1)
12.8 Fractional Differential Equations of Non-Commensurate Order
254(1)
12.9 Indexed Fractional Differential Equations: Multiple Solutions
255(1)
12.10 Discussion
256(3)
13 Fractional Trigonometric Systems 259(12)
13.1 The R-Function as a Linear System
259(1)
13.2 R-System Time Responses
260(1)
13.3 R-Function-Based Frequency Responses
260(1)
13.4 Meta-Trigonometric Function-Based Frequency Responses
261(3)
13.5 Fractional Meta-Trigonometry
264(2)
13.6 Elementary Fractional Transfer Functions
266(1)
13.7 Stability Theorem
266(1)
13.8 Stability of Elementary Fractional Transfer Functions
267(1)
13.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions
268(2)
13.9.1 Complexity Function Stability
268(1)
13.9.2 Parity Function Stability
269(1)
13.10 Discussion
270(1)
14 Numerical Issues and Approximations in the Fractional Trigonometry 271(12)
14.1 R-Function Convergence
271(1)
14.2 The Meta-Trigonometric Function Convergence
272(1)
14.3 Uniform Convergence
273(1)
14.4 Numerical Issues in the Fractional Trigonometry
274(1)
14.5 The R2 COS- and R2Sin-Function Asymptotic Behavior
275(1)
14.6 R-Function Approximations
276(3)
14.7 The Near-Order Effect
279(2)
14.8 High-Precision Software
281(2)
15 The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry 283(26)
15.1 The Fractional Spiral Functions
283(5)
15.2 Analysis of Spirals
288(15)
15.2.1 Descriptions of Spirals
288(6)
15.2.1.1 Polar Description
289(1)
15.2.1.2 Parametric Description
290(3)
15.2.1.3 Definitions
293(1)
15.2.1.4 Alternate Definitions
294(1)
15.2.1.5 Examples
294(1)
15.2.2 Spiral Length and Growth/Decay Rates
294(2)
15.2.2.1 Spiral Length
294(1)
15.2.2.2 Spiral Growth Rates for ccw Spirals
295(1)
15.2.2.3 Component Growth Rates
296(1)
15.2.3 Scaling of Spirals
296(3)
15.2.3.1 Uniform Rectangular Scaling
297(1)
15.2.3.2 Nonuniform Rectangular Scaling
297(1)
15.2.3.3 Polar Scaling
298(1)
15.2.3.4 Radial Scaling
298(1)
15.2.3.5 Angular Scaling
298(1)
15.2.4 Spiral Velocities
299(2)
15.2.5 Referenced Spirals: Retardation
301(2)
15.3 Relation to the Classical Spirals
303(4)
15.3.1 Classical Spirals
303(4)
15.4 Discussion
307(2)
16 Fractional Oscillators 309(8)
16.1 The Space of Linear Fractional Oscillators
309(5)
16.1.1 Complexity Function-Based Oscillators
310(1)
16.1.2 Parity Function-Based Oscillators
311(1)
16.1.3 Intrinsic Oscillator Damping
312(2)
16.2 Coupled Fractional Oscillators
314(3)
17 Shell Morphology and Growth 317(24)
17.1 Nautilus pompilius
317(12)
17.1.1 Introduction
317(1)
17.1.2 Nautilus Morphology
318(7)
17.1.2.1 Fractional Differential Equations
323(2)
17.1.3 Spiral Length
325(1)
17.1.4 Morphology of the Siphuncle Spiral
325(1)
17.1.5 Fractional Growth Rate
325(3)
17.1.6 Nautilus Study Summary
328(1)
17.2 Shell 5
329(1)
17.3 Shell 6
330(2)
17.4 Shell 7
332(1)
17.5 Shell 8
332(4)
17.6 Shell 9
336(1)
17.7 Shell 10
336(3)
17.8 Ammonite
339(1)
17.9 Discussion
340(1)
18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341(30)
18.1 Introduction
341(1)
18.2 Background-Fractional Spirals for Galactic Classification
342(5)
18.3 Classification Process
347(3)
18.3.1 Symmetry Assumption
347(1)
18.3.2 Galaxy Image to Spiral or Spiral to Galaxy Image
347(1)
18.3.3 Inclination
347(1)
18.3.4 Data Presentation
348(2)
18.4 Mathematical Classification of Selected Galaxies
350(12)
18.4.1 NGC 4314 SB(rs)a
350(1)
18.4.2 NGC 1365 SBb/SBc/SB(s)b/SBb(s)
350(1)
18.4.3 M95 SB(r)b/SBb(r)/SBa/SBb
350(3)
18.4.4 NGC 2997 Sc/SAB(rs)c/Sc(s)
353(1)
18.4.5 NGC 4622 (R')SA(r)a pec/Sb
353(1)
18.4.6 M 66 or NGC 3627 SAB(s)b/Sb(s)/Sb
353(2)
18.4.7 NGC 4535 SAB(s)c/SB(s)c/Sc/SBc
355(1)
18.4.8 NGC 1300 SBc/SBb(s)/SB(rs)bc
355(3)
18.4.9 Hoag's Object
358(1)
18.4.10 M 51 Sa + Sc
358(1)
18.4.11 AM 0644-741 Sc/Strongly peculiar/
359(1)
18.4.12 ESO 269-G57 (R')SAB(r)ab/Sa(r)
360(2)
18.4.13 NGC 1313 SBc/SB(s)d/SB(s)d
362(1)
18.4.14 Carbon Star 3068
362(1)
18.5 Analysis
362(5)
18.5.1 Fractional Differential Equations
366(1)
18.5.2 Alternate Classification Basis
366(1)
18.6 Discussion
367(3)
18.6.1 Benefits
368(2)
18.7 Appendix: Carbon Star
370(1)
18.7.1 Carbon Star AFGL 3068 (IRAS 23166 + 1655)
370(1)
19 Hurricanes, Tornados, and Whirlpools 371(10)
19.1 Hurricane Cloud Patterns
371(2)
19.1.1 Hurricane Fran
371(1)
19.1.2 Hurricane Isabel
371(2)
19.2 Tornado Classification
373(2)
19.2.1 The k Index
374(1)
19.2.2 Tornado Morphology Animation
374(1)
19.2.3 Tornado Morphology Classification
375(1)
19.3 Low-Pressure Cloud Pattern
375(1)
19.4 Whirlpool
375(4)
19.5 Order in Physical Systems
379(2)
20 A Look Forward 381(8)
20.1 Properties of the R-Function
382(1)
20.2 Inverse Functions
382(2)
20.3 The Generalized Fractional Trigonometries
384(1)
20.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments
384(1)
20.5 Applications: Fractional Field Equations
385(2)
20.6 Fractional Spiral and Nonspiral Properties
387(1)
20.7 Numerical Improvements for Evaluation to Larger Values of atq
387(1)
20.8 Epilog
388(1)
A Related Works 389(4)
A.1 Introduction
389(1)
A.2 Miller and Ross
389(1)
A.3 West, Bologna, and Grigolini
390(1)
A.4 Mittag-Leffler-Based Fractional Trigonometric Functions
390(1)
A.5 Relationship to Current Work
391(2)
B Computer Code 393(6)
B.1 Introduction
393(1)
B.2 Matlab® R-Function
393(1)
B.3 Matlab® R-Function Evaluation Program
394(1)
B.4 Matlab® Meta-Cosine Function
395(1)
B.5 Matlab® Cosine Evaluation Program
395(1)
B.6 Maple® 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine
396(3)
C Tornado Simulation 399(2)
D Special Topics in Fractional Differintegration 401(12)
D.1 Introduction
401(1)
D.2 Fractional Integration of the Segmented tn-Function
401(3)
D.3 Fractional Differentiation of the Segmented tn-Function
404(2)
D.4 Fractional Integration of Segmented Fractional Trigonometric Functions
406(2)
D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions
408(5)
E Alternate Forms 413(4)
E.1 Introduction
413(1)
E.2 Reduced Variable Summation Forms
414(1)
E.3 Natural Quency Simplification
415(2)
References 417(8)
Index 425
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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