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.

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 xv

Acknowledgments xix

About the Companion Website xxi

1 Introduction 1

1.1 Background 2

1.2 The Fractional Integral and Derivative 3

1.2.1 Grûnwald Definition 3

1.2.2 Riemann–Liouville Definition 4

1.2.3 The Nature of the Fractional-Order Operator 5

1.3 The Traditional Trigonometry 6

1.4 Previous Efforts 8

1.5 Expectations of a Generalized Trigonometry and Hyperboletry 8

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation 9

2.1 The Fundamental Fractional Differential Equation 9

2.2 The Generalized Impulse Response Function 10

2.3 Relationship of the F-function to the Mittag-Leffler Function 11

2.4 Properties of the F-Function 12

2.5 Behavior of the F-Function as the Parameter a Varies 13

2.6 Example 16

3 The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus 19

3.1 Introduction 19

3.2 Functions for the Fractional Calculus 19

3.2.1 Mittag-Leffler’s Function 20

3.2.2 Agarwal’s Function 20

3.2.3 Erdelyi’s Function 20

3.2.4 Oldham and Spanier’s, Hartley’s, and Matignon’s Function 20

3.2.5 Robotnov’s Function 21

3.2.6 Miller and Ross’s Function 21

3.2.7 Gorenflo and Mainardi’s, and Podlubny’s Function 21

3.3 The R-Function: A Generalized Function 22

3.4 Properties of the R_q,v(a, t)-Function 23

3.4.1 Differintegration of the R-Function 23

3.4.2 Relationship Between R_q,mq and R_q,0 25

3.4.3 Fractional-Order Impulse Function 27

3.5 Relationship of the R-Function to the Elementary Functions 27

3.5.1 Exponential Function 27

3.5.2 Sine Function 27

3.5.3 Cosine Function 28

3.5.4 Hyperbolic Sine and Cosine 28

3.6 R-Function Identities 29

3.6.1 Trigonometric-Based Identities 29

3.6.2 Further Identities 30

3.7 Relationship of the R-Function to the Fractional Calculus Functions 31

3.7.1 Mittag-Leffler’s Function 31

3.7.2 Agarwal’s Function 31

3.7.3 Erdelyi’s Function 31

3.7.4 Oldham and Spanier’s, and Hartley’s Function 31

3.7.5 Miller and Ross’s Function 32

3.7.6 Robotnov’s Function 32

3.7.7 Gorenflo and Mainardi’s, and Podlubny’s Function 32

3.8 Example: Cooling Manifold 32

3.9 Further Generalized Functions: The G-Function and the H-Function 34

3.9.1 The G-Function 34

3.9.2 The H-Function 36

3.10 Preliminaries to the Fractional Trigonometry Development 38

3.11 Eigen Character of the R-Function 38

3.12 Fractional Differintegral of the TimeScaled R-Function 39

3.13 R-Function Relationships 39

3.14 Roots of Complex Numbers 40

3.15 Indexed Forms of the R-Function 41

3.15.1 R-Function with Complex Argument 41

3.15.2 Indexed Forms of the R-Function 42

3.15.2.1 Complexity Form 42

3.15.2.2 Parity Form 43

3.16 Term-by-Term Operations 44

3.17 Discussion 46

4 R-Function Relationships 47

4.1 R-Function Basics 47

4.2 Relationships for R_m,0in Terms of R_1,048

4.3 Relationships for R_1∕m,0 in Terms of R_1,0 50

4.4 Relationships for the Rational Form R_m∕p,0 in Terms of R_1∕p,0 51

4.5 Relationships for R_1∕p,0 in Terms of R_m∕p,0 53

4.6 Relating R_m∕p,0 to the Exponential Function R_1,0 (b, t) = e^bt 54

4.7 Inverse Relationships–Relationships for R_1,0 in Terms of R_m,k 56

4.8 Inverse Relationships–Relationships for R_1,0 in Terms of R_1∕m,0 57

4.9 Inverse Relationships–Relationships for e^at = R_1,0(a, t) in Terms of R_m∕p,0 59

4.10 Discussion 61

5 The Fractional Hyperboletry 63

5.1 The Fractional R₁-Hyperbolic Functions 63

5.2 R₁-Hyperbolic Function Relationship 72

5.3 Fractional Calculus Operations on the R₁-Hyperbolic Functions 72

5.4 Laplace Transforms of the R₁-Hyperbolic Functions 73

5.5 Complexity-Based Hyperbolic Functions 73

5.6 Fractional Hyperbolic Differential Equations 74

5.7 Example 76

5.8 Discussions 77

6 The R₁-Fractional Trigonometry 79

6.1 R₁-Trigonometric Functions 79

6.1.1 R₁-Trigonometric Properties 81

6.2 R₁-Trigonometric Function Interrelationship 88

6.3 Relationships to R₁-Hyperbolic Functions 89

6.4 Fractional Calculus Operations on the R₁-Trigonometric Functions 89

6.5 Laplace Transforms of the R₁-Trigonometric Functions 90

6.5.1 Laplace Transform of R₁Cos_{q. v}(a, k, t) 90

6.5.2 Laplace Transform of R₁Sin_{q. v}(a, k, t) 91

6.6 Complexity-Based R₁-Trigonometric Functions 92

6.7 Fractional Differential Equations 94

7 The R₂-Fractional Trigonometry 97

7.1 R₂-Trigonometric Functions: Based on Real and Imaginary Parts 97

7.2 R₂-Trigonometric Functions: Based on Parity 102

7.3 Laplace Transforms of the R₂-Trigonometric Functions 111

7.3.1 R₂Cos_q,v(a, k, t) 111

7.3.2 R₂Sin_q,v(a, k, t) 112

7.3.3 L{R₂Cofl_q,v(a, k, t)} 113

7.3.4 L{R₂Flut_q,v(a, k, t)} 113

7.3.5 L{R₂Covib_q,v(a, k, t)} 113

7.3.6 L{R₂Vib_q,v(a, k, t)} 113

7.4 R₂-Trigonometric Function Relationships 113

7.4.1 R₂Cos_q,v(a, k, t) and R₂Sin_q,v(a, k, t) Relationships and Fractional Euler Equation 114

7.4.2 R₂Rot_q,v(a, t) and R₂Cor_q,v(a, t) Relationships 116

7.4.3 R₂Cofl_q,v(a, t) and R₂Flut_q,v(a, t) Relationships 116

7.4.4 R₂Covib_q,v(a, t) and R₂Vib_q,v(a, t) Relationships 118

7.5 Fractional Calculus Operations on the R₂-Trigonometric Functions 119

7.5.1 R₂Cos_q,v(a, k, t) 119

7.5.2 R₂Sin_q,v(a, k, t) 121

7.5.3 R₂Cor_q,v(a, t) 122

7.5.4 R₂Rot_q,v(a, t) 122

7.5.5 R₂Coflut_q,v(a, t) 122

7.5.6 R₂Flut_q,v(a, k, t) 123

7.5.7 R₂Covib_q,v (a, k, t) 123

7.5.8 R₂Vib_q,v(a, k, t) 124

7.5.9 Summary of Fractional Calculus Operations on the R₂-Trigonometric Functions 124

7.6 Inferred Fractional Differential Equations 127

8 The R₃-Trigonometric Functions 129

8.1 The R₃-Trigonometric Functions: Based on Complexity 129

8.2 The R₃-Trigonometric Functions: Based on Parity 134

8.3 Laplace Transforms of the R₃-Trigonometric Functions 140

8.4 R₃-Trigonometric Function Relationships 141

8.4.1 R₃Cosq,v(a, t) and R₃Sin_q,v(a, t) Relationships and Fractional Euler Equation 142

8.4.2 R₃Rotq,v(a, t) and R₃Cor_q,v(a, t) Relationships 143

8.4.3 R₃Cofl_q,v(a, t) and R₃Flut_q,v(a, t) Relationships 144

8.4.4 R₃Covib_q,v(a, t) and R₃Vib_q,v(a, t) Relationships 145

8.5 Fractional Calculus Operations on the R₃-Trigonometric Functions 146

8.5.1 R₃Cos_q,v(a, k, t) 146

8.5.2 R₃Sin_q,v(a, k, t) 148

8.5.3 R₃Cor_q,v(a, t) 149

8.5.4 R₃Rot_q,v(a, t) 150

8.5.5 R₃Coflut_q,v(a, k, t) 150

8.5.6 R₃Flut_q,v(a, k, t) 152

8.5.7 R₃Covib_q,v(a, k, t) 153

8.5.8 R₃Vib_q,v(a, k, t) 154

8.5.9 Summary of Fractional Calculus Operations on the R₃-Trigonometric Functions 157

9 The Fractional Meta-Trigonometry 159

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity 160

9.1.1 Alternate Forms 161

9.1.2 Graphical Presentation–Complexity Functions 161

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity 166

9.3 Commutative Properties of the Complexity and Parity Operations 179

9.3.1 Graphical Presentation–Parity Functions 181

9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions 188

9.5 R-Function Representation of the Fractional Meta-Trigonometric Functions 192

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions 195

9.6.1 Cos_q,v(a, 𝛼, 𝛽, k, t) 195

9.6.2 Sin_q,v(a, 𝛼, 𝛽, k, t) 197

9.6.3 Cor_q,v(a, 𝛼, 𝛽, t) 198

9.6.4 Rot_q,v(a, 𝛼, 𝛽, t) 198

9.6.5 Coflut_q,v(a, 𝛼, 𝛽, k, t) 199

9.6.6 Flut_q,v(a, 𝛼, 𝛽, k, t) 200

9.6.7 Covib_q,v(a, 𝛼, 𝛽, k, t) 202

9.6.8 Vib_q,v(a, 𝛼, 𝛽, k, t) 203

9.6.9 Summary of Fractional Calculus Operations on the Meta-Trigonometric Functions 204

9.7 Special Topics in Fractional Differintegration 206

9.8 Meta-Trigonometric Function Relationships 206

9.8.1 Cos_q,v(a, 𝛼, 𝛽, t) and Sin_q,v(a, 𝛼, 𝛽, t) Relationships 206

9.8.2 Cor_q,v(a, 𝛼, 𝛽, t) and Rot_q,v(a, 𝛼, 𝛽, t) Relationships 207

9.8.3 Covib_q,v(a, 𝛼, 𝛽, t) and Vib_q,v(a, 𝛼, 𝛽, t) Relationships 208

9.8.4 Cofl_q,v(a, 𝛼, 𝛽, t) and Flut_q,v(a, 𝛼, 𝛽, t) Relationships 208

9.8.5 Cofl_q,v(a, 𝛼, 𝛽, t) and Vib_q,v(a, 𝛼, 𝛽, t) Relationships 209

9.8.6 Cos_q,v(a, 𝛼, 𝛽, t) and Sin_q,v(a, 𝛼, 𝛽, t) Relationships to Other Functions 211

9.8.7 Meta-Identities Based on the Integer-order Trigonometric Identities 211

9.8.7.1 The cos(−x)=cos(x)-Based Identity for Cos_q,v(a, 𝛼, 𝛽, t) 211

9.8.7.2 The sin(−x)=−sin(x)-Based Identity for Sin_q,v(a, 𝛼, 𝛽, t) 212

9.8.7.3 The Cos_q,v(a, 𝛼, 𝛽, t)⇔Sin_q,v(a, 𝛼, 𝛽, t) Identity 212

9.8.7.4 The sin(x)=sin(x±m𝜋/2)-Based Identity for Sin_q,v(a, 𝛼, 𝛽, t) 213

9.9 Fractional Poles: Structure of the Laplace Transforms 214

9.10 Comments and Issues Relative to the Meta-Trigonometric Functions 214

9.11 Backward Compatibility to Earlier Fractional Trigonometries 215

9.12 Discussion 215

10 The Ratio and Reciprocal Functions 217

10.1 Fractional Complexity Functions 217

10.2 The Parity Reciprocal Functions 219

10.3 The Parity Ratio Functions 221

10.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions 225

10.5 Relationships 226

10.6 Discussion 227

11 Further Generalized Fractional Trigonometries 229

11.1 The G-Function-Based Trigonometry 229

11.2 Laplace Transforms for the G-Trigonometric Functions 230

11.3 The H-Function-Based Trigonometry 234

11.4 Laplace Transforms for the H-Trigonometric Functions 235

Introduction to Applications 241

12 The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry 243

12.1 Fractional Differential Equations 243

12.2 Fundamental Fractional Differential Equations of the First Kind 245

12.3 Fundamental Fractional Differential Equations of the Second Kind 246

12.4 Preliminaries–Laplace Transforms 246

12.4.1 Fractional Cosine Function 246

12.4.2 Fractional Sine Function 248

12.4.3 Higher-Order Numerator Dynamics 248

12.4.3.1 Fractional Cosine Function 248

12.4.3.2 Fractional Sine Function 248

12.4.4 Parity Functions–The Flutter Function 249

12.4.5 Additional Transform Pairs 250

12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots 250

12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots 252

12.6.1 Repeated Real Fractional Roots 252

12.6.2 Repeated Complex Fractional Roots 253

12.7 Fractional Differential Equations Containing Repeated Roots 253

12.8 Fractional Differential Equations of Non-Commensurate Order 254

12.9 Indexed Fractional Differential Equations: Multiple Solutions 255

12.10 Discussion 256

13 Fractional Trigonometric Systems 259

13.1 The R-Function as a Linear System 259

13.2 R-System Time Responses 260

13.3 R-Function-Based Frequency Responses 260

13.4 Meta-Trigonometric Function-Based Frequency Responses 261

13.5 Fractional Meta-Trigonometry 264

13.6 Elementary Fractional Transfer Functions 266

13.7 Stability Theorem 266

13.8 Stability of Elementary Fractional Transfer Functions 267

13.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions 268

13.9.1 Complexity Function Stability 268

13.9.2 Parity Function Stability 269

13.10 Discussion 270

14 Numerical Issues and Approximations in the Fractional Trigonometry 271

14.1 R-Function Convergence 271

14.2 The Meta-Trigonometric Function Convergence 272

14.3 Uniform Convergence 273

14.4 Numerical Issues in the Fractional Trigonometry 274

14.5 The R₂Cos- and R₂Sin-Function Asymptotic Behavior 275

14.6 R-Function Approximations 276

14.7 The Near-Order Effect 279

14.8 High-Precision Software 281

15 The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry 283

15.1 The Fractional Spiral Functions 283

15.2 Analysis of Spirals 288

15.2.1 Descriptions of Spirals 288

15.2.1.1 Polar Description 289

15.2.1.2 Parametric Description 290

15.2.1.3 Definitions 293

15.2.1.4 Alternate Definitions 294

15.2.1.5 Examples 294

15.2.2 Spiral Length and Growth/Decay Rates 294

15.2.2.1 Spiral Length 294

15.2.2.2 Spiral Growth Rates for ccw Spirals 295

15.2.2.3 Component Growth Rates 296

15.2.3 Scaling of Spirals 296

15.2.3.1 Uniform Rectangular Scaling 297

15.2.3.2 Nonuniform Rectangular Scaling 297

15.2.3.3 Polar Scaling 298

15.2.3.4 Radial Scaling 298

15.2.3.5 Angular Scaling 298

15.2.4 Spiral Velocities 299

15.2.5 Referenced Spirals: Retardation 301

15.3 Relation to the Classical Spirals 303

15.3.1 Classical Spirals 303

15.4 Discussion 307

16 Fractional Oscillators 309

16.1 The Space of Linear Fractional Oscillators 309

16.1.1 Complexity Function-Based Oscillators 310

16.1.2 Parity Function-Based Oscillators 311

16.1.3 Intrinsic Oscillator Damping 312

16.2 Coupled Fractional Oscillators 314

17 Shell Morphology and Growth 317

17.1 Nautilus pompilius 317

17.1.1 Introduction 317

17.1.2 Nautilus Morphology 318

17.1.2.1 Fractional Differential Equations 323

17.1.3 Spiral Length 325

17.1.4 Morphology of the Siphuncle Spiral 325

17.1.5 Fractional Growth Rate 325

17.1.6 Nautilus Study Summary 328

17.2 Shell 5 329

17.3 Shell 6 330

17.4 Shell 7 332

17.5 Shell 8 332

17.6 Shell 9 336

17.7 Shell 10 336

17.8 Ammonite 339

17.9 Discussion 340

18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341

18.1 Introduction 341

18.2 Background–Fractional Spirals for Galactic Classification 342

18.3 Classification Process 347

18.3.1 Symmetry Assumption 347

18.3.2 Galaxy Image to Spiral or Spiral to Galaxy Image 347

18.3.3 Inclination 347

18.3.4 Data Presentation 348

18.4 Mathematical Classification of Selected Galaxies 350

18.4.1 NGC 4314 SB(rs)a 350

18.4.2 NGC 1365 SBb/SBc/SB(s)b/SBb(s) 350

18.4.3 M95 SB(r)b/SBb(r)/SBa/SBb 350

18.4.4 NGC 2997 Sc/SAB(rs)c/Sc(s) 353

18.4.5 NGC 4622 (R′)SA(r)a pec/Sb 353

18.4.6 M 66 or NGC 3627 SAB(s)b/Sb(s)/Sb 353

18.4.7 NGC 4535 SAB(s)c/SB(s)c/Sc/SBc 355

18.4.8 NGC 1300 SBc/SBb(s)/SB(rs)bc 355

18.4.9 Hoag’s Object 358

18.4.10 M 51 Sa+Sc 358

18.4.11 AM 0644-741 Sc/Strongly peculiar/ 359

18.4.12 ESO 269-G57 (R′)SAB(r)ab/Sa(r) 360

18.4.13 NGC 1313 SBc/SB(s)d/SB(s)d 362

18.4.14 Carbon Star 3068 362

18.5 Analysis 362

18.5.1 Fractional Differential Equations 366

18.5.2 Alternate Classification Basis 366

18.6 Discussion 367

18.6.1 Benefits 368

18.7 Appendix: Carbon Star 370

18.7.1 Carbon Star AFGL 3068 (IRAS 23166+1655) 370

19 Hurricanes, Tornados, and Whirlpools 371

19.1 Hurricane Cloud Patterns 371

19.1.1 Hurricane Fran 371

19.1.2 Hurricane Isabel 371

19.2 Tornado Classification 373

19.2.1 The k Index 374

19.2.2 Tornado Morphology Animation 374

19.2.3 Tornado Morphology Classification 375

19.3 Low-Pressure Cloud Pattern 375

19.4 Whirlpool 375

19.5 Order in Physical Systems 379

20 A Look Forward 381

20.1 Properties of the R-Function 382

20.2 Inverse Functions 382

20.3 The Generalized Fractional Trigonometries 384

20.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments 384

20.5 Applications: Fractional Field Equations 385

20.6 Fractional Spiral and Nonspiral Properties 387

20.7 Numerical Improvements for Evaluation to Larger Values of at^q 387

20.8 Epilog 388

A Related Works 389

A.1 Introduction 389

A.2 Miller and Ross 389

A.3 West, Bologna, and Grigolini 390

A.4 Mittag-Leffler-Based Fractional Trigonometric Functions 390

A.5 Relationship to Current Work 391

B Computer Code 393

B.1 Introduction 393

B.2 Matlab^Ⓡ R-Function 393

B.3 Matlab^Ⓡ R-Function Evaluation Program 394

B.4 Matlab^Ⓡ Meta-Cosine Function 395

B.5 Matlab^Ⓡ Cosine Evaluation Program 395

B.6 Maple^Ⓡ 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine 396

C Tornado Simulation 399

D Special Topics in Fractional Differintegration 401

D.1 Introduction 401

D.2 Fractional Integration of the Segmented t^p-Function 401

D.3 Fractional Differentiation of the Segmented t^p-Function 404

D.4 Fractional Integration of Segmented Fractional Trigonometric Functions 406

D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions 408

E Alternate Forms 413

E.1 Introduction 413

E.2 Reduced Variable Summation Forms 414

E.3 Natural Quency Simplification 415

References 417

Index 425