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 xv

Acknowledgments xix

About the Companion Website xxi

1 Introduction 1

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

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

4 R-Function Relationships 47

5 The Fractional Hyperboletry 63

6 The R1-Fractional Trigonometry 79

7 The R2-Fractional Trigonometry 97

8 The R3-Trigonometric Functions 129

9 The Fractional Meta-Trigonometry 159

10 The Ratio and Reciprocal Functions 217

11 Further Generalized Fractional Trigonometries 229

Introduction to Applications 241

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

13 Fractional Trigonometric Systems 259

14 Numerical Issues and Approximations in the Fractional Trigonometry 271

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

16 Fractional Oscillators 309

17 Shell Morphology and Growth 317

18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341

19 Hurricanes, Tornados, and Whirlpools 371

20 A Look Forward 381

A RelatedWorks 389

B Computer Code 393

C Tornado Simulation 399

D Special Topics in Fractional Differintegration 401

E Alternate Forms 413

References 417

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.