The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science [1 ed.] 1119139406, 9781119139409

Table of contents :
Content: Preface xv Acknowledgments xix About the Companion Website xxi 1 Introduction 1 1.1 Background 2 1.2 The Fractional Integral and Derivative 3 1.3 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 3.3 The R-Function: A Generalized Function 22 3.4 Properties of the Rq,v(a, t)-Function 23 3.5 Relationship of the R-Function to the Elementary Functions 27 3.6 R-Function Identities 29 3.7 Relationship of the R-Function to the Fractional Calculus Functions 31 3.8 Example: Cooling Manifold 32 3.9 Further Generalized Functions: The G-Function and the H-Function 34 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 3.16 Term-by-Term Operations 44 3.17 Discussion 46 4 R-Function Relationships 47 4.1 R-Function Basics 47 4.2 Relationships for Rm,0 in Terms of R1,0 48 4.3 Relationships for R1Mm,0 in Terms of R1,0 50 4.4 Relationships for the Rational Form RmMp,0 in Terms of R1Mp,0 51 4.5 Relationships for R1Mp,0 in Terms of RmMp,0 53 4.6 Relating RmMp,0 to the Exponential Function R1,0(b, t) = ebt 54 4.7 Inverse Relationships Relationships for R1,0 in Terms of Rm,k 56 4.8 Inverse Relationships Relationships for R1,0 in Terms of R1Mm,0 57 4.9 Inverse Relationships Relationships for eat = R1,0(a, t) in Terms of RmMp,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 6.2 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 6.6 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 7.4 R2-Trigonometric Function Relationships 113 7.5 Fractional Calculus Operations on the R2-Trigonometric Functions 119 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 8.5 Fractional Calculus Operations on the R3-Trigonometric Functions 146 9 The Fractional Meta-Trigonometry 159 9.1 The FractionalMeta-Trigonometric Functions: Based on Complexity 160 9.2 The Meta-Fractional Trigonometric Functions: Based on Parity 166 9.3 Commutative Properties of the Complexity and Parity Operations 179 9.4 Laplace Transforms of the FractionalMeta-Trigonometric Functions 188 9.5 R-Function Representation of the FractionalMeta-Trigonometric Functions 192 9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions 195 9.7 Special Topics in Fractional Differintegration 206 9.8 Meta-Trigonometric Function Relationships 206 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 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 12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots 250 12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots 252 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 FractionalMeta-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 13.10 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 15.3 Relation to the Classical Spirals 303 15.4 Discussion 307 16 Fractional Oscillators 309 16.1 The Space of Linear Fractional Oscillators 309 16.2 Coupled Fractional Oscillators 314 17 Shell Morphology and Growth 317 17.1 Nautilus pompilius 317 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 18.4 Mathematical Classification of Selected Galaxies 350 18.5 Analysis 362 18.6 Discussion 367 18.7 Appendix: Carbon Star 370 19 Hurricanes, Tornados, and Whirlpools 371 19.1 Hurricane Cloud Patterns 371 19.2 Tornado Classification 373 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 atq 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 CurrentWork 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 tp-Function 401 D.3 Fractional Differentiation of the Segmented tp-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

Citation preview

The Fractional Trigonometry

The Fractional Trigonometry With Applications to Fractional Differential Equations and Science

Carl F. Lorenzo National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio

Tom T. Hartley The University of Akron Akron, Ohio

Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved Published simultaneously in Canada Published by John Wiley & Sons, Inc., Hoboken, New Jersey No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data: Names: Lorenzo, Carl F. | Hartley, T. T. (Tom T.), 1964Title: The fractional trigonometry : with applications to fractional differential equations and science / Carl F. Lorenzo, National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, Tom T. Hartley, The University of Akron, Akron, Ohio. Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016027837 (print) | LCCN 2016028337 (ebook) | ISBN 9781119139409 (cloth) | ISBN 9781119139423 (pdf ) | ISBN 9781119139430 (epub) Subjects: LCSH: Fractional calculus. | Trigonometry. Classification: LCC QA314 .L67 2017 (print) | LCC QA314 (ebook) | DDC 516.24–dc23 LC record available at https://lccn.loc.gov/2016027837

Set in 10/12pt Warnock by SPi Global, Chennai, India Printed in the United States of America

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v

Contents Preface xv Acknowledgments xix About the Companion Website xxi 1

Introduction 1

1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.4 1.5

Background 2 The Fractional Integral and Derivative 3 Grunwald ̆ Definition 3 Riemann–Liouville Definition 4 The Nature of the Fractional-Order Operator 5 The Traditional Trigonometry 6 Previous Efforts 8 Expectations of a Generalized Trigonometry and Hyperboletry

2

The Fractional Exponential Function via the Fundamental Fractional Differential Equation 9

2.1 2.2 2.3 2.4 2.5 2.6

The Fundamental Fractional Differential Equation 9 The Generalized Impulse Response Function 10 Relationship of the F-function to the Mittag-Leffler Function 11 Properties of the F-Function 12 Behavior of the F-Function as the Parameter a Varies 13 Example 16

3

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

3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7

Introduction 19 Functions for the Fractional Calculus 19 Mittag-Leffler’s Function 20 Agarwal’s Function 20 Erdelyi’s Function 20 Oldham and Spanier’s, Hartley’s, and Matignon’s Function 20 Robotnov’s Function 21 Miller and Ross’s Function 21 Gorenflo and Mainardi’s, and Podlubny’s Function 21

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Contents

3.3 3.4 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6 3.6.1 3.6.2 3.7 3.7.1 3.7.2 3.7.3 3.7.4 3.7.5 3.7.6 3.7.7 3.8 3.9 3.9.1 3.9.2 3.10 3.11 3.12 3.13 3.14 3.15 3.15.1 3.15.2 3.15.2.1 3.15.2.2 3.16 3.17

The R-Function: A Generalized Function 22 Properties of the Rq,v (a, t)-Function 23 Differintegration of the R-Function 23 Relationship Between Rq,mq and Rq,0 25 Fractional-Order Impulse Function 27 Relationship of the R-Function to the Elementary Functions 27 Exponential Function 27 Sine Function 27 Cosine Function 28 Hyperbolic Sine and Cosine 28 R-Function Identities 29 Trigonometric-Based Identities 29 Further Identities 30 Relationship of the R-Function to the Fractional Calculus Functions 31 Mittag-Leffler’s Function 31 Agarwal’s Function 31 Erdelyi’s Function 31 Oldham and Spanier’s, and Hartley’s Function 31 Miller and Ross’s Function 32 Robotnov’s Function 32 Gorenflo and Mainardi’s, and Podlubny’s Function 32 Example: Cooling Manifold 32 Further Generalized Functions: The G-Function and the H-Function 34 The G-Function 34 The H-Function 36 Preliminaries to the Fractional Trigonometry Development 38 Eigen Character of the R-Function 38 Fractional Differintegral of the TimeScaled R-Function 39 R-Function Relationships 39 Roots of Complex Numbers 40 Indexed Forms of the R-Function 41 R-Function with Complex Argument 41 Indexed Forms of the R-Function 42 Complexity Form 42 Parity Form 43 Term-by-Term Operations 44 Discussion 46

4

R-Function Relationships

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

47 R-Function Basics 47 Relationships for Rm,0 in Terms of R1,0 48 Relationships for R1∕m,0 in Terms of R1,0 50 Relationships for the Rational Form Rm∕p,0 in Terms of R1∕p,0 51 Relationships for R1∕p,0 in Terms of Rm∕p,0 53 Relating Rm∕p,0 to the Exponential Function R1,0 (b, t) = ebt 54 Inverse Relationships–Relationships for R1,0 in Terms of Rm,k 56 Inverse Relationships–Relationships for R1,0 in Terms of R1∕m,0 57 Inverse Relationships–Relationships for eat = R1,0 (a, t) in Terms of Rm∕p,0 Discussion 61

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Contents

5

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6

6.1 6.1.1 6.2 6.3 6.4 6.5 6.5.1 6.5.2 6.6 6.7 7

7.1 7.2 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3 7.5.4 7.5.5 7.5.6 7.5.7 7.5.8 7.5.9 7.6

63 The Fractional R1 -Hyperbolic Functions 63 R1 -Hyperbolic Function Relationship 72 Fractional Calculus Operations on the R1 -Hyperbolic Functions 72 Laplace Transforms of the R1 -Hyperbolic Functions 73 Complexity-Based Hyperbolic Functions 73 Fractional Hyperbolic Differential Equations 74 Example 76 Discussions 77 The Fractional Hyperboletry

79 R1 -Trigonometric Functions 79 R1 -Trigonometric Properties 81 R1 -Trigonometric Function Interrelationship 88 Relationships to R1 -Hyperbolic Functions 89 Fractional Calculus Operations on the R1 -Trigonometric Functions 89 Laplace Transforms of the R1 -Trigonometric Functions 90 Laplace Transform of R1 Cosq. v (a, k, t) 90 Laplace Transform of R1 Sinq. v (a, k, t) 91 Complexity-Based R1 -Trigonometric Functions 92 Fractional Differential Equations 94 The R1 -Fractional Trigonometry

97 R2 -Trigonometric Functions: Based on Real and Imaginary Parts 97 R2 -Trigonometric Functions: Based on Parity 102 Laplace Transforms of the R2 -Trigonometric Functions 111 R2 Cosq,v (a, k, t) 111 R2 Sinq,v (a, k, t) 112 L{R2 Coflq,v (a, k, t)} 113 L{R2 Flutq,v (a, k, t)} 113 L{R2 Covibq,v (a, k, t)} 113 L{R2 Vibq,v (a, k, t)} 113 R2 -Trigonometric Function Relationships 113 R2 Cosq,v (a, k, t) and R2 Sinq,v (a, k, t) Relationships and Fractional Euler Equation 114 R2 Rotq,v (a, t) and R2 Corq,v (a, t) Relationships 116 R2 Coflq,v (a, t) and R2 Flutq,v (a, t) Relationships 116 R2 Covibq,v (a, t) and R2 Vibq,v (a, t) Relationships 118 Fractional Calculus Operations on the R2 -Trigonometric Functions 119 R2 Cosq,v (a, k, t) 119 R2 Sinq,v (a, k, t) 121 R2 Corq,v (a, t) 122 R2 Rotq,v (a, t) 122 R2 Coflutq,v (a, t) 122 R2 Flutq,v (a, k, t) 123 R2 Covibq,v (a, k, t) 123 R2 Vibq,v (a, k, t) 124 Summary of Fractional Calculus Operations on the R2 -Trigonometric Functions 124 Inferred Fractional Differential Equations 127 The R2 -Fractional Trigonometry

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8.1 8.2 8.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.5.5 8.5.6 8.5.7 8.5.8 8.5.9

129 The R3 -Trigonometric Functions: Based on Complexity 129 The R3 -Trigonometric Functions: Based on Parity 134 Laplace Transforms of the R3 -Trigonometric Functions 140 R3 -Trigonometric Function Relationships 141 R3 Cosq,v (a, t) and R3 Sinq,v (a, t) Relationships and Fractional Euler Equation 142 R3 Rot q,v (a, t) and R3 Corq,v (a, t) Relationships 143 R3 Coflq,v (a, t) and R3 Flut q,v (a, t) Relationships 144 R3 Covibq,v (a, t) and R3 Vibq,v (a, t) Relationships 145 Fractional Calculus Operations on the R3 -Trigonometric Functions 146 R3 Cosq,v (a, k, t) 146 R3 Sinq,v (a, k, t) 148 R3 Corq,v (a, t) 149 R3 Rot q,v (a, t) 150 R3 Coflut q,v (a, k, t) 150 R3 Flut q,v (a, k, t) 152 R3 Covibq,v (a, k, t) 153 R3 Vibq,v (a, k, t) 154 Summary of Fractional Calculus Operations on the R3 -Trigonometric Functions 157

9

The Fractional Meta-Trigonometry

8

9.1 9.1.1 9.1.2 9.2 9.3 9.3.1 9.4 9.5 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.6.5 9.6.6 9.6.7 9.6.8 9.6.9 9.7 9.8 9.8.1 9.8.2 9.8.3 9.8.4 9.8.5

The R3 -Trigonometric Functions

159 The Fractional Meta-Trigonometric Functions: Based on Complexity 160 Alternate Forms 161 Graphical Presentation–Complexity Functions 161 The Meta-Fractional Trigonometric Functions: Based on Parity 166 Commutative Properties of the Complexity and Parity Operations 179 Graphical Presentation–Parity Functions 181 Laplace Transforms of the Fractional Meta-Trigonometric Functions 188 R-Function Representation of the Fractional Meta-Trigonometric Functions Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions 195 Cosq,v (a, 𝛼, 𝛽, k, t) 195 Sinq,v (a, 𝛼, 𝛽, k, t) 197 Corq,v (a, 𝛼, 𝛽, t) 198 Rot q,v (a, 𝛼, 𝛽, t) 198 Coflut q,v (a, 𝛼, 𝛽, k, t) 199 Flut q,v (a, 𝛼, 𝛽, k, t) 200 Covibq,v (a, 𝛼, 𝛽, k, t) 202 Vibq,v (a, 𝛼, 𝛽, k, t) 203 Summary of Fractional Calculus Operations on the Meta-Trigonometric Functions 204 Special Topics in Fractional Differintegration 206 Meta-Trigonometric Function Relationships 206 Cosq,v (a, 𝛼, 𝛽, t) and Sinq,v (a, 𝛼, 𝛽, t) Relationships 206 Corq,v (a, 𝛼, 𝛽, t) and Rot q,v (a, 𝛼, 𝛽, t) Relationships 207 Covibq,v (a, 𝛼, 𝛽, t) and Vibq,v (a, 𝛼, 𝛽, t) Relationships 208 Coflq,v (a, 𝛼, 𝛽, t) and Flut q,v (a, 𝛼, 𝛽, t) Relationships 208 Coflq,v (a, 𝛼, 𝛽, t) and Vibq,v (a, 𝛼, 𝛽, t) Relationships 209

192

Contents

9.8.6 9.8.7 9.8.7.1 9.8.7.2 9.8.7.3 9.8.7.4 9.9 9.10 9.11 9.12

Cosq,v (a, 𝛼, 𝛽, t) and Sinq,v (a, 𝛼, 𝛽, t) Relationships to Other Functions 211 Meta-Identities Based on the Integer-order Trigonometric Identities 211 The cos(−x) = cos(x)-Based Identity for Cosq,v (a, 𝛼, 𝛽, t) 211 The sin(−x) = − sin(x)-Based Identity for Sinq,v (a, 𝛼, 𝛽, t) 212 The Cosq,v (a, 𝛼, 𝛽, t) ⇔ Sinq,v (a, 𝛼, 𝛽, t) Identity 212 The sin(x) = sin(x ± m𝜋/2)-Based Identity for Sinq,v (a, 𝛼, 𝛽, t) 213 Fractional Poles: Structure of the Laplace Transforms 214 Comments and Issues Relative to the Meta-Trigonometric Functions 214 Backward Compatibility to Earlier Fractional Trigonometries 215 Discussion 215

10

The Ratio and Reciprocal Functions

10.1 10.2 10.3 10.4 10.5 10.6 11

11.1 11.2 11.3 11.4

217 Fractional Complexity Functions 217 The Parity Reciprocal Functions 219 The Parity Ratio Functions 221 R-Function Representation of the Fractional Ratio and Reciprocal Functions Relationships 226 Discussion 227

225

229 The G-Function-Based Trigonometry 229 Laplace Transforms for the G-Trigonometric Functions 230 The H-Function-Based Trigonometry 234 Laplace Transforms for the H-Trigonometric Functions 235 Further Generalized Fractional Trigonometries

Introduction to Applications

241

12

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

12.1 12.2 12.3 12.4 12.4.1 12.4.2 12.4.3 12.4.3.1 12.4.3.2 12.4.4 12.4.5 12.5 12.6 12.6.1 12.6.2 12.7 12.8 12.9 12.10

Fractional Differential Equations 243 Fundamental Fractional Differential Equations of the First Kind 245 Fundamental Fractional Differential Equations of the Second Kind 246 Preliminaries–Laplace Transforms 246 Fractional Cosine Function 246 Fractional Sine Function 248 Higher-Order Numerator Dynamics 248 Fractional Cosine Function 248 Fractional Sine Function 248 Parity Functions–The Flutter Function 249 Additional Transform Pairs 250 Fractional Differential Equations of Higher Order: Unrepeated Roots 250 Fractional Differential Equations of Higher Order: Containing Repeated Roots Repeated Real Fractional Roots 252 Repeated Complex Fractional Roots 253 Fractional Differential Equations Containing Repeated Roots 253 Fractional Differential Equations of Non-Commensurate Order 254 Indexed Fractional Differential Equations: Multiple Solutions 255 Discussion 256

252

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Contents

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.9.1 13.9.2 13.10

259 The R-Function as a Linear System 259 R-System Time Responses 260 R-Function-Based Frequency Responses 260 Meta-Trigonometric Function-Based Frequency Responses 261 Fractional Meta-Trigonometry 264 Elementary Fractional Transfer Functions 266 Stability Theorem 266 Stability of Elementary Fractional Transfer Functions 267 Insights into the Behavior of the Fractional Meta-Trigonometric Functions 268 Complexity Function Stability 268 Parity Function Stability 269 Discussion 270

14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Numerical Issues and Approximations in the Fractional Trigonometry 271 R-Function Convergence 271 The Meta-Trigonometric Function Convergence 272 Uniform Convergence 273 Numerical Issues in the Fractional Trigonometry 274 The R2 Cos- and R2 Sin-Function Asymptotic Behavior 275 R-Function Approximations 276 The Near-Order Effect 279 High-Precision Software 281

15

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

15.1 15.2 15.2.1 15.2.1.1 15.2.1.2 15.2.1.3 15.2.1.4 15.2.1.5 15.2.2 15.2.2.1 15.2.2.2 15.2.2.3 15.2.3 15.2.3.1 15.2.3.2 15.2.3.3 15.2.3.4 15.2.3.5 15.2.4 15.2.5 15.3 15.3.1 15.4

The Fractional Spiral Functions 283 Analysis of Spirals 288 Descriptions of Spirals 288 Polar Description 289 Parametric Description 290 Definitions 293 Alternate Definitions 294 Examples 294 Spiral Length and Growth/Decay Rates 294 Spiral Length 294 Spiral Growth Rates for ccw Spirals 295 Component Growth Rates 296 Scaling of Spirals 296 Uniform Rectangular Scaling 297 Nonuniform Rectangular Scaling 297 Polar Scaling 298 Radial Scaling 298 Angular Scaling 298 Spiral Velocities 299 Referenced Spirals: Retardation 301 Relation to the Classical Spirals 303 Classical Spirals 303 Discussion 307

13

Fractional Trigonometric Systems

Contents

16.1 16.1.1 16.1.2 16.1.3 16.2

309 The Space of Linear Fractional Oscillators 309 Complexity Function-Based Oscillators 310 Parity Function-Based Oscillators 311 Intrinsic Oscillator Damping 312 Coupled Fractional Oscillators 314

17

Shell Morphology and Growth 317

17.1 17.1.1 17.1.2 17.1.2.1 17.1.3 17.1.4 17.1.5 17.1.6 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9

Nautilus pompilius 317 Introduction 317 Nautilus Morphology 318 Fractional Differential Equations 323 Spiral Length 325 Morphology of the Siphuncle Spiral 325 Fractional Growth Rate 325 Nautilus Study Summary 328 Shell 5 329 Shell 6 330 Shell 7 332 Shell 8 332 Shell 9 336 Shell 10 336 Ammonite 339 Discussion 340

18

Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341

18.1 18.2 18.3 18.3.1 18.3.2 18.3.3 18.3.4 18.4 18.4.1 18.4.2 18.4.3 18.4.4 18.4.5 18.4.6 18.4.7 18.4.8 18.4.9 18.4.10 18.4.11 18.4.12 18.4.13 18.4.14 18.5

Introduction 341 Background–Fractional Spirals for Galactic Classification Classification Process 347 Symmetry Assumption 347 Galaxy Image to Spiral or Spiral to Galaxy Image 347 Inclination 347 Data Presentation 348 Mathematical Classification of Selected Galaxies 350 NGC 4314 SB(rs)a 350 NGC 1365 SBb/SBc/SB(s)b/SBb(s) 350 M95 SB(r)b/SBb(r)/SBa/SBb 350 NGC 2997 Sc/SAB(rs)c/Sc(s) 353 NGC 4622 (R′ )SA(r)a pec/Sb 353 M 66 or NGC 3627 SAB(s)b/Sb(s)/Sb 353 NGC 4535 SAB(s)c/SB(s)c/Sc/SBc 355 NGC 1300 SBc/SBb(s)/SB(rs)bc 355 Hoag’s Object 358 M 51 Sa + Sc 358 AM 0644-741 Sc/Strongly peculiar/ 359 ESO 269-G57 (R′ )SAB(r)ab/Sa(r) 360 NGC 1313 SBc/SB(s)d/SB(s)d 362 Carbon Star 3068 362 Analysis 362

16

Fractional Oscillators

342

xi

xii

Contents

18.5.1 18.5.2 18.6 18.6.1 18.7 18.7.1

Fractional Differential Equations 366 Alternate Classification Basis 366 Discussion 367 Benefits 368 Appendix: Carbon Star 370 Carbon Star AFGL 3068 (IRAS 23166 + 1655) 370

19

Hurricanes, Tornados, and Whirlpools

19.1 19.1.1 19.1.2 19.2 19.2.1 19.2.2 19.2.3 19.3 19.4 19.5 20 20.1 20.2 20.3 20.4

20.5 20.6 20.7 20.8 A A.1 A.2 A.3 A.4 A.5

371 Hurricane Cloud Patterns 371 Hurricane Fran 371 Hurricane Isabel 371 Tornado Classification 373 The k Index 374 Tornado Morphology Animation 374 Tornado Morphology Classification 375 Low-Pressure Cloud Pattern 375 Whirlpool 375 Order in Physical Systems 379

381 Properties of the R-Function 382 Inverse Functions 382 The Generalized Fractional Trigonometries 384 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments 384 Applications: Fractional Field Equations 385 Fractional Spiral and Nonspiral Properties 387 Numerical Improvements for Evaluation to Larger Values of at q 387 Epilog 388 A Look Forward

Related Works 389

Introduction 389 Miller and Ross 389 West, Bologna, and Grigolini 390 Mittag-Leffler-Based Fractional Trigonometric Functions 390 Relationship to Current Work 391

B.1 B.2 B.3 B.4 B.5 B.6

393 Introduction 393 MatlabⓇ R-Function 393 MatlabⓇ R-Function Evaluation Program 394 MatlabⓇ Meta-Cosine Function 395 MatlabⓇ Cosine Evaluation Program 395 MapleⓇ 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine 396

C

Tornado Simulation

D

Special Topics in Fractional Differintegration

D.1

Introduction

B

Computer Code

401

399 401

Contents

D.2 D.3 D.4 D.5

Fractional Integration of the Segmented t p -Function 401 Fractional Differentiation of the Segmented t p -Function 404 Fractional Integration of Segmented Fractional Trigonometric Functions 406 Fractional Differentiation of Segmented Fractional Trigonometric Functions 408

E

413 Introduction 413 Reduced Variable Summation Forms 414 Natural Quency Simplification 415

E.1 E.2 E.3

Alternate Forms

References 417 Index 425

xiii

xv

Preface There has been a strong resurgence of interest in the fractional calculus over the last two or three decades. This expansion of the classical calculus to derivatives and integrals of fractional order has given rise to the hope of a new understanding of the behavior of the physical world. The hope is that problems that have resisted solution by the integer-order calculus will yield to this greatly expanded capability. As a result of our work in the fractional calculus, and more particularly, in functions for the solutions of fractional differential equations, an interest was fostered in the behavior of generalized exponential functions for this application. Our work with the fundamental fractional differential equation had developed a function we named the F-function. This function, which had previously been mentioned in a footnote by Oldham and Spanier, acts as the fractional exponential function. It was a natural step from there to an interest in a fractional trigonometry. At that time, only a few pages of work were available in the literature and were based on the Mittag-Leffler function. These are shown in Appendix A. This book brings together our research in this area over the past 15 years and adds much new unpublished material. The classical trigonometry plays a very important role relative to the integer-order calculus; that is, it, together with the common exponential function, provides solutions for linear differential equations. We will find that the fractional trigonometry plays an analogous role relative to the fractional calculus by providing solutions to linear fractional differential equations. The importance of the classical trigonometry goes far beyond the solutions of triangles. Its use in Fourier integrals, Fourier series, signal processing, harmonic analysis, and more provided great motivation for the development of a fractional trigonometry to expand application to the fractional calculus domain. Because we are engineers, this book has been written in the style of the engineering mathematical books rather than the more rigorous and compact style of definition, theorem, and proof, found in most mathematical texts. We, of course, have made every effort to assure the derivations to be correct and are hopeful that the style has made the material accessible to a larger audience. We are also hopeful that this will not detract the interest of the mathematical community in the area since their skills will be needed to develop this important new area. Most of the materials of this book should be accessible to an undergraduate student with a background in Laplace transforms. After an introductory chapter, which offers a brief insight into the fractional calculus, the book is organized in two major parts. In Chapters 2–11, the definitions and theory of the fractional exponential and the fractional trigonometry are developed. Chapters 12–19 provide insight into various areas of potential application. Chapter 2 develops the F-function from consideration of the fundamental fractional differential equation. It generalizes the common exponential function for application in the fractional calculus. The F-function, the fractional eigenfunction, together with its generalization, the

xvi

Preface

R-function (Chapter 3), will later form the theoretical basis of the fractional trigonometry. Properties of these functions are developed in these two chapters. Their relationship to other functions for the fractional calculus is presented. An important characteristic of the R-function is that it contains the F-function as a special case and also contains its derivatives and integrals. In later chapters, it is shown that many of the newly defined fractional trigonometric functions inherit this important property. Chapter 4 further develops properties of the R-function that expose the character of this fractional exponential function. In Chapter 5, the R-function, Rq,v (a, t), with real arguments for a and t, is used to define the fractional hyperbolic functions. These functions generalize the classical hyperbolic functions. Fractional exponential forms of the hyperbolic functions are derived as well as their Laplace transforms. Furthermore, fractional differintegrals of the functions are determined. An example demonstrates the use of the Laplace transform in the solution of fractional hyperbolic differential equations. The fractional hyperbolic functions are found to be closely related to the R1 -trigonometric functions defined in Chapter 6. Chapters 6–8 present three fractional trigonometries. We have tried to make each of these chapters as stand-alone developments, at the expense of minor repetition. Chapter 6 develops the R1 -trigonometry. It is based on the R-function with imaginary parameter a, namely Rq,v (ia, t). Multiplication of the parameter by i toggles the R1 -hyperbolic functions to the R1 -trigonometric functions, and so on. A fractional trigonometry, the R2 -trigonometry based on an imaginary time variable, Rq,v (a, it), is developed in Chapter 7. It is found that these functions are characterized by their attraction to circles when plotted in phase plane format. The obvious extension of these two trigonometries, the R3 -trigonometry of Chapter 8, sets both the a parameter and the t variable to be imaginary, Rq,v (ia, it). It was thought at the time that this trigonometry would behave as an hyperbolic analog to the R2 -trigonometry. However, such simple relationships between the two were not found. Chapter 9 presents the heart of the fractional trigonometry, namely the fractional meta-trigonometry. Here, both a and t are allowed to be fully complex, by choosing as the basis Rq,v (i𝛼 a, i𝛽 t). This chapter generalizes the results of the previous four chapters. Laplace transforms for the generalized functions are determined along with their fractional differintegrals. Fractional exponential forms for the functions are also determined. In Chapter 10, the ratio and reciprocal functions associated with the generalized fractional sines and cosines of Chapter 9, that is, Sinq,v (a, 𝛼, 𝛽, k, t) and Cosq,v (a, 𝛼, 𝛽, k, t), as well as the generalized parity functions are considered. The parity functions represent a new set of fractional trigonometric functions with no counterpart in the classical trigonometry. Because of the large number of possible ratio and reciprocal functions, the treatment in this chapter is cursory. Because of the newness of this material, we have tried to be generous with the graphic forms of the functions. In spite of this attitude, we have found that because of different behavior over various domains of the functions and the number of parameters in the functions that complete coverage in this regard to be impossible and the reader is encouraged to program some of the functions and to experiment for themselves. In Chapter 3, two new functions, the G- and H-functions, are introduced. These functions are generalizations of the R-function with multiple real and complex roots in the denominators of their Laplace transforms. Because of the great generality of these functions, consideration of these functions as the basis for further generalization of the fractional trigonometry is discussed in Chapter 11. In Chapter 12, these functions are needed for the solution of linear fractional differential equations with repeated roots.

Preface

Part II of the book is largely dedicated to applications and potential application of the fractional trigonometry. The most important application is the use of the fractional trigonometry for the solution of linear constant-coefficient commensurate-order fractional differential equations. In Chapter 12, specialized Laplace transforms for the meta-trigonometric functions are developed and applied to the solution of these linear fractional differential equations. Examples showing the solution of fractional differential equations with unrepeated roots and with repeated real and complex roots are given. Chapter 13 studies the time- and frequency-domain responses for linear fractional systems based on the R-function and the meta-trigonometric functions. The stability of the basic fractional elements is also considered. Unlike the classical trigonometric functions, the fractional counterparts do not generally share the periodicity property. As a practical result, we are limited to evaluation of the defining infinite series for function evaluation. This presents numerical difficulties as the time and/or order variables increase. Chapter 14 discusses this problem and establishes series convergence. Phase plane plots of pairs of the fractional trigonometric functions define a new and unique family of spirals, the fractional spirals. Chapter 15 studies these spirals and their relationship to some of the classical spirals. Linear oscillators are often used in the study of ordinary differential equations and in the modeling of physical systems. Chapter 16 identifies those linear fractional trigonometric oscillators that are neutrally stable. This chapter also explores possible application of coupled fractional trigonometric oscillators. Chapters 17–19 study the possible application of the fractional spirals and thus the fractional trigonometry and fractional differential equations. The potential applications include sea shell growth and morphology, mathematical classification of spiral galaxy morphology, and various weather phenomena such as hurricanes and tornados. Finally, Chapter 20 looks at some of the many remaining challenges and opportunities relative to the fractional exponential function and the fractional trigonometry, in particular, the need for a fractional field equation as it relates to spatial fractional spiral morphology. For the professional with a background in the fractional calculus, a quick coverage of the essence of the book may be had from Chapters 2, 3, 9, and 12, with selected attention to the applications of interest contained in Chapters 15–19. CARL F. LORENZO TOM T. HARTLEY Cleveland OH, July 2016

xvii

xix

Acknowledgments It is with pleasure that the authors extend their gratitude to the National Aeronautics and Space Administration, Glenn Research Center for their long-term and continued support of our work in both fractional calculus and fractional trigonometry. While many people have helped us in many ways, we are deeply indebted to the managers that have supported our work, including Dr. Norman Wenger, Dr. Walter C. Merrill, Dr. Jih-Fen Lei, Dr. Gary T. Seng of NASA Glenn Research Center, and Dr. Alex DeAbreu of the University of Akron. Also from NASA Glenn Research Center, we are grateful to Dr. Ten-Huei Guo for his review of many of our papers, Mr. Robert C. Anderson for his help when the computers become balky, and Dr. John Lekki for his enlightening and helpful discussions on photon entanglement. We are grateful to our editors at John Wiley & Sons, Susanne Steitz-Filler and Allison McGinniss, for their support and guidance through the publication process. We are most appreciative of Dr. Mary V. Zeller from NASA Glenn Research Center, whose support and encouragement during the writing of this book were most helpful. We would like to express our gratitude to Professor Rachid Malti of Bordeaux University whose work on stability of elementary transfer functions of the second kind provides an important contribution to Chapter 13. We are grateful to Professor Diego Dominici of New York State University at New Paltz for his helpful discussions on inverse functions. We would like to thank in advance any kind readers who forward to us any suggestions for improvement or corrections in the book. Finally, we must recognize the support and love of our wives Margaret Lorenzo and Karen Hartley and our children Mary (Lorenzo) Stafford, Ann (Lorenzo) Wahlay, James Lorenzo, Carol (Lorenzo) Palko, Margaret (Lorenzo) Strekal, and John Hartley who give meaning to our work.

xxi

About the Companion Website This book is accompanied by a companion website:

www.wiley.com/go/Lorenzo/Fractional_Trigonometry The website includes: • Figures from the book appearing in color.

1

1 Introduction The ongoing development of the fractional calculus and the related development of fractional differential equations have created new opportunities and new challenges. The need for a generalized exponential function applicable to fractional-order differential equations has given rise to new functions. In the traditional integer-order calculus, the role of the exponential function and the trigonometric functions is central to the solution of linear ordinary differential equations. Such a supporting structure is also needed for the fractional calculus and fractional differential equations. The purpose of this book is the development of the fractional trigonometries and hyperboletries that generalize the traditional trigonometric and hyperbolic functions based on generalizations of the common exponential function. The fundamental idea is that through the development of the fractional calculus, which generalizes the integer-order calculus, generalizations of the exponential function have been developed. The exponential function in the integer-order calculus provides the basis for the solution of linear fractional differential equations. Also, it may be thought of as the basis of the trigonometry. A high-level summary of the flow of the development of the book is given in Figure 1.1. The generalized exponential functions that we use, the F-function and the R-function, are fractional eigenfunctions; that is, they return themselves on fractional differintegration. The F-function is the solution to the fundamental fractional differential equation q 0 dt x(t)

+ ax(t) = 𝛿(t)

when driven by a unit impulse. The R-function, Rq,v (a, t), generalizes the F-function by including its integrals and derivatives as well. First, we show that these functions provide solutions to the fundamental fractional-order differential equation. Then, we explore the properties of the generalized exponential functions and develop some properties of the functions that will aid in the development and understanding of the fractional trigonometries and hyperboletries. This development follows a few mathematical preliminaries. The R1 , R2 , and R3 trigonometries along with the R1 hyperboletry are developed by replacing a and t in the R-function with various combinations of real and purely imaginary variables. Based on the newly defined functions, a variety of basic properties and identities are determined. Furthermore, the Laplace transforms of the functions are determined and the fractional derivatives and fractional integrals of the functions elucidated. The following chapters develop an overarching fractional trigonometry called the fractional meta-trigonometry that contains all of the fractional trigonometries shown in Figure 1.1 and infinitely many more. This is accomplished by replacing a and t in the R-function with general complex variables. We find that the fractional trigonometric functions lead to a generalization

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

2

1 Introduction

Fundamental fractional differential equation

F-function R-function

R1 Hyperbolic functions

R1 Trigonometric functions

R2 Trigonometric R3 Trigonometric functions functions

Fractional meta-trigonometric functions Complexity functions

Parity functions

Solutions of linear fractional differential equations

Fractional spirals

Potential scientific applications

Fractional oscillators

Shell morphology

Galactic classification

Weather phenomenon

Figure 1.1 Overview of the development of the fractional trigonometry and its applications.

of the circular functions, which we have called the fractional spiral functions. These functions appear to model various natural phenomena, and preliminary applications of these functions to the properties of fractional oscillators, sea shells, galaxies, and more are explored. An important aspect of this modeling is that we can infer from the spiral functions the underlying fractional differential equations describing the phenomena, which is demonstrated. More importantly, the new fractional functions provide the solutions to classes of linear fractional differential equations.

1.1 Background Because of the close association of the fractional calculus and the fractional trigonometry to be developed, we present here a brief introduction to the concepts of the fractional calculus for the reader who is unfamiliar with the area. Several important textbooks have been written that are extremely helpful to someone entering the field. Perhaps the most useful from the engineering and scientific viewpoint, are the textbooks by Oldham and Spanier, “The Fractional Calculus” [104], and by Igor Podluby entitled “Fractional Differential Equations” [109]. A more mathematically oriented treatment is given

1.2 The Fractional Integral and Derivative

in the book by Miller and Ross [95]. An encyclopedic reference volume written by Samko et al. [114] has also been published. Furthermore, a very good engineering book has been written by Oustaloup [105] and is available in French and Bush [19]. There are a growing number of physical systems whose behavior can be compactly described using fractional differential equations theory. Areas include long lines, electrochemical processes, diffusion, dielectric polarization, noise, viscoelasticity, chaos, creep, rheology, capacitors, batteries, heat conduction, percolation, cylindrical waves, cylindrical diffusion, water through a weir notch, Boussinesq shallow water waves, financial systems, biological systems, semiconductors, control systems, electrical machinery, and more.

1.2 The Fractional Integral and Derivative The first question we need to address is “just what is a fractional derivative?” There are two separate but equivalent definitions of the fractional differintegral (Oldham and Spanier [104]), known as the Grünwald definition and the Riemann–Liouville definition. We present the Grünwald definition first, as it most recognizably generalizes the standard calculus. We then follow with the Riemann–Liouville definition as it is most easily used in practice. 1.2.1

Grünwald Definition

The Grünwald definition of the fractional-order differintegral is essentially a generalization of the derivative definition that most of us learned in introductory calculus, namely ( )−q t−a N−1 )) ( q ∑ Γ(j − q) ( d f (t) || N t−a , (1.1) ≡ lim f t − j | [d(t − a)]q |GRUN N→∞ Γ(−q) j=0 Γ(j + 1) N or in a slightly more familiar form N−1 ∑ f (t − jΔtN ) dq f (t) || ≡ lim b (q) , | [d(t − a)]q |GRUN N→∞ j=0 j (ΔtN )q

where ΔtN =

t−a , N

bj (q) =

(1.2)

Γ(j − q) . Γ(−q) Γ(j + 1)

In this definition, q is not limited to the integers and may be any real or complex number, and a is the starting time of the fractional differintegration, not to be confused with a in the differential equation in the introduction. Also, q > 0 defines differentiation, and q < 0 integration. Furthermore, Γ(∘) is the gamma function, or the generalized factorial function. It basically acts as a calibration constant here to properly interpolate the operators for values of q between the integers. In terms of notation, Oldham and Spanier [104] provide a development of equation (1.2) and generalize the fractional differintegral as dq x(t) , [d(t − a)] q

(1.3)

where it should be noticed that the differential in the denominator starts at some time a, and ends at a final time t. Thus, we see that the fractional derivative is defined on an interval and is no longer a local operator except for integer orders. Interestingly, the gamma functions force the series to terminate with a finite number of terms whenever q is any integer greater than or

3

4

1 Introduction

equal to zero, which represent the usual integer-order derivatives. When q is a negative integer, this series also contains the single and multiple integrals as well (which have always had infinite memory). The important aspect to be recognized is that there exists an uncountable infinity of fractional derivatives and integrals between the integers. The Grünwald definition is also equivalent to the more often used Riemann–Liouville definition, which is discussed in the following section. 1.2.2

Riemann–Liouville Definition

The Riemann–Liouville definition is easiest to present for fractional integrals first, and then generalize that to the fractional derivatives. The qth-order integral is defined as (see, e.g., Oldham and Spanier [104], Podlubny [109]) t

−q a dt x(t)

(t − 𝜏)q−1 d−q x(t) ≡ ≡ x(𝜏) d𝜏, [d(t − a)]−q ∫ Γ(q)

t ≥ a,

(1.4)

a

It is important to note that this is the key definition of the fractional integral and is used by most investigators. Miller and Ross [95] provide four separate developments of this important equation. It can be shown that whenever q is a positive integer, this equation becomes a standard integer-order multiple integral. The Riemann–Liouville fractional derivative is defined as the integer-order derivative of a fractional integral dm q−m ( d x(t)), t ≥ a, (1.5) dt m a t where m is typically chosen as the smallest integer such that q − m is negative, and the integer-order derivatives are those as defined in the traditional calculus. These equations define the uninitialized fractional integral and derivative. For most engineering problems, system components, by virtue of their histories, are placed into some initial configuration or are initialized. Using mechanical systems as an example, the initial conditions are often mass positions and velocities at time zero. Fractional-order components, however, require a time-varying initialization Lorenzo [77] and Hartley [85], as they inherently have a fading infinite memory. Considering the aforementioned fractional-order integral, we assume that the fractional-order integration was started in the past, beginning at some time a, while the given problem begins at time c > a, where c is usually taken to be zero. Consider two fractional integrals of the same order acting on x(t), where x(t) and all of its derivatives are zero for all t < a. If the integral starting at c is to continue the integral starting at a, we must add an initialization 𝜓, thus q a dt x(t)

≡

t

−q a dt x(t)

=

−q c dt x(t)

+𝜓 ⇒

(t − 𝜏)q−1 x(𝜏) d𝜏 ∫ Γ(q) a

t

=

(t − 𝜏)q−1 x(𝜏) d𝜏 + 𝜓, ∫ Γ(q)

t ≥ c,

q > 0,

(1.6)

c

therefore t

𝜓=

t c (t − 𝜏)q−1 (t − 𝜏)q−1 (t − 𝜏)q−1 x(𝜏) d𝜏 − x(𝜏) d𝜏 = x(𝜏) d𝜏, ∫c ∫a ∫ Γ(q) Γ(q) Γ(q) a

t ≥ c, q > 0, (1.7)

1.2 The Fractional Integral and Derivative

clearly a time-varying function. This term represents the historical effect (Lorenzo and Hartley [68, 71]) or the initialization required for the fractional integral. The initialized fractional-order integration operator then is defined as −q c Dt x(t)

−q

≡ c dt x(t) + 𝜓(xi , −q, a, c, t),

where

c

𝜓(xi , −q, a, c, t) ≡

∫a

t ≥ c,

(t − 𝜏)q−1 xi (𝜏) d𝜏, Γ(q)

(1.8)

t ≥ c.

(1.9)

𝜓(xi , −q, a, c, t) is called the initialization function and is generally a time-varying function that must be added to the fractional-order operator to account for the effects of the past. This is a generalization of the constant of integration that is usually added to the normal order-one integral. The subscript i is appended to x to indicate that xi is not necessarily the same as x. −q −q Clearly then, c Dt x(t) = a dt x(t), t ≥ c. The initialization function is a time-varying function and is required to properly bring the historical effects of the fractional-order integral into the future. Similar considerations also apply for fractional-order derivatives [68, 71], that is, for any real value of q. Again, for convenience, c = 0 is typically chosen. 1.2.3

The Nature of the Fractional-Order Operator

The important properties of integer-order integration and differentiation have been shown to hold for initialized fractional-order operators (Lorenzo and Hartley [68] and [71]), including linearity and the index law. Physical insight into the nature of the fractional operators may be found in Hartley and Lorenzo [44, 47]. The fractional differintegral operator is a linear operator, and all the properties associated with linear operators hold for them. Also of considerable importance is the index law; that is, a dtu+v x(t) = a dtu a dtv x(t). The index law essentially allows us to state, for example, that the half-derivative of the half-derivative of a function is the same as the first-derivative of that function. Laplace transforms are standard tools for integer-order operators and can still be readily used for fractional-order operators. In this regard, the Laplace transform of the initialized fractional-order differintegral is shown in Lorenzo and Hartley [68, 71] to be } { q } { q L 0 Dt x(t) = L 0 dt x(t) + 𝜓(x, q, a, 0, t) = sq X(s) + L {𝜓(x, q, a, 0, t)} for all real q. (1.10) q It is important to note that L{0 dt x(t)} = sq X(s), for all q, as this is the generalization of the derivative rule for the integer-order situation. Also, note that L−1 {s−q } = t q−1 ∕Γ(q), q > 0, which leads directly to the Riemann–Liouville definition via convolution t

−q 0 dt x(t)

t

(t)q−1 (t − 𝜏)q−1 ⇔ s X(s) ⇔ x(t − 𝜏) d𝜏 = x(𝜏) d𝜏. ∫ Γ(q) ∫ Γ(q) −q

0

(1.11)

0

The Laplace transform for the fractional integral is given [78] as { −q } { −q } { } L 0 Dt h(t) = L 0 dt f (t) + L 𝜓 (fi , −q, −a, 0, t) 0

1 1 e−𝜏 s Γ(q + 1, −𝜏 s) fi (𝜏)d𝜏. = q L{f (t)} + q s s Γ(q) ∫ −a

where q ≥ 0 and

{ fi (t) h(t) = f (t)

− a < t ≤ 0, 0 < t,

q ≥ 0,

(1.12)

5

6

1 Introduction

and where fi may differ from f during the initialization period. More detailed forms are presented in Ref. [78]. The transform for the fractional derivative of order u, where u = n − q, is given by L

{

n−1 } { } ∑ { } u n−q D f (t) = s L f (t) − sn−1−j 𝜓 (j) (fi , −q, −a, 0, t)|t=0+ + sn L 𝜓(fi , −q, −a, 0, t) , 0 t j=0

(1.13) where u = n − q ≥ 0, n = 1, 2, 3, …, q ≥ 0, fi (t) = 0, ∀t < −a, and sn−q−1 [ as e Γ(q + 1, as) fi (−a) − Γ(q + 1)fi (0) sn L{𝜓 (fi , −q, −a, 0, t)} = Γ(q + 1) 0

+

] e−𝜏 s Γ(q + 1, −𝜏 s)fi′ (𝜏)d𝜏 .

∫

(1.14)

−a

In this relationship, 𝜓 (fi , −q, −a, 0, t) is the initialization function for the fractional integral part of the operator. An alternative form of equation (1.14) where the integration is based on fi (t) rather than fi ′ (t) is given by L

{

}

u 0 Dt f (t)

n−q

=s

L{f (t)} −

n−1 ∑

0 n−1−j

s

j=0

sn−q 𝜓 (fi , −q, −a, 0, t)|t=0+ + e−𝜏 s Γ(q, −𝜏 s) fi (𝜏)d𝜏, Γ(q) ∫ (j)

−a

(1.15) where u = n − q ≥ 0, n = 1, 2, 3, … , q ≥ 0, fi (t) = 0, ∀t < −a. These forms simplify for constant initialization [78], that is, when fi = constant = b, ] [ as u n−q n−q−1 e Γ(q − n + 1, as) L{0 Dt f (t)} = s L{f (t)} + b s −1 , Γ(q − n + 1) q not integer, 0 ≤ u = (n − q) ≤ n , n = 1, 2, 3, … .

(1.16)

1.3 The Traditional Trigonometry The application of the traditional integer-order trigonometry to analysis as well as engineering and science goes well beyond the calculation of triangles and triangulation. The applications include Fourier analysis, spectral analysis, solutions to ordinary and partial differential equations, and more. The trigonometric functions are found in nearly every branch of mathematics. The traditional trigonometry was originated for the solution of plane triangles. However, an additional way of interpreting the integer-order trigonometry is based on its relationship to the exponential function. The connections between the trigonometric functions and the exponential functions are very close. These relationships center on the Euler equation; that is, for z = x + iy (1.17) ez = ex ei y = ex (cos y + i sin y), as well as the definitions

eit + e−it ∑ (−1)n t 2n = 2 (2n) ! n=0

(1.18)

eit − e−it ∑ (−1)n t 2n+1 = 2i (2n + 1) ! n=0

(1.19)

∞

cos(t) ≡ and

∞

sin(t) ≡

1.3 The Traditional Trigonometry

for the sine and cosine functions. In fact, the exponential and trigonometric functions are fundamental to complex numbers and complex computation. The hyperbolic functions are also based on the exponential function; these are given in the following relationships: ∞ et + e−t ∑ t 2n cosh(t) ≡ = , (1.20) 2 (2n) ! n=0 and

et − e−t ∑ t 2n+1 = . 2 (2n + 1) ! n=0 ∞

sinh(t) ≡

(1.21)

The development of the fractional calculus has involved new functions that generalized the common exponential function. These functions allow the opportunity to generalize both the hyperbolic functions and the trigonometric functions to “fractional” or “generalized” versions. Two of these functions, to be detailed later in the book, are the F-function (Hartley and Lorenzo [45]), which is the solution of the fundamental fractional differential equation q c Dt x(t)

(1.22)

= −ax(t) + bu(t)

and its generalization, the R-function (Lorenzo and Hartley [69, 70]). They are defined as ∞ ∑ an t (n+1)q−1 Fq (a, t) ≡ , Γ((n + 1)q) n=0

t>0

(1.23)

and its vth differintegral Rq,v (a, t) ≡

∞ ∑ an t (n+1)q−1−v , Γ((n + 1)q − v) n=0

t > 0.

(1.24)

The Laplace transforms of these functions are determined in Ref. [69] as { } } { 1 sv L Fq (a, t) = q and L Rq,v (a, t) = q , Re(q − v) ≥ 0. (1.25) s −a s −a It can be seen from the series definitions of these functions that they contain the exponential function ∞ ∑ (at)n (at)2 (at)3 eat = 1 + at + + +···= (1.26) 2! 3! Γ(n + 1) 0 as the q = 1, v = 0 special case. This result and the fact that the F- and R-functions are eigenfunctions for the qth-order derivative are powerful drivers toward a new generalized trigonometry based on the fractional (or generalized) exponential function, that is, the F- or the R-function. The expectation and hope is that such a trigonometry will lead also to new generalizations of all the products of the integer-order trigonometry, a situation that will be broadly useful. These expectations and more derive from the usefulness of the ordinary trigonometry. It is well known, and follows from equation (1.26), that (it)2 (it)3 (it)4 eit = 1 + it + + + +··· 2! 3! 4 !} { { } t2 t4 t6 t3 t5 + − +··· +i t− + −··· . eit = 1 − 2! 4! 6! 3! 5!

(1.27) (1.28)

These series are, of course, recognized as representing the circular functions giving the well-known Euler equation (1.29) eit = cos(t) + i sin(t).

7

8

1 Introduction

It is important to note that cos(t) is a summation of terms that are simultaneously both the real part of eit and are even powers of t. Also, that sin(t) is a summation of terms that are simultaneously both the imaginary part of eit and are odd powers of t. This observation will prove important in the development to follow. Not all of the new fractional trigonometric functions will share this property. The R-function, since it includes within it the fractional differintegrals of the F-function, and is a representation of the fractional eigenfunction, is used as the generalizing basis of the exponential function. Based on the R-function, parallels with the integer-order trigonometry are used to generate related fractional trigonometries. The properties of these new trigonometries and identities flowing from the definitions are then developed. The trigonometries derived from these generalizations will be jointly termed “The Fractional Trigonometry.” The definitions for the fractional trigonometries can be based on several different parallels between various properties of the integer-order trigonometry and the proposed fractional-order trigonometries. For example, parallels based on equations (1.17)–(1.19) each provide a basis for definitions. Laplace transforms of the new functions are determined. Fractional differential equations for which the functions are solutions and various intra- and interrelationships of the new trigonometric functions are studied.

1.4 Previous Efforts There have been previous definitions offered for fractional trigonometric functions. These efforts, each amounting to a page or so of definitions, have been based on the Mittag-Leffler function and are discussed in Appendix A. In all cases, the definitions are subsets of those to be presented here.

1.5 Expectations of a Generalized Trigonometry and Hyperboletry There are some characteristics that a generalized trigonometry should have and additional characteristics that may be desirable. We require that any fractional trigonometry should contain the traditional, integer-order, plane trigonometry as a special case, have an eigenfunction basis, exhibit series compatibility between defined functions and generalized exponentials, and form a basis for the solution of fractional-order linear differential equations. These requirements are essentially self-explanatory. The first requires backward compatibility to the ordinary trigonometry. The second and fourth requirements are a way of saying that the new generalized trigonometry should be closely coupled to the solution of fractional differential equations and that the solutions should be expressible as linear combinations of the functions. The expectation flowing from this is that we expect insight into the solutions of fractional differential equations from the fractional trigonometry to be similar to that obtained from the trigonometric solutions associated with the solutions of ordinary differential equations. In general, our requirements and expectations for the generalized hyperbolic functions parallel those listed for the fractional trigonometry. For example, we require backward compatibility with the traditional hyperbolic functions, and so on. In addition, we expect to maintain or generalize the relationships between the traditional integer-order trigonometric functions and the traditional integer-order hyperbolic functions when the fractional versions are defined.

9

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation 2.1 The Fundamental Fractional Differential Equation This chapter develops the F-function as the solution of the fundamental fractional differential equation equation (2.1). A similar function was first used by Robotnov [112, 113]. The F-function was given in a footnote by Oldham and Spanier [104], p. 122, and later independently found by Hartley and Lorenzo [45, 48] as the solution to equation (2.1). This function is the foundation to the development of the fractional trigonometries. Following our study of the F-function, we introduce the R-function, which generalizes the F-function by including its fractional derivatives and integrals. The derivations of this chapter are abstracted from Hartley and Lorenzo, NASA [45]: The problem to be addressed is the solution of the uninitialized fractional-order differential equation q c Dt x(t)

q

= c dt x(t) = −ax(t) + bu(t),

q c Dt x(t)

q>0

(2.1)

q c dt x(t)

is the initialized qth-order derivative and represents the uniniwhere tialized qth-order derivative. Here, it is assumed for simplicity that the problem starts at t = 0, which sets c = 0. It is also assumed that all initial conditions, or initialization q q functions, are zero; thus, c Dt x(t) = c dt x(t). We are primarily concerned with the forced response. Rewriting equation (2.1) with these assumptions gives q 0 dt x(t)

= −ax(t) + bu(t).

(2.2)

We use Laplace transform techniques to obtain the solution of this differential equation. In order to do so for this problem, the Laplace transform of the fractional differential is required. Using equation (1.11) and ignoring initialization terms, equation (2.2) can be Laplace transformed as sq = X(s) = −aX(s) + bU(s).

(2.3)

This equation is rearranged to obtain the transfer function X(s) b = G(s) = q . U(s) s +a

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(2.4)

10

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation

This transfer function of the fundamental linear fractional-order differential equation contains the fundamental “fractional” pole (discussed later) and is a basis element for fractional differential equations of higher order. Specifically, transfer functions can be inverse transformed to obtain the impulse response of a differential equation. The impulse response can then be used with the convolution approach to obtain the solution of fractional differential equations with arbitrary forcing functions. In general, if U(s) is given, the product G(s)U(s) can be expanded using partial fractions, and the forced response obtained by inverse transforming each term separately. To accomplish these tasks, it is necessary to obtain the inverse transform of equation (2.4), which is the impulse response of the fundamental fractional-order differential equation. To obtain the solution for arbitrary q, it is necessary to derive the generalized fundamental impulse response for the fractional-order differential equation equation (2.4), as this is not available in the standard tables of transforms, such as Oberhettinger and Badii [103] or Erdelyi et al. [34]. This is derived in the following section.

2.2 The Generalized Impulse Response Function Continuing from Ref. [45]: To obtain the generalized impulse response, we expand the right-hand side of equation (2.4) in descending powers of s, and then inverse transform the series term-by-term. It is assumed that q > 0. As the constant b in equation (2.4) is a constant multiplier, it can be assumed, with no loss of generality, to be unity. Then, expanding the right-hand side of equation (2.4) about s = ∞ using long division gives a a2 1 ∑ (−a)n 1 1 . = q − 2q + 3q − · · · = q q s +a s s s s n=0 snq ∞

F(s) =

(2.5)

This power series, in s−q , can now be inverse transformed term-by-term using the transform pair L{t k−1 } = Γ(k)∕sk for k > 0. The result is } { a a2 1 −1 −1 L {F(s)} = L − + −··· sq s2q s3q t q−1 at 2q−1 a2 t 3q−1 = − + − · · · , q > 0. (2.6) Γ(q) Γ(2q) Γ(3q) The right-hand side can now be collected into a summation and used as the definition of the generalized impulse response function Fq [a, t] ≡ t

q−1

∞ ∑ n=0

(a)n t nq , Γ((n + 1)q)

q > 0.

(2.7)

This function is the generalization of the common exponential function that is needed for the fractional calculus. Furthermore, the important Laplace transform identity 1 , q>0 (2.8) sq − a has been established. When u(t) in equation (2.2) is a unit impulse, x(t) = bFq [−a, t] is seen to be the forced response of the fundamental fractional differential equation. L{Fq [a, t]} =

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

1.5

Fq[–1, t]

1

q = 2.0

0.5

0

q = 0.25

–0.5 –1

0

2

4

6

8

10

t-Time

Figure 2.1 The Fq [−1, t]-function versus time as q varies from 0.25 to 2.0 in 0.25 increments.

This section has established the F-function as the impulse response of the fundamental linear fractional-order differential equation. The F-function generalizes the usual exponential function and is the fractional eigenfunction. The solution of equation (2.2), with b = 1, the F-function, is shown for various values of q in Figure 2.1. We note that Fq [a, t] is a generalization of the exponential function, since for q = 1, F1 [a, t] =

∞ ∑ n=0

(at)n ≡ eat , Γ(n + 1)

t > 0.

(2.9)

This generalization, Fq [a, t], is the basis for the solution of linear fractional-order differential equations composed of combinations of fractional poles of the type of equation (2.8).

2.3 Relationship of the F-function to the Mittag-Leffler Function The F-function is closely related to the Mittag-Leffler function, Eq [at q ] [96–98], which, at this time is commonly used in the fractional calculus. For this reason, we discuss the Mittag-Leffler function briefly; from Ref. [45], we have the following: This function is defined as Eq [x] ≡

∞ ∑ n=0

xn , Γ(nq + 1)

q > 0,

(2.10)

(Erdelyi et al. [34, 35]). Letting x = −at q , this becomes ∞ [ ] ∑ (−a)n t nq Eq −at q ≡ , Γ(nq + 1) n=0

q > 0,

(2.11)

which is similar to, but not the same as equation (2.7). The Laplace transform of this equation (2.11) can also be obtained via term-by-term transformation, that is { } ]} { [ 1 at q a2 t 2q q =L − + +··· L Eq −at Γ(1) Γ(1 + q) Γ(1 + 2q)

11

12

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation

1 a2 a − q+1 + 2q+1 + · · · , s s s

(2.12)

] [ ∞ ( )n ]} 1 { [ a2 1 ∑ −a a q = L Eq −at . 1 − q + 2q − · · · = s s s s n=0 sq

(2.13)

= or equivalently

Using equation (2.5), equation (2.13) may be written as ]} 1 [ sq ] 1 q { [ = [s L{Fq [−a, t]}]. L Eq −at q = s sq + a s Thus, the Laplace transform of the Mittag-Leffler function can be written as

(2.14)

{ [ ]} sq−1 L Eq at q = q , q > 0. (2.15) s −a More importantly, from equation (2.14), the E-function and the F-function are related as follows: q−1 q (2.16) 0 dt Fq [a, t] = Eq [at ]. Functions similar to the F-function are mentioned by other authors as well. Oldham and Spanier [104], p. 122 mention it in passing in a footnote discussing eigenfunctions. Also, Robotnov [112, 113] studied a closely related function extensively with respect to hereditary integrals; he calls it the ϶-function. Also, other authors have used less direct methods to solve equation (2.2). Bagley and Calico [11] obtain a solution in terms of Mittag-Leffler functions. Miller and Ross’s [95] solution is in terms of the fractional derivative of the exponential function. They use the function Et (𝜐, a) ≡ 0 dt−𝜐 eat ,

(2.17)

whose Laplace transform is { } s−𝜐 L Et (𝜐, a) = . (2.18) s−a Also, Glockle and Nonnenmacher [38] obtain a solution in terms of the more complicated Fox Functions.

2.4 Properties of the F-Function In this section, the eigenfunction property of the F-function is derived. This essentially means that the qth-derivative of the function Fq [a, t q ], returns the same function Fq [a, t q ] for t > 0 (see equation 2.21). Then, from Ref. [45]: q

Taking the uninitialized qth-derivative (0 dt ) in the Laplace domain by multiplying by sq gives { q } s q = 0 dt Fq [−a, t]. (2.19) L−1 q s +a This equation can also be rewritten as } { q } { s a = L−1 1 − q = 𝛿(t) − aFq [−a, t], (2.20) L−1 q s +a s +a

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

where the delta function is recognized as the unit impulse function. Now comparing equations (2.19) and (2.20), it can be seen that q 0 dt Fq [−a, t]

= 𝛿(t) − aFq [−a, t].

(2.21)

This equation demonstrates the eigenfunction property of returning the same function upon qth-order differentiation for t > 0. This is a generalization of the exponential function in integer-order calculus. It is now easy to show that the F-function is the impulse response of the differential equation (2.2). Referring back to equation (2.2), and setting u(t) = 𝛿(t) and b = 1, yields q 0 dt x(t)

= −ax(t) + 𝛿(t).

(2.22)

For the F-function to be the impulse response, it must be the solution to equation (2.22), that is x(t) = Fq [−a, t]. Substituting this into equation (2.22) gives q 0 dt Fq [−a, t]

= −aFq [−a, t] + 𝛿(t).

(2.23)

This equation has been obtained by direct substitution into the differential equation. Referring back to equation (2.21), however, shows that the qth-derivative of the F-function on the left-hand side is in fact equal to the right-hand side of equation (2.23). Thus, it is shown by direct substitution that the F-function is indeed the impulse response of the differential equation (2.22). These sections have developed the F-function and the properties that are key to the development of the fractional trigonometry. The following section develops further important properties of the F-function and illustrates its application to the solution of fractional differential equations.

2.5 Behavior of the F-Function as the Parameter a Varies This section studies the F-function with complex values for the parameter a. A linear system theory approach provides further understanding of the properties of the F-function, when its Laplace transform is considered as a transfer function of a physical system. From Ref. [45]: Thus, we consider the Laplace transform of equation (2.22) as the transfer function of a related linear system 1 , q > 0. (2.24) L{Fq [a, t]} = q s −a In general, to understand the dynamics of any particular system, we often consider the nature of the s-domain singularities. We define s = rei𝜃 in what follows. For a < 0, the particular function of equation (2.24) does not have any poles on the primary Riemann sheet of the s-plane (|𝜃| < 𝜋), as it is impossible to force the denominator of the right-hand side of equation (2.24) to zero. Note, however, that it is possible to force the denominator to zero if secondary Riemann sheets are considered. For example, the denominator of the Laplace transform 1 , (2.25) L{F1∕2 [−1, t]} = 1∕2 s +1 does not go to zero anywhere on the primary sheet of the s-plane (|𝜃| < 𝜋). It does go to zero on the secondary sheet, however. With s = e±i2𝜋 , the denominator is indeed zero. Thus, this Laplace transform has a pole at s = e±i2𝜋 , which is at s = 1 + i 0 on the second

13

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation

Magnitude 1/(s0.5+1) (dBs)

4 3 2 Secondary sheet 1 0 –1 2 0 Imag (s)

–2

–1

–2

2

1

0 Real (s)

Figure 2.2 Both sheets of the Laplace transform of the F-function in the s-plane. Source: Hartley and Lorenzo 1998 [45]. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

Riemann sheet. This is shown in Figure 2.2, where |1∕s1∕2 + 1| is plotted as a function of Real(s) and Imaginary(s). As it is difficult to visualize multiple Riemann sheets, following LePage [65], it is useful to perform a conformal transformation into a new plane. Let w = sq .

(2.26)

The transform in equation (2.24) then becomes 1 1 L{Fq [−a, t]} = q ⇔ . (2.27) s +a w+a With this transformation, we study the w-plane poles. Once we understand the time-domain responses that correspond to the w-plane pole locations, we will be able Figure 2.3 The w-plane for q = 1∕2, with w = s1∕2 and s = a + ib. Source: Hartley and Lorenzo 1998 [45]. Public domain.

3

2

a= –4 b=4

1 Imaginary (w)

14

b=2 a= 0 a= 2 b=0

b=0 0 b=0 –1

–2

–3 –3

a= –2 Secondary Riemann sheet

–2

–1

a= 4

a= 0 b= –2 b = –4

a= –4

0 Real (w)

1

2

3

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

to clearly understand the implications of this new complex plane. To accomplish this, it is necessary to map the s-plane, along with the time-domain function properties associated with each point, into the new complex w-plane. To simplify the discussion, we limit the order of the fractional operator to 0 < q ≤ 1. Let w = 𝜌ei𝜑 = 𝛼 + i𝛽.

(2.28)

Then, referring to equation (2.26) w = sq = (rei𝜃 )q = rq eiq𝜃 = 𝜌ei𝜑 .

(2.29)

Thus, 𝜌 = rq and 𝜑 = q𝜃. With this equation, it is possible to map either lines of constant radius or lines of constant angle from the s-plane into the w-plane. Of particular interest is the image of the line of s-plane stability (the imaginary axis), that is, s = re±iq𝜋∕2 . The image of this line in the w-plane is w = r q e±iq𝜋∕2 ,

(2.30)

which is the pair of lines at 𝜑 = ±q𝜋∕2. Thus, the right half of the s-plane maps into a wedge in the w-plane of angle less than ±90q degrees, that is, the right half s-plane maps into |𝜑| < q𝜋∕2. (2.31) For example, with q = 1∕2, the right half of the s-plane maps into the wedge bounded by −𝜋∕4 < 𝜑 < 𝜋∕4; see Figure 2.3. It is also important to consider the mapping of the negative real s-plane axis, s = re±i𝜋 . The image is (2.32) w = r q e±iq𝜋 . Thus, the entire primary sheet of the s-plane maps into a w-plane wedge of angle less than ±180q degrees. For example, if q = 1∕2, then the negative real s-plane axis maps into the w-plane lines at ±90 degrees; see Figure 2.3. Continuing with the q = 1∕2 example, and referring to Figure 2.3, it should now be clear that the right half of the w-plane corresponds to the primary sheet of the Laplace s-plane. The time responses we are familiar with from integer-order systems have poles that are in the right half of the w-plane. The left half of the w-plane, however, corresponds to the secondary Riemann sheet of the s-plane. A pole at w = −1 + i0 lies at s = +1 + i0, on the secondary Riemann sheet of the s-plane. This point in the s-plane is really not in the right half s-plane, corresponding to instability, but rather is “underneath” the primary s-plane Riemann sheet. As the corresponding time responses must then be even more than overdamped, we call any time response whose pole is on a secondary Riemann sheet, “hyperdamped.” It should now be easy for the reader to extend this analysis to other values of q. To summarize this, the shape of the F-function time response, Fq [−a, t], depends upon both q and the parameter −a, which is the pole of equation (2.27). This is shown in Figure 2.4. For a fixed value of q, the angle 𝜑 of the parameter −a, as measured from the positive real w-axis, determines the type of response to expect. For small angles, |𝜑| < q𝜋∕2, the time response will be unstable and oscillatory, corresponding to poles in the right half s-plane. For larger angles, q𝜋∕2 < |𝜑| < q𝜋, the time response will be stable and oscillatory, corresponding to poles in the left half s-plane. For even larger angles, |𝜙| > q𝜋, the time response will be hyperdamped, corresponding to poles on secondary Riemann sheets.

15

16

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation

i iβ

α

Figure 2.4 Step responses corresponding to various pole locations in the w-plane, for q = 1∕2. Source: Hartley and Lorenzo 1998 [45]. Public domain.

It is now possible to do fractional system analysis and design, for commensurate-order fractional systems, directly in the w-plane. To do this, it is necessary to first choose the greatest common fraction (q) of a particular system (clearly nonrationally related powers are an important problem although a close approximation of the irrational number will be sufficient for practical application). Once this is done, all powers of sq are replaced by powers of w. Then the standard pole-zero analysis procedures can be done with the w-variable, being careful to recognize the different areas of the particular w-plane. This analysis includes root finding, partial fractions (note that complex conjugate w-plane poles still occur in pairs), root locus, compensation, and so on. We have now characterized the possible behaviors for fractional commensurate-order systems in a new complex w-plane; that is, given a set of w-plane poles, the corresponding time-domain functions are known both quantitatively and qualitatively. Although most of the discussion has actually been for 0 < q ≤ 1∕2, it is reasonably applicable to larger values of q with the appropriate modifications for many-to-many mappings.

2.6 Example In this example, we consider the impulse response of the inductor–supercapacitor pair shown in Figure 2.5. In the study of Hartley et al. [49], it was shown that a particular

2.6 Example

sL +

Figure 2.5 Supercapacitor circuit example. +

R α

Vo(s)

Vi(s)

√s

1 sc –

–

commercial supercapacitor is accurately modeled with the impedance transfer function Z(s) = R + √𝛼 + sC1 . The voltage across this device is chosen as the output voltage for this s example. Then, in Figure 2.5, the voltage transfer function from the input terminals to the supercapacitor terminals is found to be √ R + √𝛼 + sC1 RCs + 𝛼C s + 1 Vo (s) s = . (2.33) = √ Vi (s) sL + R + √𝛼 + 1 LCs2 + RCs + 𝛼C s + 1 s

For this example, we let RC = 1,

sC

𝛼C = 1,

and LC = 1. Then, √ s+ s+1 Vo (s) . = √ Vi (s) s2 + s + s + 1

(2.34)

The poles and zeros of this transfer function are plotted on the w-plane in Figure 2.6, which also shows the 45∘ stability lines for q = 0.5. Clearly, there are two poles in the right half of the w-plane, but to the left of the stability boundary. These pole locations correspond to complex stable poles in the s-plane and imply a damped oscillatory impulse response. We obtain the impulse response of equation (2.34) using F-functions. First, it is necessary to define the new Figure 2.6 The w-plane stability diagram for supercapacitor.

1.5 1

Imag (w)

0.5 0 –0.5 –1 –1.5 –1.5

–1

–0.5

0

0.5

Real (w)

1

1.5

2

17

2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation

complex variable w = becomes

√

s. Then, assuming an impulse input, the output of the transfer function

w2 + w + 1 + w2 + w + 1 −0.0570 − 0.4334i −0.0570 + 0.4334i = + w − 0.5474 − 1.1209i w − 0.5474 + 1.1209i 0.0570 − 0.1308i 0.0570 + 0.1308i + + . w + 0.5474 − 1.1209i w + 0.5474 + 1.1209i √ Inverse transforming the partial fractions with w = s yields Vo (s) =

w4

(2.35)

vo (t) = (−0.0570 − 0.4334i)F0.5 [0.5474 + 1.1209i, t] + (−0.0570 + 0.4334i)F0.5 [0.5474 − 1.1209i, t] + (0.0570 − 0.4334i)F0.5 [−0.5474 + 1.1209i, t] + (0.0570 + 0.4334i)F0.5 [−0.5474 − 1.1209i, t].

(2.36)

This impulse response is plotted in Figure 2.7. Some damped oscillations are observed as expected from the pole-zero plot. Figure 2.7 Supercapacitor impulse response.

1.4 1.2 1 Output voltage

18

0.8 0.6 0.4 0.2 0 –0.2 0

1

2

3

4

5

6

t-Time (s)

This simple example is presented to demonstrate the use of the F-function to obtain the solution of a physical fractional-order system. In this chapter, the fundamental linear fractional-order differential equation has been considered and its impulse response has been obtained as the F-function. This function most directly generalizes the exponential function for application to fractional differential equations. It is at the heart of our development of the fractional trigonometry. Also, several properties of this function have been presented and discussed. In particular, the Laplace transform properties of the F-function have been discussed using multiple Riemann sheets and a conformal mapping into a more readily useful complex w-plane. It is felt that this generalization of the exponential function, the F-function, is the most easily understood and most readily implemented of the several other generalizations presented in the literature.

19

3 The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus 3.1

Introduction

This chapter generalizes the F-function, that is, the eigenfunction solution of the fundamental fractional differential equation, to the R-function, which carries in it the derivatives and integrals of the F-function. The R-function then becomes the basis for the development of the fractional trigonometric and fractional hyperbolic functions. This chapter also compares the properties of these functions with other important functions that have been associated with the fractional calculus. The previous chapter developed the F-function, Fq [a, t], for the solution of fractional differential equations. This function provided direct solution and important understanding for the fundamental linear fractional-order differential equation and for the related initial value problem (Hartley and Lorenzo [46]). This chapter presents functions commonly used in the fractional calculus, and their Laplace transforms. A new function called the R-function and two related generalized functions, the G-function and the H-function, are presented for consideration. In the sequel, we will see that these functions are important in the development and application of the fractional trigonometries and are, therefore, useful in the solution of fractional differential equations. The R-function, Rq,v (a, t), contains the vth-order derivatives and integrals of the F-function. With v = 0, the R-function becomes the F-function and thus the Rq,0 -function also returns itself on qth-order differintegration. The G- and H-functions are needed for the analysis of repeated and partially repeated fractional poles. An example application of the R-function is provided. Sections 3.10–3.16 present some preliminaries to the development of the fractional trigonometries. The derivations of Sections 3.2–3.9 are adapted from Lorenzo and Hartley [69].

3.2 Functions for the Fractional Calculus This section summarizes a number of functions that have been found useful in the solution of problems of the fractional calculus and more particularly in the solution of fractional differential equations.

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

20

3 The Generalized Fractional Exponential Function

3.2.1

Mittag-Leffler’s Function

The Mittag-Leffler function [96, 97, 98] is given by the following equation: Eq [t] =

∞ ∑ n=0

tn , Γ(nq + 1)

q > 0.

(3.1)

This function often appears with the argument −at q , and its Laplace transform then is given as {∞ } ∑ (−a)n t nq sq q L{Eq [−at ]} = L = q , q > 0. (3.2) Γ(nq + 1) s(s + a) n=0 3.2.2

Agarwal’s Function

The Mittag-Leffler function is generalized by Agarwal [5, 6] as follows: (

𝛽−1

)

∞ m+ 𝛼 ∑ t E𝛼,𝛽 (t) = . Γ(𝛼 m + 𝛽) m=0

(3.3)

This function is particularly interesting to the fractional-order system theory due to its Laplace transform, given by Agarwal as L{E𝛼,𝛽 [t 𝛼 ]} =

s𝛼−𝛽 . s𝛼 − 1

(3.4)

This function is the (𝛼 − 𝛽)-order fractional derivative of the F-function with argument a = 1. 3.2.3

Erdelyi’s Function

Erdelyi et al. [34] has studied the following related generalization of the Mittag-Leffler function: E𝛼,𝛽 (t) =

∞ ∑

tm , Γ(𝛼 m + 𝛽) m=0

𝛼, 𝛽 > 0,

(3.5)

where the powers of t are integer. The Laplace transform of this function is given by L{E𝛼,𝛽 (t)} =

∞ ∑

Γ(m + 1) , Γ(𝛼 m + 𝛽)sm+1 m=0

𝛼 > 1, 𝛽 > 0 .

(3.6)

As this function cannot be easily generalized, it is not considered further. 3.2.4

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

To effect the direct solution of the fundamental linear fractional-order differential equation (Chapter 2), the following function was used (Hartley and Lorenzo [45]): Fq [−a, t] = t

q−1

∞ ∑ (−a)n t nq , Γ(nq + q) n=0

q > 0.

(3.7)

This function had been mentioned earlier in a footnote by Oldham and Spanier [104], p. 122. The important feature of this function is the power and simplicity of its Laplace transform,

3.2 Functions for the Fractional Calculus

namely

1 , q > 0. (3.8) sq − a This function, with a = 1, is the fractional eigenfunction in that it returns itself on qth-order differintegration. We note that Matignon [92] has also recognized this function as the fractional eigenfunction. L{Fq [a, t]} =

3.2.5

Robotnov’s Function

Robotnov [112, 113] used the function 𝜉q (a, t) =

∞ ∑ n=0

an t (n+1)(q+1)−1 Γ((n + 1)(q + 1))

in his study of hereditary integrals. The Laplace transform of this function is 1 L{𝜉q (a, t)} = q+1 . s −a Using Robotnov’s function, 𝜉q−1 (1, t) is the eigenfunction. Continuing from [69]: 3.2.6

(3.9)

(3.10)

Miller and Ross’s Function

Miller and Ross [95], pp. 80 and 309–351 introduce another function as the basis of the solution of the fractional-order initial value problem. It is defined as the vth integral of the exponential function, that is, ∞ ∑ (at)k d−v Et (v, a) = −v eat = t v eat 𝛾 ∗ (v, at) = t v , (3.11) dt Γ(v + k + 1) k=0 where 𝛾 ∗ (v, at) is the incomplete gamma function. The Laplace transform of equation (3.11) follows directly as s−v L{Et (v, a)} = , Re (v) > 1 . (3.12) s−a Miller and Ross then show that } { q ∑ 1 1 1 1 aj−1 Et (jv − 1, aq ) = v , q = 1, 2, 3, … , v = = 1, , , … , (3.13) L s − a q 2 3 j=1 which is a special case of the F-function. 3.2.7

Gorenflo and Mainardi’s, and Podlubny’s Function

Gorenflo and Mainardi [40] and Podlubny [109] use the function 𝜉q (t, a, q, v) =

∞ ∑ an t qn+v−1 . Γ(nq + v) n=0

(3.14)

This convenient function has the Laplace transform sq−v . (3.15) −a Table 3.1 presents a summary of the defining series and respective Laplace transforms for these important functions discussed here and shows the relation of their Laplace transforms to L{𝜉q (t, a, q, v)} =

sq

21

22

3 The Generalized Fractional Exponential Function

Table 3.1 Special fractional calculus functions. Function

Time expression

Mittag-Leffler (1903) [96, 97]

Eq [at q ] =

∞ ∑ n=0

Agarwal (1953) [5, 6]

Eq,𝛽 [t q ] =

Erdelyi et al. (1954) [34]

Eq,𝛽 [t q ] =

an t nq Γ(nq + 1)

∞ ∑ t (n+(𝛽−1)∕q)q Γ(nq + 𝛽) n=0 ∞ ∑ tn

Γ(nq + 𝛽) n=0 ∞ ∑ an t (n+1)q−1 Oldham and Spanier (1974) [104], Fq [a, t] = Γ((n + 1)q) n=0 Hartley and Lorenzo (1998) [44], Matignon (1998) [92] ∞ ∑ an t n+v Miller and Ross (1993)[95] Et [v, a] = Γ(v + n + 1) n=0 ∞ ∑ an t (n+1)q−1−v Lorenzo and Hartley (1999) [69] Rq,v [a, t] = Γ((n + 1)q − v) n=0 Robotnov (1969) [112]

𝜉q (a, t) =

∞ ∑ n=0

Gorenflo and Mainardi (1997) [40], Podlubny (1999) [109, 110]

𝜉(t, a, q, v) =

Laplace transform

sq − a)

s(sq

Remarks

(q − 1) differintegral of eigenfunction

sq−𝛽 − 1) ∞ ∑ Γ(n + 1) (sq

n=0

Γ(qn + 𝛽)sn+1

1 sq − a

eigenfunction

s−v (s − a) (sq

sv − a)

an t (n+1)(1+q)−1 Γ((n + 1)(1 + q))

1 sq+1 − a

∞ ∑ an t qn+v−1 Γ(nq + v) n=0

sq−v (sq − a)

vth differintegral of eigenfunction 𝜉q−1 (a, t) is eigenfunction (q − v) differintegral of eigenfunction

The G- and H-functions may be found in Sections 3.9.1 and 3.9.2. Source: Adapted from Lorenzo and Hartley 1999 [69].

that of the F-function. Clearly, many of these functions are useful for the solution of various fractional differential equations, but the F-function presented in the previous chapter appears to most properly generalize the exponential function.

3.3 The R-Function: A Generalized Function It is of great interest to develop a generalized function which, when fractionally differintegrated by any order, returns itself (with a new parameter). A function of this type would be useful for the solution of fractional-order differential equations. The following form is proposed [69, 80, 81]. Consider the function ∞ ∑ (a)n t (n+1)q−1−v . (3.16) Rq,v [a, t] = Γ((n + 1)q − v) n=0 For t < 0, R will be complex except for the cases when the exponent ((n + 1)q − 1 − v) is integer. Clearly, when v = 0 in equation (3.16), the R-function becomes the F-function (equation (2.7)); that is, Rq,0 [a, t] = Fq [a, t]. The Laplace transform of such a function will facilitate the solution of fractional-order differential equations. The Laplace transform of the R-function is determined as follows: } ∞ { ∑ (a)n t (n+1)q−1−v , t > 0. (3.17) L{Rq,v [a, t]} = L Γ((n + 1)q − v) n=0

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

Then, we have L{Rq,v [a, t]} =

∞ ∑

{ n

(a) L

n=0

(t)(n+1)q−1−v Γ((n + 1)q − v)

} .

(3.18)

From the transform tables, we have L{ t v } = Γ(v + 1)s−v−1 ,

Re (v) > −1, Re(s) > 0 .

(3.19)

Then, equation (3.18) becomes L{Rq,v [a, t]} =

∞ ∑

(a)n

n=0

L{Rq,v [a, t]} =

1 , s(n+1)q−v

∞ 1 ∑ (a)n , s−v n=0 s(n+1)q

Re((n + 1)q − v) > 0, Re (s) > 0, t > 0 . Re(q − v) > 0, Re(s) > 0, t > 0.

(3.20) (3.21)

This geometric series converges when | a∕sq | < 1. By long division, we have sv , t > 0, Re(q − v) > 0, Re(s) > 0 . sq − a We have from Ref. [69], for a time-shifted starting point

(3.22)

L{Rq,v (a, t)} =

L{Rq,v (a, t − c)} =

e−cs sv , sq − a

c ≥ 0, Re ((n + 1)q − v) > 0, Re (s) > 0.

(3.23)

3.4 Properties of the Rq,v (a, t)-Function Continuing from Ref. [69]: The time-domain character of the R-function is shown in Figures 3.1–3.3. Figure 3.1 shows the effect of variations in q with v = 0 and a = ±1. The exponential character of the function is readily observed (see q = 1). Figure 3.2 shows the effect of v on the behavior of the R-function. The effect of the characteristic time a is shown in Figure 3.3. The characteristic time is 1∕aq . For q = 1, 1∕a is the time constant, when q = 2 we have the natural frequency, when q takes on other values we have the generalized characteristic time (or generalized time constant). The eigen-property for Rq,0 [a, t] = Fq [a, t] is proved in Section 3.11. 3.4.1 Differintegration of the R-Function

It is of interest to determine the differintegral of the R-function, that is, u u 0 dt Rq,v [a, t] = 0 dt

∞ ∞ ∑ ∑ (a)n 0 dtu t (n+1)q−1−v (a)n t (n+1)q−1−v = , Γ((n + 1)q − v) n=0 Γ((n + 1)q − v) n=0

t > 0.

(3.24)

Oldham and Spanier [104], p. 67 prove the following differentiation form: v a dx [x

− a]p =

Γ(p + 1)[x − a]p−v , Γ(p − v + 1)

p > −1 ,

(3.25)

which is applied to equation (3.24) to yield u 0 dt Rq,v [a, t] =

∞ ∑ n=0

(a)n t (n+1)q−1−(v+u) , u ≥ 0, q − v > 0, t > 0. Γ((n + 1)q − (v + u))

(3.26)

23

3 The Generalized Fractional Exponential Function

10 q = 0.25 8

Rq,0 [1, t]

6 q = 0.50 4

2

q=1 q = 2.5

0

0

0.5

1

2

1.5 t-Time (a)

2.5

3

2 1.5 q = 2.5

q=1

1 0.5 Rq,0 [–1, t]

24

0

q = 0.25

–0.5 –1 –1.5 –2 0

1

2

3

4

5

t-Time (b)

Figure 3.1 Effect of q on Rq,0 (1,t), v = 0.0, a = ±1.0, 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].

Thus, we have the useful result u 0 dt Rq,v [a, t]

= Rq,v+u [a, t],

u ≥ 0, q > v, t > 0 .

(3.27)

That is, u order differintegration of the R-function returns another R-function. Furthermore, it can be easily shown that u 0 dt Rq,v [a, bt]

= bu Rq,v+u [a, bt], u ≥ 0,

t > 0, q > v .

(3.28)

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

2

R-function, R0.25,𝜈 [–1, t]

1.5

1

𝜈 = –0.50

0.5

𝜈 = 0.00 0 𝜈 = 0.50 –0.5

–1 0

0.2

0.6

0.4

0.8

1

t-Time (a) 2

R-function, R0.75,𝜈 [–1, t]

1.5

1 𝜈 = –0.50 0.5 𝜈 = 0.00 0 𝜈 = 0.50 –0.5 0

0.2

0.6

0.4

0.8

1

t-Time (b)

Figure 3.2 Effect of v on Rq,v (−1, t), v = −0.50 to 0.50 in steps of 0.25, a = −1.0. (a) q = 0.25. (b) q = 0.75.

3.4.2

Relationship Between Rq,mq and Rq,0

Substituting v = mq into the definition of R, we have

Rq,mq (a, t) =

∞ ∞ ∑ ∑ (a)n t (n+1)q−1−mq (a)n t (n+1)q−1 = t −mq , Γ((n + 1)q − mq) Γ((n − m + 1)q) n=0 n=0

t > 0.

(3.29)

25

3 The Generalized Fractional Exponential Function

R-function, R0.25,0 [a, t]

2

1.5 a = 0.50 1

0.5

0

a = –1.50 0

0.5

1

1.5

2

2.5

3

2

2.5

3

t-Time (a) 2

R-function, R0.75,0 [a, t]

26

a = 0.50

1.5

a = 0.0

1

0.5

a = –1.50 0 0

0.5

1

1.5 t-Time (b)

Figure 3.3 Effect of a on (a) R0.25,0 (a, t), q = 0.25, and (b) R0.75,0 (a, t), q = 0.75, v = 0, a = −1.5 to 0.50 in steps of 0.5. (a, b) Source: Adapted from Lorenzo and Hartley 1999 [69].

Taking n − m = r, m = 1, 2, 3, … gives Rq,mq (a, t) = t −mq

∞ ∑ (a)r+m t (r+m+1)q−1 , Γ((r + 1)q) r=−m

or Rq,mq (a, t) = (a)m

∞ ∑ (a)r t (r+1)q−1 r=0

Γ((r + 1)q)

+ (a)m

−1 ∑ (a)r t (r+1)q−1 . Γ((r + 1)q) r=−m

(3.30)

3.5 Relationship of the R-Function to the Elementary Functions

The first summation on the right-hand side is Rq,0 (a, t); thus, we have −1 ∑ (a)r t (r+1)q−1 , m = 1, 2, 3, …, Rq,mq (a, t) = (a) Rq,0 (a, t) + (a) Γ((r + 1)q) r=−m m

m

t > 0.

(3.31)

When (r + 1)q ≤ 0 and integer, the elements of the summation term vanish. 3.4.3 Fractional-Order Impulse Function

Continuing from Ref. [69]: Consider the function Rq,0 (0, t), then we can write Rq,0 (0, t) = lim Rq,0 (a, t) = lim a→0

a→0

∞ ∑ (a)n t (n+1)q−1 n=0

Γ((n + 1)q)

,

t > 0.

(3.32)

In the limit, the n > 0 terms of the summation vanish; thus, (a)0 t q−1 t q−1 = , a→0 Γ(q) Γ(q)

Rq,0 (0, t) = lim

t > 0.

From equation (3.22), the associated Laplace transform pair is given by 1 , t > 0, Re(q) > 0, Re(s) > 0, sq that is, the response of a qth-order fractional integration.

(3.33)

L{Rq,0 (0, t)} =

3.5 Relationship of the R-Function to the Elementary Functions Continuing from Ref. [69]: Many of the elementary functions are special cases of the R-function. Some of these are illustrated here. 3.5.1

Exponential Function

Consider R1,0 (a, t), by definition, we have R1,0 (a, t) =

∞ ∞ ∑ ∑ (a)n t n (at)n = , Γ(n + 1) n=0 n! n=0

t > 0.

(3.34)

Thus, R1,0 (a, t) = eat ,

t > 0.

(3.35)

Clearly then R1,0 (1, 1) = e; thus, the generalized exponential constant for any order q is given by Rq,0 (1, 1). A plot of these values is presented in Figure 3.4. 3.5.2

Sine Function

Consider a R2,0 (−a2 , t), by definition, we have a R2,0 (−a , t) = a 2

∞ ∑ (−a2 )n t (n+1)2−1 n=0

(2n + 1) !

{ =a

a 2 t 3 a4 t 5 t− + −··· 3! 5!

} .

(3.36)

27

3 The Generalized Fractional Exponential Function

20 Generalized exponential const. Rq,0 [1,1]

28

q

Rq,0 [1,1]

1/4 1/3 1/2 2/3 3/4 1.0 3/2 2.0

15

10

11.2102 8.3953 5.5732 4.1544 3.6786 2.7183=e 1.7232 1.1752

5

0

0

1

2

3

5

4

Order q

Figure 3.4 Generalized exponential constant Rq,0 (1, 1) versus q.

Thus, a R2,0 (−a2 , t) = sin (at), 3.5.3

t > 0.

(3.37)

Cosine Function

The cosine function relates to R2,1 (−a2 , t), again by definition R2,1 (−a2 , t) =

∞ ∑ (−a2 )n t (n+1)2−1−1 n=0

{ =

Γ((n + 1)2 − 1)

=

∞ ∑ (−a2 )n t 2 n

(2n) !

n=0

} a2 t 2 a 4 t 4 + −··· . 1− 2! 4!

(3.38)

Thus, R2,1 (−a2 , t) = cos (at), 3.5.4

t > 0.

(3.39)

Hyperbolic Sine and Cosine

Consider a R2,0 (a2 , t), by definition, we have a R2,0 (a2 , t) = a

∞ ∑ (a2 )n t (n+1) 2−1 n=0

Γ((n + 1)2)

=a

∞ ∑ (a2 )n t 2 n+1 n=0

(2n + 1) !

{ } a3 t 3 a 5 t 5 = at + + +··· . 3! 5!

(3.40)

Thus, a R2,0 (a2 , t) = sinh(at),

t > 0.

(3.41)

R2,1 (a2 , t) = cosh (at),

t > 0.

(3.42)

In a similar manner,

3.6 R-Function Identities

3.6 R-Function Identities 3.6.1

Trigonometric-Based Identities

Continuing from Ref. [69]: A number of identities involving the R-function may be readily shown based on the elementary functions. The exponential function (equation (3.35)) R1,0 (a, x) = eax ,

x > 0,

(3.43)

ei x = R1,0 (i, x),

x > 0.

(3.44)

may be expressed as Then, from equation (3.37), sin (ax) = a R2,0 (−a2 , x),

x > 0,

(3.45)

and expressing the sine function in complex exponential terms gives 1 ix (e − e−i x ), x > 0 . 2i Combining equations (3.44)–(3.46), then yields the identity sin (x) =

1 (R (i, x) − R1,0 (−i, x)), x > 0 . 2 i 1,0 In a similar manner, using the cosine function (equation (3.39)) R2,0 (−1, x) =

cos (x) = R2,1 (−1, x) =

1 ix (e + e−ix ), 2

x > 0,

(3.46)

(3.47)

(3.48)

from which

1 (3.49) (R (i, x) + R1,0 (−i, x)), x > 0 . 2 1,0 The hyperbolic functions may also be used as a basis using the sinh function, which yields R2,1 (−1, x) =

R2,0 (1, x) =

1 (R (1, x) − R1,0 (−1, x)), 2 1,0

x > 0.

(3.50)

The cosh function gives 1 (3.51) (R (1, x) + R1,0 (−1, x)), x > 0 . 2 1,0 Many other identities may be found based on the known trigonometric identities, a few examples follow, from (3.52) sin2 (x) + cos2 (x) = 1 , R2,1 (1, x) =

we have R22,0 (−1, x) + R22,1 (−1, x) = 1,

x > 0.

(3.53)

From the identity sin (2x) = 2 sin (x) cos (x)

(3.54)

derives R2,0 (−1, 2x) = 2R2,0 (−1, x)R2,1 (−1, x),

x>0.

(3.55)

From the trigonometric identity sin (3x) = 3 sin (x) − 4 sin 3 (x) ,

(3.56)

29

30

3 The Generalized Fractional Exponential Function

we determine the identity R2,0 (−1, 3x) = 3 R2,0 (−1, x) − 4R32,0 (−1, x), 3.6.2

x > 0.

(3.57)

Further Identities

Other identities may be derived as follows. Let v = q − p, then the Laplace transform of the R-function (equation (3.22)) may be written as sq−p 1 L{Rq,q−p (−a, t)} = q = p−q q , Re(p) > 0, Re(s) > 0. (3.58) s + a s (s + a) This may be rearranged to give ] [ −a 1 1 . (3.59) = p 1+ q p−q q s (s + a) s s +a Inverse transforming gives the identity Rq,q−p (−a, t) = Rp,0 (0, t) − aRq,−p (−a, t), t > 0.

(3.60)

Another set of identities follows by factoring the denominator of the Laplace transform; thus, [ ] sv 1 v =s . (3.61) Rq,v (a, t) ⇔ q s −a (sq∕2 − a1∕2 ) (sq∕2 + a1∕2 ) Now, a partial fraction expansion of the denominator gives 1 1 1 ⎡ ⎤ ⎡ 1 sv sv ⎤ 1∕2 1∕2 ⎢ 2a1∕2 ⎥ ⎢ 2a1∕2 ⎥ 2a 2a − q∕2 − q∕2 = s ⎢ q∕2 ⎥ = ⎢ q∕2 ⎥, 1∕2 1∕2 1∕2 1∕2 s + a ⎥ ⎢s − a s +a ⎥ ⎢s − a ⎣ ⎦ ⎣ ⎦ v

q > 2v.

(3.62)

Taking the inverse transform yields 1 Rq,v (a, t) = 1∕2 {Rq∕2,v (a1∕2 , t) − Rq∕2,v (−a1∕2 , t)}, t > 0 . (3.63) 2a The following relationships of the R-function [70] with advanced functions are based on expansions for the defined functions (see, e.g., Spanier and Oldham [116]). The product of the exponential function and the complementary error function is given by √ ∞ ∑ √ (∓ x)n (3.64) exp(x) erfc(± x) = = R1∕2,−1∕2 (∓1, x), x > 0. Γ(1 + n∕2) n=0 The error function is given by erf(x) = exp(−x2 )

∞ ∑ n=0

x2n+1 = R1,0 (−1, 0, x2 )R1,−1∕2 (1, x2 ), Γ(n + 3∕2)

The expansion for Dawson’s integral becomes √ √ ∞ √ 𝜋 𝜋x ∑ (−x)n (−1, x), = R daw( x) = 2 n=0 Γ(n + 3∕2) 2 1,−1∕2

x > 0.

x > 0.

(3.65)

(3.66)

Many distributions may be in terms of exponential of powers of x (see Spanier and Oldham [116], p. 260). Since R1,0 (a, t) = R1,0 (1, at) = ea t , these distributions may also be expressed as R-functions. Many more such identities are possible, indeed because of the generality of the R-function, powerful meta-identities may be possible. More general R-function identities are derived in Chapter 4.

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

3.7 Relationship of the R-Function to the Fractional Calculus Functions Continuing from Ref. [69]: The generality of the R-function allows it to be related to many other functions. In this section, it is related to the important functions discussed in the introductory section of the paper. The Laplace transform facilitates determination of the desired relationships. The double arrow is used to indicate the transform pairs, thus for the R-function Rq,v (a, t) ⇔

3.7.1

sv , sq − a

Re(q − v) > 0 .

(3.67)

Mittag-Leffler’s Function

The Mittag-Leffler function and its transform relate to the R-function as sq−1 ⇔ Rq,q−1 (−a, t), sq + a

Eq [−at q ] ⇔

t > 0.

(3.68)

The time-domain relationship is Rq,q−1 (−a, t) = Eq [−a t q ] =

∞ ∑ (−a)n t nq , Γ(n q + 1) n=0

t > 0.

(3.69)

Also, because 0 dt𝛼 Rq,v [−a, t] = Rq,v+𝛼 [−a, t], it follows that q−1 Rq,0 (−a, t) 0 dt

3.7.2

= Eq [−at q ],

t > 0.

(3.70)

Agarwal’s Function

The Agarwal function and its transform relate to the R-function as follows: Eq,p [t q ] ⇔

sq−p ⇔ Rq,q−p (1, t), sq − 1

t > 0.

(3.71)

The time-domain relationship is Rq,q−p (1, t) = Eq,p [t q ] = 3.7.3

∞ ∑ t nq−1+p , Γ(nq + p) n=0

t > 0.

(3.72)

Erdelyi’s Function

The relationship between the Erdelyi function and the R-function is given by Rq,q−𝛽 (1, t) = t

𝛽−1

q

Eq,𝛽 [t ] = t

𝛽−1

∞ ∑ n=0

3.7.4

t nq , Γ(n q + 𝛽)

t > 0.

(3.73)

Oldham and Spanier’s, and Hartley’s Function

The F-function and its transform relate to the R-function as follows: 1 Fq [−a, t] ⇔ q ⇔ Rq,0 (−a, t), t > 0. s +a

(3.74)

31

32

3 The Generalized Fractional Exponential Function

The time series common to these functions is given as Rq,0 (−a, t) = Fq [−a, t] =

∞ ∑ (−a)n t (n+1)q−1 n=0

Γ((n + 1)q)

,

t > 0.

(3.75)

3.7.5 Miller and Ross’s Function

The Miller and Ross’s function and its transform relate to the R-function as follows: s−v Et (v, a) ⇔ ⇔ R1,−v (a, t), t > 0. s−a The time series common to these functions is given as R1,−v (a, t) = Et (v, a) =

∞ ∑ n=0

3.7.6

(a)n t n+v , Γ(n + v + 1)

t > 0.

(3.76)

(3.77)

Robotnov’s Function

The R-function may be written as Rq+1,0 (a, t) =

∞ ∑ (a)n t (n+1)(q+1)−1 = 𝜉q (a, t), Γ((n + 1)(q + 1)) n=0

t > 0,

(3.78)

which is the function of Robotnov. 3.7.7

Gorenflo and Mainardi’s, and Podlubny’s Function

Consider the following form of the R-function, Rq,q−v (a, t) =

∞ ∑ n=0

∑ (a)n t nq−1+v (a)n t (n+1)q−1−(q−v) = = 𝜉(t, a, q, v), Γ((n + 1)q − (q − v)) n=0 Γ(nq + v)) ∞

t > 0,

(3.79)

which is the desired function. The Laplace transform for this function is given Table 3.1.

3.8 Example: Cooling Manifold To demonstrate the use of the R-function in the solution of fractional differential equations, we consider the transient response of a cooling manifold (Figure 3.5). The cooling manifold could apply to a heat engine or to heat sink for an integrated circuit, or to other similar situations. Here, a source temperature Ts , drives the heat flow Qin, , into the manifold by convective heat transfer; thus, Qin (t) = hA(Ts (t) − Tm (t)),

(3.80)

where h A is the product of the convection heat transfer coefficient and the surface area. The wall temperature is Tm . Cooling fins, which are considered here to be of semi-infinite length for the timescale of interest, conduct the heat flow Qf away from the wall. The fin heat flow then is described by ( ( )) k k 1∕2 1∕2 Qf (t) = √ 0 Dt Tm (t) = √ 0 dt Tm (t) + 𝜓 Tm , 1∕2, a, t , (3.81) 𝛼 𝛼

3.8 Example: Cooling Manifold

Figure 3.5 Cooling manifold.

Tm Ts

Qin

Qf

1∕2

where k is the thermal conductivity and )𝛼 is the thermal diffusivity, 0 dt Tm (t) is the unini( tialized semiderivative and 𝜓 Tm , 1∕2, a, t is the initialization function [71, 78]. This function brings in the effect of the initial temperature distribution along the fin. For simplicity, we consider Tm to be uniform along the manifold wall and therefore the analysis of a single cooling fin will expose the basic behavior of the process. The manifold temperature results from the heat balance equation: t

Tm (t) =

1 (Qin (t) − Qf (t))dt + Tmi , wcv ∫0

(3.82)

where wcv is the product of the manifold mass and the specific heat of the material. Taking the Laplace transform of these equations yields Qin (s) = hA(Ts (s) − Tm (s)), k Qf (s) = √ (s1∕2 Tm (s) + 𝜓(s)), 𝛼 ( ) T 1 1 Tm (s) = (Qin (s) − Qf (s)) + mi . wcv s s

(3.83) (3.84) (3.85)

Solving for Tm (s) yields Tm (s) =

T − b 𝜓(s) cTs (s) + mi 1∕2 , 1∕2 s + bs + c (s + bs + c)

(3.86)

√ where b = k∕wcv 𝛼 and c = hA∕wcv . The denominator of the first term on the right is factored and expanded in partial fractions, yielding 1 1 ⎛ ⎞ 𝛽1 − 𝛽2 ⎟ ⎜ 𝛽 2 − 𝛽1 1 =⎜ + ⎟, s + bs1∕2 + c ⎜ s1∕2 + 𝛽1 s1∕2 + 𝛽2 ⎟ ⎝ ⎠

(3.87)

√ √ where 𝛽1 = b2 + 12 b2 − 4c and 𝛽2 = b2 − 12 b2 − 4c, and it is assumed that b2 > 4c.. Note, if this is not the case, then the methods of Chapter 12 may be applied to solve with quadratic forms

33

34

3 The Generalized Fractional Exponential Function

in equation (3.87). Then, Tm (s) may be written as 1 1 1 1 ⎞ ⎞ ⎛ ⎛ 𝛽2 − 𝛽1 ⎟ 𝛽2 − 𝛽1 ⎟ ⎜ 𝛽2 − 𝛽1 ⎜ 𝛽 2 − 𝛽1 − 1∕2 − 1∕2 Tm (s) = c ⎜ 1∕2 ⎟ Ts (s) − b ⎜ 1∕2 ⎟ 𝜓(s) ⎜ s + 𝛽1 s + 𝛽2 ⎟ ⎜ s + 𝛽1 s + 𝛽2 ⎟ ⎠ ⎠ ⎝ ⎝ 1 1 ⎞ ⎛ 𝛽 2 − 𝛽1 ⎟ ⎜ 𝛽2 − 𝛽1 − 1∕2 + ⎜ 1∕2 ⎟ Tmi . ⎜ s + 𝛽1 s + 𝛽 2 ⎟ ⎠ ⎝

(3.88)

Then, assuming a unit impulse for Ts (s) and applying the convolution theorem for the 𝜓(s) terms, the solution may be inverse transformed directly in terms of R-functions as c Tm (t) = (R (−𝛽1 , t) − R1∕2,0 (−𝛽2 , t)) 𝛽2 − 𝛽1 1∕2,0 ) ( t b − (R1∕2,0 (−𝛽1 , t − 𝜏) − R1∕2,0 (−𝛽2 , t − 𝜏))𝜓(𝜏) d𝜏 𝛽2 − 𝛽1 ∫0 Tmi (R (−𝛽1 , t) − R1∕2,0 (−𝛽2 , t)). (3.89) + 𝛽2 − 𝛽1 1∕2,0

3.9 Further Generalized Functions: The G-Function and the H-Function Continuing from Ref. [69]: 3.9.1

The G-Function

Functions more general than the R-function are needed. Two such functions are derived here. It is simpler here to work backward from the s-domain to the time domain. Thus, we consider the following function: s𝜈 G(s) = q , (3.90) (s − a)r where 𝜈, q, and r are real and are not constrained to be integers. Then, this may be written as ( ) a −r . (3.91) G(s) = sv−qr 1 − q s Now, the parenthetical expression may be expanded using the binomial theorem to give ∞ ( )j ∑ | a| Γ(1 − r) −a | | < 1, (3.92) G(s) = s𝜈−q r , | sq | q Γ(1 + j)Γ(1 − j − r) s | | j=0 or G(s) =

∞ ∑ j=0

Γ(1 − r) (−a)j s𝜈−q r−q j . Γ(1 + j)Γ(1 − j − r)

(3.93)

This expression may be term-by-term inverse transformed, yielding for t > 0, Gq,𝜈,r [a, t] =

∞ ∑ j=0

Γ(1 − r) (−a)j t (r+j)q−𝜈−1 , Γ(1 + j)Γ(1 − j − r)Γ((r + j)q − 𝜈)

|a| Re(qr − v) > 0, Re(s) > 0, || q || < 1 . |s |

(3.94)

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

Thus, we have the following transform pair: { } s𝜈 L Gq,𝜈,r [a, t] = q , Re(qr − v) > 0, Re(s) > 0, (s − a)r

| a| | | > 0. (3.95) | sq | | | The form of equation (3.94) presents evaluation difficulties, since when r is an integer, Γ(1 − r) and Γ(1 − j − r) can become infinite. Equation (3.94) maybe rewritten as follows (from Spanier and Oldham (p. 414, eq. 43:5:6) [116]): Γ(x − n) =

(−1)n Γ(x) Γ(x) , = (x − 1)(x − 2) · · · (x − n) (1 − x)n

n = 0, 1, 2, … .

(3.96)

where (1 − x)n is the Pochhammer polynomial. From this result with x = 1 − r, we can write Γ(1 − j − r) =

(−1)j Γ(1 − r) Γ(1 − r) , = (−r)(−1 − r) · · · (1 − r − j) (r)j

j = 0, 1, 2, … .

Substituting this result in equation (3.94) yields the following more computable result: ∞ ∑ {(−r)(−1 − r) · · · (1 − j − r)} (−a)j t (r+j)q−𝜈−1 Gq,𝜈,r [a, t] = , j! Γ((r + j)q − 𝜈) j=0

t > 0.

(3.97)

In terms of the Pochhammer polynomial Gq,𝜈,r [a, t] =

∞ ∑ (r)j (a)j t (r+j)q−𝜈−1 j=0

| a| , Re(qr − v) > 0, Re(s) > 0, || q || < 1, j! Γ((r + j)q − 𝜈) |s |

t > 0.

(3.98)

The Laplace transform of the G-function (equation (3.90)) indicates that this function will be useful for the solution of fractional differential equations with repeated real roots and is used in Chapter 12. Figures 3.6–3.8 show the G-function for q = 0.75, 1.0, and 1.50 each with r varying between 0.25 and 2.0. Podlubny [109] presents a form that is a special case of the G-function where r is constrained to be an integer. It is also clear that taking r = 1 specializes the G-function into the R-function. 1

G0.75,0,r (–1, t)

0.8

0.6

0.4

0.2

r = 2.0

0

r = 0.25 0

2

4

6

8

10

t-Time

Figure 3.6 Gq,v,r (a, t) versus t for q = 0.75, a = −1.0, v = 0, and r = 0.25–2.0 in steps of 0.25.

35

3 The Generalized Fractional Exponential Function

1.2 1

Gq,v,r (–1, t)

0.8 0.6 0.4 0.2

r = 2.0

0 –0.2

r = 0.25 0

4

2

6

8

10

t-Time

Figure 3.7 Gq,v,r (a, t) versus t for q = 1.0, a = −1.0, v = 0, and r = 0.25–2.0 in steps of 0.25. 1.2 1 0.8 Gq,v,r (–1, t)

36

0.6 r = 2.0 0.4 0.2 0 r = 0.25

–0.2 –0.4

0

2

4

6

8

10

t-Time

Figure 3.8 Gq,v,r (a, t) versus t for q = 1.5, a = −1.0, v = 0, and r = 0.25–2.0 in steps of 0.25.

3.9.2

The H-Function

Repeated complex roots may be analyzed by Laplace transforms of the form Hq,v,r (c0 , c1 , s) =

(s2q

s𝜈 , + c1 sq + c0 )r

(3.99)

where at this point there is no constraint on the size of v. From this transform, the time-domain H-function is derived. This equation may be rewritten as ( c0 )−r c1 s𝜈 𝜈−2qr 1 + = s + (3.100) Hq,v,r (c0 , c1 , s) = ( ( c ))r c sq s2q s2q 1 + q1 + 2q0 s s

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

The Binomial theorem is applied to give ) ∞ ( ∑ c0 )n −r ( c1 Hq,v,r (c0 , c1 , s) = s𝜈−2qr + , n sq s2q n=0

| | c1 | + c0 | < 1, | sq s2q | | | ( ( ))n ∞ ∑ c0 c1 Γ(1 − r) Hq,v,r (c0 , c1 , s) = s𝜈−2qr , 1 + Γ(1 − n − r)Γ(n + 1) sq c1 sq n=0 ) ∞ ( c )n ( ∑ c0 n Γ(1 − r) 1 Hq,v,r (c0 , c1 , s) = s𝜈−2qr . 1 + Γ(1 − n − r)Γ(n + 1) sq c1 sq n=0

The Binomial theorem is applied again to the last term, giving ) ∞ ∞ ( )( ( c )n ∑ ∑ c0 k Γ(1 − r) n 1 𝜈−2qr , Hq,v,r (c0 , c1 , s) = s k Γ(1 − n − r)Γ(n + 1) sq c1 sq n=0 k=0 | | | | c1 | + c0 | < 1, | c0 | < 1. | c sq | | sq s2q | | | 1 | | n ∞ ( ) ∑ c1 Γ(1 − r) 1 Hq,v,r (c0 , c1 , s) = 2qr−v s Γ(1 − n − r)Γ(n + 1) sq n=0 ( ) ∞ ∑ c0 k Γ(n + 1) × , Γ(n − k + 1)Γ(k + 1) c1 sq k=0 Hq,v,r (c0 , c1 , s) =

∞ ∑ n=0

∞ ck0 cn−k Γ(1 − r) ∑ 1 1 , q(2r+n+k)−v Γ(1 − n − r) k=0 Γ(n − k + 1)Γ(k + 1) s

(3.101)

(3.102)

2qr > v.

(3.103)

Inverse transforming term-by-term gives Hq,v,r (c0 , c1 , t) =

∞ ∞ ∑ ∑ n=0 k=0

ck0 cn−k Γ(1 − r) t q(2r+n+k)−v−1 1 , Γ(1 − n − r) Γ(n − k + 1) Γ(k + 1) Γ(q(2r + n + k) − v)

t > 0.

(3.104) As in the G-function case, Γ(1 − r), Γ(1 − n − r), and Γ(n − k + 1) can become infinite when r is integer; therefore, applying the form of equation (3.96), we have Γ(1 − r) = (−1)n (r)n , Γ(1 − n − r)

n = 0, 1, 2, … ,

and (1 − (n + 1))k (−1)k (−n)k 1 = , = k Γ(n − k + 1) (−1) Γ(n + 1) Γ(n + 1)

k = 0, 1, 2, … .

Therefore, Hq,v,r (c0 , c1 , t) =

∞ ∞ ∑ ∑ (−1)n (r)n (−1)k (−n)k ck0 cn−k 1 n=0 k=0

k! n!

t q(2r+n+k)−v−1 , Γ(q(2r + n + k) − v)

t > 0,

(3.105)

where again (r)n and (−n)k are Pochhammer polynomials. Now, from Ref. [116], p. 151 (18:5:1) (−n)k = (−1)k (n − k + 1)k , therefore Hq,v,r (c0 , c1 , t) =

∞ ∞ q(2r+n+k)−v−1 ∑ ∑ (−1)n (r)n (n − k + 1)k ck0 cn−k 1 t n=0 k=0

k! n! Γ(q(2r + n + k) − v)

,

t > 0.

(3.106)

37

38

3 The Generalized Fractional Exponential Function

Thus, we have the transform pair L{Hq,v,r (c0 , c1 , t)} =

(s2q

s𝜈 , + c1 sq + c0 )r

| | c1 | + c0 | < 1, | sq s2q | | |

| c0 | | | | c sq | < 1, | 1 |

2qr − v > 0,

(3.107)

where Hq,v,r (c0 , c1 , t) is given by equation (3.106). It is clear from the Laplace transforms (3.90) and (3.99) that the H-function can be simplified to the G-function by setting c0 = 0. Table 3.1 summarizes the advanced functions studied in previous sections along with their defining series and Laplace Transforms.

3.10 Preliminaries to the Fractional Trigonometry Development The previous sections of this chapter have considered the R-function as it relates to the other important functions of the fractional calculus and to the traditional elementary functions. Furthermore, various identities have been shown. Some specific tools and properties, however, are still needed. These include the eigen-property and some additional relationships. The remainder of this chapter presents material that will be useful in the development of the fractional trigonometries that follow.

3.11 Eigen Character of the R-Function The fundamental property of the exponential function that makes it so useful is the eigen-property, namely d x (3.108) e = ex , dx that is, the function returns itself on differentiation (and integration). In other words, the function is an iteration operator. That is, repeated differintegrations still return the same form. It is this property that underlies its central role in the solution of integer-order ordinary and partial differential equations. The following development follows that of Lorenzo and Hartley [69] also in Ref. [80]. The R-function with v = 0 is the F-function and has the eigenfunction character under qth-order differintegration. Consider q 0 dt Rq,0 (a, t)

=

q ∞ ∑ (a)n 0 dt t (n+1)q−1 n=0

Γ((n + 1)q)

.

(3.109)

Oldham and Spanier [104], p. 67 prove the following useful form: v p a dx [x − a] =

Γ(p + 1)[x − a]p−v , Γ(p − v + 1)

p > −1.

Applying this equation, we have q

0 dt Rq,0 (a, t) =

∞ ∑ (a)n t nq−1 n=0

Γ(nq)

,

q > 0.

Now, let n = m + 1, then, q

0 dt Rq,0 (a, t) =

∞ ∑ (a)m+1 t (m+1)q−1 , Γ((m + 1)q) m=−1

q > 0.

(3.110)

3.13 R-Function Relationships

or q 0 dt Rq,0 (a, t)

(a)m t (m+1)q−1 , q > 0. m→−1 Γ((m + 1)q)

= (a)Rq,0 (a, t) + a lim

(3.111)

The right most term in this equation is zero for t > 0 and is at most an impulse function at t = 0 (see equation (2.21)). Thus, we have q 0 dt Rq,0 (a, t)

t > 0, q > 0 .

= (a)Rq,0 (a, t),

(3.112)

Thus, for a = 1, the function returns itself under qth-order differintegration.

3.12 Fractional Differintegral of the Timescaled R-Function It is sometimes necessary to differintegrate the R-function in more general form than that of Section 3.4; then, ∞ ∞ 𝜎 ∑ an (bt)(n+1)q−1−v ∑ an b(n+1)q−1−v 0 dt t (n+1)q−1−v 𝜎 𝜎 d R (a, bt) = d = . (3.113) 0 t q,v 0 t Γ((n + 1)q − v) n=0 Γ((n + 1)q − v) n=0 On applying equation (3.110) to equation (3.108), we obtain 𝜎 0 dt Rq,v (a, bt) =

∞ ∑ an b𝜎 (bt)(n+1)q−1−v−𝜎 n=0

Γ((n + 1)q − v − 𝜎)

,

q > v, t > 0,

thus yielding the final result 𝜎 0 dt Rq,v (a, bt)

= b𝜎 Rq,v+𝜎 (a, bt),

q > v, t > 0,

(3.114)

which will be found useful in the sequel.

3.13 R-Function Relationships The following relationships are particularly useful for the developments that follow. Consider Rq,v (ab𝛼 , t) =

∞ ∑ (ab𝛼 )n t (n+1)q−1−v n=0

Γ((n + 1)q − v)

Rq,v (ab𝛼 , t) = t q−1−v

∞ ∑ (a)n (b𝛼∕q t)(n+1)q−1−v n=0

For 𝛼 = q, we have

t > 0,

(3.115)

∞ ∑ (a)n (b𝛼∕q t)nq , Γ((n + 1)q − v) n=0

Rq,v (ab𝛼 , t) = b−𝛼(q−1−v)∕q Thus, we have

,

Γ((n + 1)q − v)

(3.116)

.

Rq,v (ab𝛼 , t) = b−𝛼(q−1−v)∕q Rq,v (a, b𝛼∕q t). Rq,v (abq , t) = b−(q−1−v) Rq,v (a, bt),

t > 0.

(3.117)

(3.118) (3.119)

From equation (3.118), we also have the following important special cases: Rq,v (a b, t) = b−(q−1−v)∕q Rq,v (a, b1∕q t), t > 0, Rq,v (a (±i)𝛼 , t) = (±i)−𝛼(q−1−v)∕q Rq,v (a, (±i)𝛼∕q t), t > 0.

(3.120) (3.121)

39

40

3 The Generalized Fractional Exponential Function

Because of the fractional nature of the exponents, these functions can be multivalued. When appropriate, we will deal with the principal functions.

3.14 Roots of Complex Numbers There will be a number of occasions in which we will require various roots of complex numbers. This section presents a short review of this topic in the context needed for the development to follow. The general expression for the roots of a complex number follow (see, e.g., Churchill [22]). We have for z = rei𝜃 = x + i y the general roots are given by { [ ] [ ]} √ m m n zm∕n = rm cos (𝜃 + 2 𝜋 k) + i sin (𝜃 + 2 𝜋 k) , k = 0, 1, 2, … , n − 1 . (3.122) n n Now, for z = i, we have { [ ( )] [ ( )]} m 𝜋 m 𝜋 im∕n = cos + 2 𝜋 k + i sin + 2𝜋 k , n 2 n 2

k = 0, 1, 2, … , n − 1 .

(3.123)

By Euler equation, we have ei y = cos y + i sin y; therefore, we have the useful form [ ( )] [ ( )] [ ( )] m 𝜋 m 𝜋 i m 𝜋 +2𝜋k = cos + 2𝜋k + i sin + 2𝜋k , im∕n = e n 2 n 2 n 2 k = 0, 1, 2, … , n − 1 .

(3.124)

In a similar manner,

( ( [ )] [ )] m 𝜋 m 𝜋 = cos − + 2𝜋k + i sin − + 2𝜋k n 2 n 2 )] [ ( )] [ ( m 𝜋 m 𝜋 + 2𝜋k − i sin + 2𝜋k , k = 0, 1, 2, … , n − 1 , = cos n 2 n 2 [ (

i

−m∕n

=e

−i

m n

and

𝜋 2

)]

+2𝜋k

[ (

(−i)

m∕n

=e

i

m n

− 𝜋2 +2𝜋k

[

)]

= cos

(3.125)

( )] [ ( )] 𝜋 𝜋 m m − + 2𝜋k + i sin − + 2𝜋k , n 2 n 2 k = 0, 1, 2, … , n − 1 .

(3.126)

Letting k = −k ∗ [

(−i)

m∕n

=e

−i

( m n

)] 𝜋 ( ( )] [ )] [ + 2𝜋k ∗ m 𝜋 m 𝜋 2 = cos − + 2𝜋k ∗ + i sin − + 2𝜋k ∗ , n 2 n 2

k ∗ = 0, 1, 2, … , n − 1 .

(3.127)

Thus, we can see that equations (3.125) and (3.126) are the same but for the direction that the roots are enumerated. They are precisely the same when k = 0 = −k ∗ . Thus, we may write [ ( )] [ ( )] [ ( )] m 𝜋 m 𝜋 −i m 𝜋 ±2𝜋k = cos ± 2𝜋k − i sin ± 2𝜋k , (−i)m∕n = i−m∕n = e n 2 n 2 n 2 k = 0, 1, 2, … , n − 1 , (3.128) with the understanding that the individual roots are enumerated backward in one case relative to the other.

3.15 Indexed Forms of the R-Function

3.15 Indexed Forms of the R-Function 3.15.1 R-Function with Complex Argument

Here, the indexed forms of the R-function are developed. Previous studies of the R-function [69, 70] have been limited to positive real values for the t argument. Consider the R-function with complex time argument, that is, z = tei𝜆 . Rq,v (a, z) =

∞ ∑ an (z)(n+1)q−1−v . Γ((n + 1)q − v) n=0

(3.129)

Now, for q and v rational, we may write nq + q − 1 − v =

nNq Dv + Nq Dv − Dq Dv − Nv Dq Dq Dv

=

M D

where q = Nq ∕Dq and v = Nv ∕Dv are ratios of integers. Then, equation (3.129) becomes ∞ (( ) ) (( ) )} ∑ an (t)(n+1)q−1−v { M M cos (𝜆 + 2𝜋 k) + i sin (𝜆 + 2𝜋 k) , Rq,v (a, k, te ) = Γ((n + 1)q − v) D D n=0 (3.130) with k = 0, 1, … , Dq Dv − 1 and where k has been introduced into the R-function argument to reflect its presence on the right-hand side of the equation. For imaginary time z = it ⇒ 𝜆 = 𝜋∕2, and equation (3.130) becomes i𝜆

Rq,v (a, k, it) =

∞ ∑ an (t)(n+1)q−1−v Γ((n + 1)q − v) n=0 )) (( ) ( ))} { (( ) ( 𝜋 𝜋 M M + 2𝜋 k + i sin + 2𝜋 k , × cos D 2 D 2 k = 0, 1, … , Dq Dv − 1.

(3.131)

For real time z = t ⇒ 𝜆 = 0, and equation (3.130) becomes Rq,v (a, k, t) =

∞ ∑ an (t)(n+1)q−1−v Γ((n + 1)q − v) n=0 ) (( ) )} { (( ) M M (2𝜋 k) + i sin (2𝜋 k) , × cos D D

k = 0, 1, … , Dq Dv − 1. (3.132)

It is useful to consider Rq,v (a, k, ±t) for t > 0. Then, equation (3.130) may be written as Rq,v (a, k, t) =

∞ ∑ an (t)(n+1)q−1−v cos(𝜆k ) Γ((n + 1)q − v) n=0 ∞ ∑ an (t)(n+1)q−1−v +i sin(𝜆k ), Γ((n + 1)q − v) n=0

k = 0, 1, … , Dq Dv − 1,

(3.133)

where 𝜆k = (nq + q − 1 − v)(2𝜋k) for Rq,v (a, k, t) and 𝜆k = (nq + q − 1 − v)(𝜋 + 2𝜋k) for Rq,v (a, k, −t). When k = 0, the principal function is notated as Rq,v (a, t). Thus, we see that for negative values of t the R-function becomes complex with Dq Dv branches. Also, we observe that even with t a positive real number the function is multivalued. Equation (3.130) together with equation (1.24) provides a more complete definition for the R-function. Figure 3.9 shows the principal real and imaginary parts of R-function for various values of q and real t.

41

3 The Generalized Fractional Exponential Function

Real [Rq,0 (1, t)]

10

q = 0.2 q = 0.4

5 q = 0.6, 0.8

q = 1.0

q = 0.2 0

q = 0.4

–2

–1

0

1

2

t-Time 1 Imag [Rq,0 (1, t)]

42

q = 1.0

0

q = 0.2, 0.4, 0.6, 0.8, 1 q = 0.2

–1

q = 0.8

–2 –3 –4 –2

–1

0 t-Time

1

2

Figure 3.9 R-Function Rq,0 (1, t). Real and imaginary parts for q = 0.2–1.0 in steps of 0.2.

Thus, from these results we make the observation that the fundamental fractional differential equation (1.22) considered earlier will have Dq Dv many solutions. It follows that all fractional differential equations can have multiple solutions. It may also be true that the initialization (FDE history) may determine a single solution. These issues yet need to be investigated. 3.15.2

Indexed Forms of the R-Function

In the traditional trigonometry, the introduction of i or − i into the exponent of the common exponential produced the Euler equation (1.29), which opens a path to the traditional trigonometric functions. Similarly, introduction of i or − i into the arguments of the R-function leads to various fractional trigonometries that are studied in future chapters. However, because of the fractional exponents of t, introduction of i or − i into the arguments of the R-function results in indexed versions of the R-function. 3.15.2.1

Complexity Form

There are a variety of special arguments of the R-function that are considered in the chapters that follow. We now consider the real and imaginary parts of the R-function with imaginary arguments, and ∞ ∑ (ai𝜃 )n (i𝜆 t)(n+1)q−1−v , t > 0, (3.134) Rq,v (ai𝜃 , i𝜆 t) = Γ((n + 1)q − v) n=0 or Rq,v (ai𝜃 , i𝜆 t) =

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

in(𝜃+𝜆)+𝜆(q−1−v) ,

t > 0.

(3.135)

3.15 Indexed Forms of the R-Function

Now, for q, v, 𝜃, and 𝜆 real, rational, and in minimal form, we may write n 𝜃 + n 𝜆 + 𝜆q − 𝜆 − 𝜆v nN𝜃 Dq Dv D𝜆 + nN𝜆 Dq Dv D𝜃 + N𝜆 Nq Dv D𝜃 − N𝜆 Dq Dv D𝜃 − N𝜆 Nv Dq D𝜃 M , = = D Dq Dv D𝜃 D𝜆 where q = Nq ∕Dq , v = Nv ∕Dv , 𝜃 = N𝜃 ∕D𝜃 , 𝜆 = N𝜆 ∕D𝜆 , and M/D are ratios of integers with no common factors. Then, by equation (3.123), equation (3.135) becomes ∞ ( ) ∑ (a)n (t)(n+1)q−1−v 𝜋 Rq,v (ai𝜃 , k, i𝜆 t) = cis((n(𝜃 + 𝜆) + 𝜆(q − 1 − v)) + 2𝜋k) , Γ((n + 1)q − v) 2 n=0 t > 0, k = 0, 1, 2, … , D − 1,

(3.136)

where cis(•) = cos(•) + i sin(•). This form of the R-function allows various arguments of interest to be evaluated. Table 3.2 lists various complexity-based indexed forms of the R-function that may be used as a basis for a fractional trigonometry. 3.15.2.2

Parity Form

The R-function may also be indexed by separation into even and odd powers of a or tq . Collecting the even and odd powers of a from equation (3.16), the parity form of the R-function is written as ∞ ∑ (a)n t (n+1)q−1−v Rq,v [a, t] = Γ((n + 1)q − v) n=0 =

∞ ∞ ∑ ∑ (a)2n t (n+1)2q−1−v−q (a)2n+1 t (n+1)2q−1−v + , Γ((n + 1)2q − v − q) n=0 Γ((n + 1)2q − v) n=0

Table 3.2 Special case coefficients for indexed forms of the R-function using equation (3.136) or (3.140). R

𝜽

𝝀

Rq,v (a, k, t)

0

0

(3.141)

Rq,v (−a, k, t)

2

0

(3.142)

Equation number

Rq,v (a, k, −t)

0

2

(3.143)

Rq,v (−a, k, −t)

2

2

(3.144)

Rq,v (ia, k, t)

1

0

(3.145)

Rq,v (−ia, k, t)

3

0

(3.146)

Rq,v (ia, k, −t)

1

2

(3.147)

Rq,v (−ia, k, −t)

3

2

(3.148)

Rq,v (a, k, it)

0

1

(3.149)

Rq,v (a, k, −it)

2

1

(3.150)

Rq,v (−a, k, −it)

2

3

(3.151)

Rq,v (a, k, −it)

0

3

(3.152)

Rq,v (ia, k, it)

1

1

(3.153)

Rq,v (ia, k, −it)

1

3

(3.154)

Rq,v (−ia, k, it)

3

1

(3.155)

Rq,v (−ia, k, −it)

3

3

(3.156)

t > 0.

(3.137)

43

44

3 The Generalized Fractional Exponential Function

Again this form is generalized by now considering the R-function with complex arguments, ai𝜃 and i𝜆 t, then ∞ ∑ (ai𝜃 )2n (i𝜆 t)(n+1)2q−1−v−q Rq,v [ai𝜃 , i𝜆 t] = Γ((n + 1)2q − v − q) n=0 +

∞ ∑ (ai𝜃 )2n+1 (i𝜆 t)(n+1)2q−1−v n=0

Γ((n + 1)2q − v)

,

t > 0,

(3.138)

∞ ∑ (a)2n (t)(n+1)2q−1−v−q 2n𝜃+𝜆((n+1)2q−1−v−q) i Rq,v [ai , i t] = Γ((n + 1)2q − v − q) n=0 𝜃

𝜆

+

∞ ∑ (a)2n+1 (t)(n+1)2q−1−v n=0

Γ((n + 1)2q − v)

i𝜃(2n+1)+𝜆((n+1)2q−1−v) ,

t > 0,

(3.139)

with q, v, 𝜃, and 𝜆 rational and in minimal form. As mentioned earlier, we may write M M 2n 𝜃 + 2n 𝜆q + 𝜆q − 𝜆 − 𝜆v = 1 and 2n 𝜃 + 2n 𝜆q + 𝜃 + 2𝜆q − 𝜆 − 𝜆v = 2 , D D where q = Nq ∕Dq , v = Nv ∕Dv , 𝜃 = N𝜃 ∕D𝜃 , and 𝜆 = N𝜆 ∕D𝜆 are ratios of integers. Then, equation (3.139) becomes Rq,v [ai𝜃 , i𝜆 t] =

∞ { ( )} ∑ (a)2n (t)(n+1)2q−1−v−q 𝜋 cis [2n𝜃 + 𝜆((n + 1)2q − 1 − v − q)] + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 ∞ { ( )} 2n+1 (n+1)2q−1−v ∑ (a) (t) 𝜋 cis [𝜃(2n + 1) + 𝜆((n + 1)2q − 1 − v)] + 2𝜋k , + Γ((n + 1)2q − v) 2 n=0

t > 0. (3.140)

Then, Table 3.2 also applies to various parity-based indexed forms of the R-function that may be used as a basis for a fractional trigonometry using equation (3.140).

3.16 Term-by-Term Operations Term-by-term operations, that is, integrations, differentiations, and more generally differintegrations of fractional-order on infinite series occur frequently throughout this book, making it prohibitive to treat the topic in detail at each occurrence. Most instances are series, composed of fractional-order terms that are special cases of those generalized meta-series found in Chapter 9. The remaining cases are infinite power series. It is important to clearly delineate the general conditions under which such term-by-term operations may be used. For term-by-term integration of an infinite series, for example, Miller [94], p. 102 provides the following applicable theorem: “Let u1 (x) + u2 (x) + · · · + un (x) + · · ·

(3.157)

be an infinite series of functions. Let the series converge uniformly to s(x). Then, if every ui (x), i = 1, 2, … is defined and integrable on [ab]; the series b

∫a

b

b

u1 (x)dx+ u (x)dx+ · · · + u (x)dx+ · · · ∫a 2 ∫a n

(3.158)

3.16 Term-by-Term Operations

is a convergent series and ∞ b∑

∫a

uk (x)dx =

k=1

∞ ∑ k=1

b

∫a

uk (x)dx.

(3.159)

For term-by-term differentiation of an infinite series, Miller [94], p. 107 provides the following theorem: “Let u1 (x) + u2 (x) + · · · + un (x) + · · ·

(3.160)

be an infinite series of functions. Let each ui (x), i = 1, 2, … be defined and differentiable on an interval [a,b]. Let the series of equation (3.160) converge to the sum s(x) and let the series of derivatives u′1 (x) + u′2 (x) + · · · + u′n (x) + · · · converge uniformly to w(x). Then, equation (3.160) converges uniformly; s(x) is differentiable and w(x) = s′ (x).” In other words, ∑ d d∑ uk (x) = u (x). dx k=1 dx k k=1 ∞

∞

(3.161)

Other formulations for term-by-term operations on infinite series with slightly different conditions may be found in Wylie [125], pp. 785, 786, Knopp [58], pp. 341, 342, and Hyslop [54], pp. 75–80. Uniform convergence for series of interest is considered in Chapter 14. Conveniently, Oldham and Spanier [104], pp. 38, 69–75 extend the classical results of Widder [123] in this area to determine the conditions required for term-by-term differintegration. Here, we summarize their results in our notation. For integer-order integration [104], p. 38, “If f0 , f1 , … are functions defined and continuous on the closed interval a ≤ x ≤ b } {∞ ∞ ∑ ∑ −1 −1 d f (x) = a ≤ x ≤ b, (3.162) a dx {fj (x)}, a x j ∑

j=0

j=0

provided fj converges uniformly (see Section 14.3) in the interval a ≤ x ≤ b.” For integer-order differentiation [104], p. 38, each fj must have continuous derivatives on a ≤ x ≤ b, “Then, {∞ } ∞ ∑ ∑ dfj d fj = , a ≤ x ≤ b, (3.163) dx dx j=0 j=0 ∑ df provided fj converges pointwise and dxj converges uniformly (see Section 14.3) on the interval a ≤ x ≤ b.” These results are then extended [104], p. 71 to the following fractional case: { } ∞ ∞ ∑ ∑ q q p j d a [x − a] aja dx [x − a]p+j , for all q. (3.164) [x − a] = a x j j=0

j=0

This result applies inside the … “interval of convergence of the differintegrable series ∞ ∑ aj [x − a]p+j .” The definition for differintegrable series and functions of Oldham and Spanier j=0

[104], p. 46 include the functions of interest of this book and more. Uniform convergence for various series is discussed in Section 14.3. It is well known that power series may be integrated and differentiated term-by-term within their radius of convergence; see, for example, Hyslop [54], pp. 79–80.

45

46

3 The Generalized Fractional Exponential Function

3.17 Discussion This chapter has presented the R-function, a new function for use in the fractional calculus. Also presented are the related G- and H-functions. The G- and H-functions are generalizations of the R-function that allow the analysis of repeated and partially repeated fractional-order poles. This generalization allows more complicated time functions. The R-function is special because it directly contains the derivatives and integrals of the F-function. The R-function, with v = 0, returns itself on qth-order differintegration, that is, it has the eigen-property. When specialized, the R-function can represent the exponential function, the sine, cosine, hyperbolic sine, and hyperbolic cosine functions. The fractional calculus functions – the Mittag-Leffler function, Agarwal’s function, Erdelyi’s function, Hartley and Lorenzo’s F-function, and Miller and Ross’s function – may all be expressed as special cases of the R-functions. Many identities are possible with the R-function; some of these have been presented here. The usefulness of the R-function is shown in the cooling manifold problem (Section 3.8). It enables the analyst to directly inverse transform the Laplace domain solution, to obtain the time-domain solution. An R-function-based trigonometry is developed in the chapters that follow. Specifically, the material that follows develops a fractional hyperboletry based on Rq,v (a, t) with both a and t real, and fractional trigonometries based on Rq,v (i a, t) and Rq,v (a, i t) each having different parallels with the integer-order trigonometry. Because of the fractional character of q and v we find each of these trigonometries to have families of functions for fractional trigonometric functions instead of a single function, as is the case for the integer-order trigonometric functions (i.e., for the sin, cos, etc.). For each trigonometry, we present defining equations for the functions, parallels to the ordinary trigonometry, interrelationships between the functions, graphic plots of the functions, Laplace transforms of the functions, and the fractional differential equations for which these functions are solutions. To minimize confusion between the various trigonometries, a numerical subscript is used to differentiate the trigonometries, for example, R1 Cosq,v and R2 Cosq,v for fractional trigonometries 1 and 2, respectively. The subscript carries no other special meaning or indication of importance. The v parameter of the R-function is included in the definitions and mathematical operations that follow with the understanding that the v = 0 case is of primary interest. It is included with the expectation that the differentiation and integration property will be inherited for the derived functions. The next chapter develops more general relationships and identities of the R-function; it is not essential to the understanding of the fractional trigonometry and may be omitted on first reading.

47

4 R-Function Relationships 4.1 R-Function Basics The F-function and, its generalization, the R-function are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of linear fractional differential equations. Because of this central role in the fractional calculus, this chapter explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, et , and its fractional derivatives are shown. Furthermore, the inverse relationships of the exponential function, et , in terms of the R-function are developed. From the relationships developed, some important approximations are presented in Chapter 14. This chapter is adapted from Lorenzo and Hartley [70] (see also Ref. [80]): The F-function (Chapter 2) is defined as Fq (a, t) ≡

∞ ∑ an t (n+1)q−1 , Γ((n + 1)q) n=0

(4.1)

and its generalization the R-function (Chapter 3) has been defined as Rq,v (a, t) ≡

∞ ∑ an (t)(n+1)q−1−v , Γ((n + 1)q − v) n=0

t > 0.

(4.2)

In this chapter, our interest is confined to t > 0, q ≥ 0, and v ≤ q. In Chapter 3, we determined a variety of relationships associated with the R-function, including those involving relationships with the circular and hyperbolic functions as well as other advanced functions. It has been shown (Chapter 2) that the solution of the fundamental linear fractional differential equation q (4.3) a Dt x(t) + a x(t) = f (t), may be expressed in terms of these functions. As in the case of ordinary differential equations, combinations and convolutions of R-functions are used to express the solutions of systems of fractional differential equations. The general character of the R-function is shown in Figure 3.1. Figure 3.1 shows the effect of variations of q with v = 0 and a = ±1. The exponential character of the function is readily observed (see q = 1). The Laplace transform of the R-function, sv , (4.4) L{Rq,v (a, t)} = q s −a The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

48

4 R-Function Relationships

is derived in Chapter 3. It is also noted that R1,0 (a, t) = R1,0 (1, at) =

∞ ∑ n=0

(at)n = ea t , Γ(n + 1)

t > 0.

(4.5)

These relationships (4.4) and (4.5) will be useful in the analysis that follows. This is special, because in general (4.6) Rq,v (a, t) ≠ Rq,v (1, at). The following useful relationship, however, applies: ∞ ∞ ∑ ∑ (a t)(n+1)q−1−v (aq )n t (n+1)q−1−v q−1−v Rq,v (1, at) = =a , Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0

t > 0.

(4.7)

Therefore, Rq,v (1, at) = aq−1−v Rq,v (aq , t),

t > 0.

(4.8)

Alternatively, this may be written as Rq,v (a, t) = a(v+1−q)∕q Rq,v (1, a1∕q t),

t > 0.

(4.9)

In what follows in this chapter, intrarelationships between R-functions of different arguments are developed. In all cases, the constraint t > 0 applies.

4.2 Relationships for Rm,0 in Terms of R1,0 This section develops the relation Rm,0 (1, t) = f (R1,0 (a, t)) = f (eat ),

m = 1, 2, 3, … .

(4.10)

We consider first the even cases, in particular m = 2. We have then R2,0 (1, t) =

∞ ∑

t 2n+1 t t3 t5 = + + + · · ·. Γ(2n + 2) Γ(2) Γ(4) Γ(6)

(4.11)

tn t t2 t3 =1+ + + + · · ·, Γ(n + 1) Γ(2) Γ(3) Γ(4)

(4.12)

n=0

Now, since R1,0 (1, t) =

∞ ∑ n=0

and R1,0 (−1, t) =

∞ ∑ (−1)n t n t t2 t3 =1− + − + · · ·, Γ(n + 1) Γ(2) Γ(3) Γ(4) n=0

(4.13)

it is readily seen by substitution that 1 et − e−t (R1,0 (1, t) − R1,0 (−1, t)) = = sinh(t). 2 2 An alternative approach to this problem is available through the Laplace transform ( ) 1 1 1 1 L{R2,0 (1, t)} = 2 = − s −1 2 s−1 s+1 ( ) 1 1∑ 1 = cis(2𝜋 k∕2) , 2 k=0 s − cis(2𝜋 k∕2) R2,0 (1, t) =

(4.14)

(4.15)

4.2 Relationships for Rm,0 in Terms of R1,0

where cis(𝜑) = cos(𝜑) + i sin(𝜑). The inverse transform of this equation, of course, yields the equation (4.14) result. The m = 4 case is now considered, then R4,0 (1, t) =

∞ ∑ n=0

t 4n+3 t3 t7 t 11 = + + +···. Γ(4n + 4) Γ(4) Γ(8) Γ(12)

(4.16)

Again by substitution, it is readily verified that 1 (R (1, t) − R1,0 (−1, t) + iR1,0 (i, t) − iR1,0 (−i, t)) , 4 1,0 3 3 1 ∑ i𝜋k∕2 1∑ cis(2𝜋 k∕4)R1,0 (cis(2𝜋 k∕4), t) = e R1,0 (ei𝜋k∕2 , t) , = 4 k=0 4 k=0

R4,0 (1, t) =

(4.17) (4.18)

1 1 sinh(t) − sin(t). (4.19) 2 2 Examining these solutions (equations (4.14) and (4.17)), it is observed that the values of the coefficients and the a parameter of the R1,0 -functions lay on the unit circle of the complex plane for their respective m values. This is validated for the m = 3 case. Using the Laplace transform, we have 1 L{R3,0 (1, t)} = 3 . (4.20) s −1 Now, the roots of s3 − 1 = 0 are ( ) ( ) ( ) 2𝜋k 2𝜋 k 2𝜋 k s = 11∕3 = cos + i sin = cis k = 0, 1, 2. (4.21) 3 3 3 =

Thus,

1 1∑ cis(2𝜋 k∕3) = 3 s − 1 3 k=0 2

L{R3,0 (1, t)} =

(

) 1 . s − cis(2𝜋 k∕3)

(4.22)

Inverse transforming yields 1∑ cis(2𝜋 k∕3) (R1,0 (cis(2𝜋 k∕3), t)). 3 k=0

(4.23)

1∑ cis(2𝜋 k∕3) ecis(2𝜋k∕3)t . 3 k=0

(4.24)

2

R3,0 (1, t) = This may also be written as

2

R3,0 (1, t) =

These results (equations (4.15), (4.18), and (4.23)) are now generalized to give 1∑ cis(2𝜋 k∕m) (R1,0 (cis(2𝜋 k∕m), t)) , m k=0 m−1

Rm,0 (1, t) = or

1 ∑ i2𝜋k∕m cis(2𝜋 k∕m)t Rm,0 (1, t) = e e , m k=0

m = 1, 2, 3, … ,

(4.25)

m−1

m = 1, 2, 3, … .

(4.26)

Thus, any Rm,0 -function may be written in terms of R1,0 for m = 1, 2, 3, …. Consideration of the principal value k = 0 gives the result Rm,0 (1, t)k=0 =

1 t e. m

(4.27)

49

50

4 R-Function Relationships

This is found to be a useful approximation for general m and t > m; see Chapter 14. The results of equations (4.25) and (4.26) may be generalized to arbitrary positive value for the parameter a. This more general result is given by 1 ∑ (1−m)∕m a cis(2𝜋 k∕m)R1,0 (a1∕m cis(2𝜋 k∕m), t) , m k=0 m−1

Rm,0 (a, t) =

a > 0, or

m = 1, 2, 3, … ,

1 ∑ (1−m)∕m i2𝜋k∕m a1∕m cis(2𝜋 k∕m)t a e e , m k=0

(4.28)

m−1

Rm,0 (a, t) =

a > 0,

m = 1, 2, 3, … .

(4.29)

For negative values of the a parameter, the following forms apply: 1 ∑ (1−m)∕m a cis(𝛼)R1,0 (a1∕m cis(𝛼), t) , m k=0 m−1

Rm,0 (−a, t) = −

t > 0,

a > 0,

m = 1, 2, 3, … ,

(4.30)

or 1 ∑ (1−m)∕m i𝛼 a1∕m cis(𝛼)t a e e , t > 0, a > 0 , m = 1, 2, 3, … , m k=0 m−1

Rm,0 (−a, t) = −

(4.31)

where 𝛼 = (2k + 1)𝜋∕m.

4.3 Relationships for R1/m,0 in Terms of R1,0 In this section, we seek to express R1∕m,0 (a1∕m , t) in terms of R1,0 (a, t), where m = 1, 2, 3, …. The initial interest will be 1∕m = 1∕2. Then, applying the Laplace transform L{R1∕2,0 (a1∕2 , t)} =

1 s1∕2 s1∕2 + a1∕2 a1∕2 = + . s−a s−a s1∕2 − a1∕2 s1∕2 + a1∕2

(4.32)

Inverse transforming gives R1∕2,0 (a1∕2 , t) = R1,1∕2 (a, t) + a1∕2 R1,0 (a, t), or

1∕2

R1∕2,0 (a1∕2 , t) = 0 dt

R1,0 (a, t) + a1∕2 R1,0 (a, t) .

(4.33) (4.34)

It terms of the exponential function 1∕2

R1∕2,0 (a1∕2 , t) = 0 dt eat + a1∕2 eat .

(4.35)

The m = 4 case is considered next. The Laplace transform is applied L{R1∕4,0 (a1∕4 , t)} = = =

1 s1∕4 + a1∕4 s1∕4 − ia1∕4 s1∕4 + ia1∕4 , s1∕4 − a1∕4 s1∕4 + a1∕4 s1∕4 − ia1∕4 s1∕4 + ia1∕4

(4.36)

s3∕4 + a1∕4 s1∕2 + a1∕2 s1∕4 + a3∕4 , s−a

(4.37)

3 ∑ a(3−k)∕4 sk∕4 k=0

s−a

.

(4.38)

4.4 Relationships for the Rational Form Rm∕p,0 in Terms of R1∕p,0

Inverse transforming yields the desired result 3 ∑

R1∕4,0 (a1∕4 , t) =

a(3−k)∕4 R1,k∕4 (a, t),

(4.39)

k∕4

(4.40)

k=0

or R1∕4,0 (a1∕4 , t) =

3 ∑

a(3−k)∕4 0 dt R1,0 (a, t).

k=0

The general results are seen to be ∑

m−1

R1∕m,0 (a1∕m , t) =

a(m−1−k)∕m R1,k∕m (a, t),

a > 0,

m = 1, 2, 3, … ,

(4.41)

k=0

and ∑

m−1

R1∕m,0 (a1∕m , t) =

k∕m

a(m−1−k)∕m 0 dt

R1,0 (a, t),

a > 0,

m = 1, 2, 3, … .

(4.42)

k=0

These results are now validated for the m = 3 case. Then, R1∕3,0 (a1∕3 , t) =

2 ∑

a(2−k)∕3 R1,k∕3 (a, t).

(4.43)

k=0

Laplace transforming this equation gives s1∕3

1 a2∕3 a1∕3 s1∕3 s2∕3 = + + , 1∕3 s−a s−a s−a −a =

(s1∕3

(4.44)

1 s2∕3 + a1∕3 s1∕3 + a2∕3 = . − a1∕3 ) (s2∕3 + a1∕3 s1∕3 + a2∕3 ) s1∕3 − a1∕3

(4.45)

The results of equations (4.41) and (4.42) are extended to the case of a negative a parameter by using the following expressions: ∑

m−1

R1∕m,0 (−a1∕m , t) =

(−1)m−1−k a(m−1−k)∕m R1, k∕m ((−1)m a, t),

k=0

t > 0,

a > 0,

m = 1, 2, 3, …

(4.46)

and ∑

m−1

R1∕m,0 (−a1∕m , t) =

k∕m

(−1)m−1−k a(m−1−k)∕m 0 dt

R1,0 ((−1)m a, t) ,

k=0

t > 0,

a > 0,

m = 1, 2, 3, … .

(4.47)

4.4 Relationships for the Rational Form Rm/p,0 in Terms of R1/p,0 In this part, we wish to relate Rm∕p,0 (1, t) to R1∕p,0 (a, t), where m and p are positive integers. This is a generalization of the preceding parts. We start with the example R3∕2 (1, t); applying the Laplace transform gives 1 . (4.48) L{R3∕2,0 (1, t)} = 3∕2 s −1

51

52

4 R-Function Relationships

) ( Now, the roots of s3∕2 − 1 = 0 are s = 12∕3 = cis 2𝜋 k 23 ; however, because of the periodicity of the roots, equivalent results are obtained from cis(2𝜋 k∕3). Thus, equation (4.48) may be written as 1 1 L{R3∕2,0 (1, t)} = 3∕2 = s − 1 (s1∕2 − c0 ) (s1∕2 − c1 ) (s1∕2 − c2 ) A A1 A2 + + , (4.49) = 1∕2 0 s − c0 s1∕2 − c1 s1∕2 − c2 where ck = cis(2𝜋 k∕3). The values for this problem are c0 = 1,

c1 = −0.5 + 0.866i ,

c2 = −0.5 − 0.866i.

The Ak are determined from partial fraction expansion to be ( ) ( ) 1 1 1 1 1 A2 = . A0 = A1 = 3 3 −0.5 − 0.866i 3 −0.5 + 0.866i

(4.50)

The Ak are recognized as Ak = ck ∕3. This gives the following for the transform: 1 s3∕2 − 1 cis(2𝜋∕3) cis(4𝜋∕3) cis(0) + + . = 1∕2 1∕2 1∕2 3(s − cis(0)) 3(s − cis(2𝜋∕3)) 3(s − cis(4𝜋∕3))

L{R3∕2,0 (1, t)} =

(4.51)

The inverse transform then is given as 1∑ cis(2𝜋 k∕3)R1∕2,0 (cis(2𝜋 k∕3), t). 3 k=0 2

R3∕2,0 (1, t) =

(4.52)

As in the previous parts, these results are generalized to the following: 1∑ cis(2𝜋 k∕m)R1∕p,0 (cis(2𝜋 k∕m), t), m k=0 m−1

Rm∕p,0 (1, t) =

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.53)

1 ∑ i2𝜋 k∕m e R1∕p,0 (ei2𝜋 k∕m , t) m = 1, 2, 3, … p = 1, 2, 3, … . m k=0

(4.54)

m−1

Rm∕p,0 (1, t) =

These results may also be generalized to include a non-unity value for the a parameter. The general form is given by 1 ∑ (1−m)∕m a cis(2𝜋 k∕m)R1∕p,0 (a1∕m cis(2𝜋 k∕m), t) , m k=0 m−1

Rm∕p,0 (a, t) =

a > 0,

m = 1, 2, 3 … , p = 1, 2, 3..,

(4.55)

and 1 ∑ (1−m)∕m i2𝜋 k∕m a e R1∕p,0 (a1∕m ei2𝜋 k∕m , t), m k=0 m−1

Rm∕p,0 (a, t) =

a > 0,

m = 1, 2, 3 … , p = 1, 2, 3 … .

(4.56)

Equations (4.55) and (4.56) allow any rational-based q = m∕p, Rm∕p,0 -function to be expressed in terms of its basis R1∕p,0 -functions. The results of equations (4.55) and (4.56) are extended to

4.5 Relationships for R1∕p,0 in Terms of Rm∕p,0

the case of negative a parameter by the following equations: 1 ∑ (1−m)∕m Rm∕p,0 (−a, t) = − a cis((2k + 1)𝜋∕m)R1∕p,0 (a1∕m cis((2k + 1)𝜋∕m), t), m k=0 m−1

t > 0,

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, ..,

(4.57)

and 1 ∑ (1−m)∕m i((2k+1)𝜋∕m) a e R1∕p,0 (a1∕m ei((2k+1)𝜋∕m) , t) , m k=0 m−1

Rm∕p,0 (−a, t) = −

t > 0,

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.58)

These results (equations (4.55) to (4.58)) are the most general direct relationships to the basis function presented in this chapter. The R1∕p,0 are seen as basis functions for any Rm∕p,0 .

4.5 Relationships for R1/p,0 in Terms of Rm/p,0 This section develops the reciprocal relation to that formed in the previous section. This form will be useful in developing the inverse relationships, which follow in later sections. Consider the case for p = 2, m = 1, then the Laplace transform is L{R1∕2,0 (a1∕2 , t)} = =

1 s1∕2 + a1∕2 s1∕2 + a1∕2 = , s−a s1∕2 − a1∕2 s1∕2 + a1∕2

(4.59)

s1∕2 a1∕2 + , s−a s−a

(4.60)

Inverse Laplace transforming gives R1∕2,0 (a1∕2 , t) = R1,1∕2 (a, t) + a1∕2 R1,0 (a, t),

(4.61)

or 1∕2

R1∕2,0 (a1∕2 , t) = 0 dt

R1,0 (a, t) + a1∕2 R1,0 (a, t).

(4.62)

The case p = 2, m = 4 is now considered. The Laplace transform is given by L{R1∕2,0 (a1∕4 , t)} =

1 s1∕2 + a1∕4 s1∕2 − ia1∕4 s1∕2 + ia1∕4 , s1∕2 − a1∕4 s1∕2 + a1∕4 s1∕2 − ia1∕4 s1∕2 + ia1∕4

(4.63)

=

s3∕2 + a1∕4 s1 + a1∕2 s1∕2 + a3∕4 ∑ a(3−k)∕4 sk∕2 = . s2 − a s2 − a k=0

(4.64)

3

Inverse transforming yields R1∕2,0 (a1∕4 , t) =

3 ∑

a(3−k)∕4 R2,k∕2 (a, t),

(4.65)

k∕2

(4.66)

k=0

or R1∕2,0 (a1∕4 , t) =

3 ∑ k=0

a(3−k)∕4 0 dt R2,0 (a, t).

53

54

4 R-Function Relationships

These results are generalized in the following forms: ∑

m−1

R1∕p,0 (a

1∕m

, t) =

a(m−1−k)∕m Rm∕p,k∕p (a, t),

k=0

a > 0,

p = 1, 2, 3, … , m = 1, 2, 3, … ,

(4.67)

and ∑

m−1

R1∕p,0 (a1∕m , t) =

k∕p

a(m−1−k)∕m 0 dt Rm∕p,0 (a, t),

k=0

a > 0,

p = 1, 2, 3, … , m = 1, 2, 3, … .

(4.68)

These results are now tested for the case p = 2, m = 3. Then, from equation (4.67) R1∕2,0 (a

1∕3

2 ∑

, t) =

a(2−k)∕3 R3∕2, k∕2 (a, t),

(4.69)

k=0

= a2∕3 R3∕2,0 (a, t) + a1∕3 R3∕2,1∕2 (a, t) + R3∕2,1 (a, t).

(4.70)

Applying the Laplace transform gives a1∕3 s1∕2 s 1 a2∕3 + + , = s1∕2 − a1∕3 s3∕2 − a s3∕2 − a s3∕2 − a = =

(s1∕2 s1∕2

s + a1∕3 s1∕2 + a2∕3 , − a1∕3 ) (s + a1∕3 s1∕2 + a2∕3 )

1 , − a1∕3

(4.71) (4.72) (4.73)

providing a validation point for the general form (equation (4.67)). The results of equations (4.67) and (4.68) may also be extended to the case for a negative a parameter. These results are given as ∑

m−1

R1∕p,0 (−a1∕m , t) =

(−1)m−1−k a(m−1−k)∕m Rm∕p,k∕p ((−1)m a, t) ,

k=0

t > 0,

a > 0,

p = 1, 2, 3, … , m = 1, 2, 3, … ,

(4.74)

and ∑

m−1

R1∕p,0 (−a

1∕m

, t) =

k∕p

(−1)m−1−k a(m−1−k)∕m 0 dt Rm∕p,0 ((−1)m a, t),

k=0

t > 0,

a > 0,

p = 1, 2, 3, … , m = 1, 2, 3, … .

(4.75)

4.6 Relating Rm/p,0 to the Exponential Function R1,0 (b, t) = ebt Using the results of the previous sections, it is now possible to express any Rq,0 (a, t) in terms of R1,0 (b, t) = ebt , for q = m∕p (rational). Two results are required: equations (4.42) and (4.56).

4.6 Relating Rm∕p,0 to the Exponential Function R1,0 (b, t) = ebt

For clarity of discussion, we rewrite equation (4.42) in the following terms: R1∕p,0 (b, t) =

p−1 ∑

j∕p

bp−1−j 0 dt R1,0 (bp , t)

j=0

=

p−1 ∑

j∕p

p

bp−1−j 0 dt eb t ,

b ≥ 0,

p = 1, 2, 3, … .

(4.76)

j=0

Now, this result may be directly substituted into equation (4.56) to give Rm∕p,0 (a, t) =

p−1 m−1 (( ) ) 1 ∑ (1−m)∕m i2𝜋 k∕m ∑ ( 1∕m i2𝜋 k∕m )p−1−j j∕p 1∕m i2𝜋 k∕m p a e a a e d R e ,t 1,0 0 t m k=0 j=0

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.77)

Now, this may be written as (( p∕m i2𝜋 kp∕m ) ) 1 ∑ ∑ (p−j−m)∕m i2𝜋 k(p−j)∕m j∕p a e ,t , a e 0 dt R1,0 m k=0 j=0 m−1 p−1

Rm∕p,0 (a, t) =

a ≥ 0,

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.78)

(( ) ) 1 ∑ ∑ (p−j−m)∕m i2𝜋 k(p−j)∕m a e R1, j∕p ap∕m ei2𝜋 kp∕m , t , m k=0 j=0 m−1 p−1

Rm∕p,0 (a, t) =

t > 0,

a ≥ 0,

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.79)

Thus, from equation (4.78), the generalized exponential function Rm∕p,0 may now be expressed as a function of fractional derivatives of the common exponential function (( p∕m i2𝜋 kp∕m ) ) 1 ∑ ∑ (p−j−m)∕m i2𝜋 k(p−j)∕m j∕p t , a e a e 0 dt exp m k=0 j=0 m−1 p−1

Rm∕p,0 (a, t) =

t > 0,

a ≥ 0,

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.80)

These results (equations (4.78)–(4.80)) contain the results of equations (4.28), (4.29), (4.41), and (4.42). The case for negative a parameter follows a similar development as mentioned earlier. Equation (4.41) is written as ( ) 1 ∑ (1−m)∕m i𝛼 a e R1∕p,0 a1∕m ei𝛼 , t , m k=0 m−1

Rm∕p,0 (−a, t) = −

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, … ,

(4.81)

where 𝛼 = (2 k + 1)𝜋∕m. Now, equation (4.76) is used to rewrite R1∕p,0 in equation (4.81) to give ( p∕m i𝛼p ) 1 ∑ ∑ i𝛼 (p−j) (p−j−m)∕m j∕p e a e ,t , 0 dt R1,0 a m k=0 j=0 m−1 p−1

Rm∕p,0 (−a, t) = −

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, … ,

(4.82)

55

56

4 R-Function Relationships

or ( ) 1 ∑ ∑ i𝛼 (p−j) (p−j−m)∕m e a R1, j∕p ap∕m ei𝛼p , t , m k=0 j=0 m−1 p−1

Rm∕p,0 (−a, t) = −

t > 0,

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, … ,

(4.83)

or (( p∕m i𝛼p ) ) 1 ∑ ∑ i𝛼 (p−j) (p−j−m)∕m j∕p t , e a a e 0 dt exp m k=0 j=0 m−1 p−1

Rm∕p,0 (−a, t) = −

t > 0,

a > 0,

m = 1, 2, 3, … , p = 1, 2, 3, … .

(4.84)

These results (equations (4.78)–(4.80), and (4.82)–(4.84)) are the most general direct expressions for the R-function in terms of the common exponential function presented here.

4.7 Inverse Relationships – Relationships for R1,0 in Terms of Rm,k In this and the following sections, inverse relationships expressing the exponential function in terms of various R-functions are developed. Consider 1 s+a = , (4.85) L{ea t } = s − a (s − a) (s + a) a s + . (4.86) = 2 s − a2 s2 − a2 Upon inverse transforming, we have ea t = R1,0 (a, t) = R2,1 (a2 , t) + a R2,0 (a2 , t).

(4.87)

Similarly, for m = 4, we have (s + a) (s − ia) (s + ia) 1 = , s − a (s − a) (s + a) (s − ia) (s + ia)

(4.88)

=

s3 + a s2 + a2 s + a3 , s4 − a4

(4.89)

=

3 ∑ a3−k sk . s4 − a4 k=0

(4.90)

L{ea t } =

Inverse transforming this result yields ea t = R1,0 (a, t) =

3 ∑

a3−k R4,k (a4 , t) =

k=0

3 ∑

( ) a3−k 0 dtk R4,0 a4 , t .

(4.91)

a > 0,

(4.92)

k=0

These results generalize to ∑

m−1

ea t = R1,0 (a, t) =

am−1−k Rm,k (am , t),

t > 0,

m = 1, 2, 3, .. ,

k=0

and ∑

m−1

ea t = R1,0 (a, t) =

k=0

am−1−k 0 dtk Rm,0 (am , t), t > 0, a > 0, m = 1, 2, 3, … .

(4.93)

4.8 Inverse Relationships – Relationships for R1,0 in Terms of R1∕m,0

These results are now used to test the m = 3 case. Then, based on equation (4.92), ea t = R1,0 (a, t) =

2 ∑

( ) a2−k R3,k a3 , t .

(4.94)

k=0

Thus, the Laplace transform provides the validation since ∑ a2−k sk 1 s2 + a s + a 2 = , = s − a k=0 s3 − a3 s3 − a3 2

=

(4.95)

s2 + a s + a2 1 = . (s − a) (s2 + a s + a2 ) s − a

(4.96)

For negative values of the a parameter, the following forms apply: ∑

m−1

e−a t = R1,0 (−a, t) =

( ) (−1)m−1−k am−1−k Rm,k (−1)m am , t ,

k=0

t > 0,

a > 0,

m = 1, 2, 3, … ,

(4.97)

and ∑

m−1

e−a t = R1,0 (−a, t) =

(−1)m−1−k am−1−k 0 dtk Rm,0 ((−1)m am , t) ,

k=0

t > 0,

a > 0,

m = 1, 2, 3, … .

(4.98)

4.8 Inverse Relationships – Relationships for R1,0 in Terms of R1/m,0 In this section, we seek to express R1,0 (1, t) in terms of R1∕m,0 , where m = 1, 2, 3 …. The initial interest will be 1∕m = 1∕2. Then, R1∕2,0 (1, t) =

∞ ∑ k=0

=

t k∕2−1∕2 Γ(k∕2 + 1∕2)

t −1∕2 t0 t 1∕2 t1 t 3∕2 + + + + +··· Γ(1∕2) Γ(1) Γ(3∕2) Γ(2) Γ(5∕2)

(4.99)

and R1∕2,0 (−1, t) = =

∞ ∑ (−1)k t k∕2−1∕2 Γ(k∕2 + 1∕2) k=0

t −1∕2 t0 t 1∕2 t1 t 3∕2 − + − + −···. Γ(1∕2) Γ(1) Γ(3∕2) Γ(2) Γ(5∕2)

(4.100)

Therefore, it is easily seen that R1,0 (1, t) =

1 (1, t) − R1∕2,0 (−1, t)). (R 2 1∕2,0

(4.101)

57

58

4 R-Function Relationships

We now consider the case 1∕m = 1∕4. Then, R1∕4,0 (1, t) =

∞ ∑ k=0

=

t k∕4−3∕4 Γ(k∕4 + 1∕4)

t −3∕4 t −1∕2 t −1∕4 t0 t 1∕4 t 1∕2 + + + + + + · · ·. Γ(1∕4) Γ(2∕4) Γ(3∕4) Γ(4∕4) Γ(5∕4) Γ(6∕4)

(4.102)

Now, it may be shown by substitution that R1,0 (1, t) =

1 (R (1, t) − R1∕4,0 (−1, t) + iR1∕4,0 (i, t) − iR1∕4,0 (−i, t)). 4 1∕4

(4.103)

As discussed in the previous section, this may be generalized as 1∑ cis(2𝜋 k∕m)R1∕m,0 (cis(2𝜋 k∕m), t) , m = 1, 2, 3, … . m k=0 m−1

R1,0 (1, t) =

(4.104)

Remembering that R1,0 (1, t) = et , this equation (4.104) is recognized as a decomposition (of the mth order m = 1, 2, 3, …) of the exponential function. That is, each of these functions is more basic than the exponential function in that the exponential function may readily be expressed in terms of the “fractional exponential (i.e., R1∕m,0 (a, t))” in closed form without differintegrating. The results of equation (4.104) may be generalized to arbitrary value for the parameter a, this more general result is given by R1,0 (1, a t) = R1,0 (a, t) = eat 1 ∑ (1−m)∕m = a cis(2𝜋 k∕m)R1∕m,0 (a1∕m cis(2𝜋 k∕m), t), m k=0 m−1

1 ∑ (1−m)∕m i2 𝜋 k∕m a e R1∕m,0 (a1∕m ei 2 𝜋 k∕m , t), m k=0 m−1

=

t > 0,

a > 0,

m = 1, 2, 3, … .

(4.105)

The approach to the solution for a negative a parameter is now demonstrated for the case 1∕m = 1∕4. Applying the Laplace transform gives 1 s+a 1 = 1∕4 , (s − c0 a1∕4 ) (s1∕4 − c1 a1∕4 ) (s1∕4 − c2 a1∕4 ) (s1∕4 − c3 a1∕4 )

L{R1,0 (−a, t)} = L{e−at } =

(4.106)

where ck = cia ((2k + 1) 𝜋 ∕4). This may be written as A3 A0 A1 A2 1 + + + . = s + a s1∕4 − c0 a1∕4 s1∕4 − c1 a1∕4 s1∕4 − c2 a1∕4 s1∕4 − c3 a1∕4

(4.107)

4.9 Inverse Relationships – Relationships for eat = R1,0 (a, t) in Terms of Rm∕p,0

The Ak are found by partial fractions to be Ak = −ck ∕(4a3∕4 ). The general form is then validated and given as R1,0 (1, −a t) = R1,0 (−a, t) = e−at 1 ∑ (1−m)∕m −a cis(𝛼)R1∕m,0 (a1∕m cis(𝛼), t), m k=0 m−1

=

1 ∑ (1−m)∕m i𝛼 = −a e R1∕m,0 (a1∕m ei𝛼 , t) , m k=0 m−1

t > 0,

a > 0,

m = 1, 2, 3, … ,

(4.108)

where 𝛼 = (2k + 1)𝜋 ∕m.

4.9 Inverse Relationships – Relationships for eat = R1,0 (a, t) in Terms of Rm∕p,0 Using the results of the previous sections, it is now possible to express eat = R1,0 (a, t) in terms of any Rq,0 (b, t), for q = m∕p (rational). Two results are required: equations (4.105) and (4.68). For clarity of discussion, we rewrite equation (4.68) in the following terms: R1∕p,0 (b, t) =

r−1 ∑

j∕p

b(r−1−j) 0 dt Rr∕p,0 (br , t),

j=0

b > 0,

p = 1, 2, 3 … , m = 1, 2, 3, … .

(4.109)

Using equation (4.109) to replace R1∕m,0 in equation (4.105) gives m−1 r−1 ∑ ( 1∕m )r−1−j 1 ∑ (1−m)∕m a cis(𝜆 ) a cis(𝜆 ) m k=0 j=0 (( 1∕m )r ) j∕m × 0 dt Rr∕m,0 a cis(𝜆 ) , t , a > 0, m = 1, 2, 3, … , (4.110)

R1,0 (a, t) = eat =

1 ∑ ∑ (r−j−m)∕m a cis(𝜆 (r − j))Rr∕m , j∕m (ar∕m cis(𝜆 r) , t), m k=0 j=0 m−1 r−1

R1,0 (a, t) = ea t = t > 0,

a > 0,

r = 1, 2, 3, … , m = 1, 2, 3, … ,

(4.111)

where 𝜆 = 2𝜋 k∕m. Alternatively, 1 ∑ ∑ (r−j−m)∕m i (r−j)𝜆 a e Rr∕m , j∕m ((ar∕m ei r 𝜆 ), t), m k=0 j=0 m−1 r−1

R1,0 (a, t) = ea t = t > 0,

a > 0,

r = 1, 2, 3, … , m = 1, 2, 3, … .

(4.112)

For the case of a negative a parameter, we use equation (4.109) in equation (4.108) to give e−at = R1,0 (1, −a t) = R1,0 (−a, t) =−

1

∑

m−1

m a1−(1∕m) k=0

a > 0,

r−1 ∑ ( 1∕m )r−1−j j∕m 1∕m a cis(𝛼) cis(𝛼) cis( 𝛼))r , t), 0 dt Rr∕m,0 ((a j=0

r = 1, 2, 3 … , m = 1, 2, 3, … ,

(4.113)

59

60

4 R-Function Relationships

Table 4.1 R-Function relationships positive parameter. 1 ∑ (1−m)∕m a cis(𝜆)R1,0 (a1∕m cis(𝜆), t) m k=0 m−1

Rm,0 (a, t) =

∑

(4.28)

m−1

R1∕m,0 (a1∕m , t) =

a(m−1−k)∕m R1,k∕m (a, t)

(4.41)

k=0

1 ∑ (1−m)∕m a cis(𝜆)R1∕p,0 (a1∕m cis(𝜆), t) m k=0 m−1

Rm∕p,0 (a, t) =

∑

(4.55)

m−1

R1∕p,0 (a1∕m , t) =

a(m−1−k)∕m Rm∕p,k∕p (a, t)

(4.67)

k=0

1 ∑ ∑ (p−j−m)∕m a cis((p − j)𝜆) R1, j∕p ((ap∕m cis(p𝜆)), t) m k=0 j=0 m−1 p−1

Rm∕p,0 (a, t) =

∑

(4.79)

m−1

R1,0 (a, t) = ea t =

am−1−k Rm,k (am , t)

(4.92)

k=0

1 ∑ (1−m)∕m a cis(𝜆)R1∕m,0 (a1∕m cis(𝜆), t) m k=0 m−1

R1,0 (1, a t) = R1,0 (a, t) = eat =

1 ∑ ∑ (r−j−m)∕m j∕m a (cis(𝜆 ))r−j 0 dt Rr∕m,0 (ar∕m (cis(𝜆 ))r , t) m k=0 j=0

(4.105)

m−1 r−1

R1,0 (a, t) = ea t =

(4.111)

For this table t > 0, a > 0, m = 1, 2, 3, … , p = 1, 2, 3, … , r = 1, 2, 3, … , 𝜆 = 2𝜋k∕m.

where 𝛼 = (2k + 1)𝜋∕m. Thus, we have R1,0 (1, −a t) = R1,0 (−a, t) = e−at 1 ∑ ∑ (r−j−m)∕m a cis(𝛼 (r − j)) Rr∕m, j∕m (ar∕m cis( 𝛼 r), t), m k=0 j=0 m−1 r−1

=−

t > 0,

a > 0,

r = 1, 2, 3 … , m = 1, 2, 3, … ,

(4.114)

or 1 ∑ ∑ (r−j−m)∕m i 𝛼 (r−j) a e Rr∕m, j∕m (ar∕m ei𝛼 r , t) m k=0 j=0 m−1 r−1

R1,0 (1, −a t) = e−a t = − t > 0,

a > 0,

r = 1, 2, 3, … , m = 1, 2, 3, … .

(4.115)

The expressions (4.111), (4.112), (4.114), and (4.115) are the most general expressions for the exponential function in terms of the rational R-function Rr∕p,0 and its fractional derivatives presented in this chapter. Tables 4.1 and 4.2 summarize the key R-function relationships developed in this chapter in a common form. Table 4.1 presents the relationships for positive a parameter in the left-hand side of Ru,v (a, t)-function, while Table 4.2 presents the relationships for a negative a parameter.

4.10 Discussion

Table 4.2 R-Function relationships negative parameter. 1 ∑ (1−m)∕m a cis(𝛼)R1,0 (a1∕m cis(𝛼), t) m k=0 m−1

Rm,0 (−a, t) = −

∑

(4.30)

m−1

R1∕m,0 (−a1∕m , t) =

(−1)m−1−k a(m−1−k)∕m R1,k∕m ((−1)m a, t)

(4.46)

k=0

1 ∑ (1−m)∕m a cis(𝛼)R1∕p,0 (a1∕m cis(𝛼), t) m k=0 m−1

Rm∕p,0 (−a, t) = −

∑

(4.57)

m−1

R1∕p,0 (−a1∕m , t) =

(−1)m−1−k a(m−1−k)∕m Rm∕p,k∕p ((−1)m a, t)

(4.74)

k=0

1 ∑ ∑ (p−j−m)∕m a cis(𝛼(p − j)) R1,j∕p (ap∕m cis(𝛼 p), t) m k=0 j=0 m−1 p−1

Rm∕p,0 (−a, t) = −

∑

(4.83)

m−1

R1,0 (−a, t) = e−a t =

(−1)m−1−k am−1−k Rm,k ((−1)m am , t)

(4.97)

k=0

R1,0 (1, −a t) = R1,0 (−a, t) = e−at = −

1 ∑ (1−m)∕m a cis(𝛼)R1∕m,0 (a1∕m cis(𝛼), t) m k=0

(4.108)

R1,0 (1, −a t) = R1,0 (−a, t) = e−a t = −

1 ∑ ∑ (r−j−m)∕m a cis(𝛼 (r − j)) Rr∕m, j∕m (ar∕m cis( 𝛼 r), t) m k=0 j=0

(4.114)

m−1

m−1 r−1

For this table, t > 0, a > 0, m = 1, 2, 3, … , p = 1, 2, 3, … , r = 1, 2, 3, … , 𝛼 = (2k + 1)𝜋∕m. Source: From Lorenzo and Hartley [70].

4.10 Discussion This chapter has presented a variety of relationships relating various R-functions. A key result was that Rq,0 , with q = m∕p and positive rational, may be written in terms of basis R-functions R1∕p,0 (equations ((4.55) to (4.58)). Also, reciprocal relationships have been developed expressing R1∕p,0 in terms of Rq,0 , with q = m∕p and positive rational (equations (4.67), (4.68), (4.74), and (4.75)). It was also determined that Rq,0 , with q = m∕p and positive rational, may be written in terms of fractional derivatives of R1,0 -functions (i.e., exponential functions; equations (4.78)–(4.81)). Furthermore, the R1,0 (exponential) functions may in turn be written as a function of basis functions R1∕p,0 (equations (4.92), (4.93), (4.97), and (4.98)). These results have allowed very general relationships to be written relating R1,0 to Rq,0 and its fractional derivatives, with q = m∕p and positive rational (equations (4.111), (4.112), (4.114), and (4.115)). We note that the R-function results containing imaginary or complex arguments may be evaluated using the fractional meta-trigonometric functions to be developed in Chapter 9. It is expected that the results presented here should be analytically useful since the Rq,v -function is the solution or solution basis of many linear fractional differential equations. It is also observed that all of the aforementioned relationships are expressed as finite series, the lengths of which depend on m and p. Finally, various approximations of R-functions with both positive and negative arguments have been developed and are presented in Section 14.5.

61

63

5 The Fractional Hyperboletry 5.1 The Fractional R1 -Hyperbolic Functions In previous chapters, we developed the R-function as a fractional generalization of the exponential function. Because of its eigen-character, the R-function may be used as the basis of a generalized hyperboletry. This chapter presents that development. The classical hyperbolic functions are based on the exponential function (equations (1.20) and (1.21)). The R-function is now applied to create fractional generalizations of the hyperbolic functions. With the exception of Section 5.7 and Figures 5.4–5.11 and the related discussions, this chapter is adapted from Lorenzo and Hartley [76], with the permission of the ASME: The basis for the fractional hyperboletry is Rq,v (a, t), with both a and t real. Therefore, expanding the R-function into a series of even and odd powers of a gives Rq,v (a, t) =

∞ ∑ (a)n (t)(n+1)q−1−v n=0 ∞

Rq,v (a, t) =

Γ((n + 1)q − v)

,

∑ (a)2n (t)(2n+1)q−1−v n=0

Γ((2n + 1)q − v)

t > 0, +

∞ ∑ (a)2n+1 (t)(2n+2)q−1−v n=0

Γ((2n + 2)q − v)

(5.1) ,

t > 0.

(5.2)

By reference to the exponential and the traditional hyperbolic functions, the even and odd terms are defined as the R1 -hyperbolic functions. Thus, the even part of Rq,v (a, t), that is, even powers of a, will become the R1 Cosh-function and the odd part of Rq,v (a, t), that is, odd powers of a, will become the R1 Sinh-function. In these series, t is raised to a fractional power and is thus multivalued. We include this multivalued character of the fractional hyperbolic functions by considering the roots of t based on equation (3.122): { ( ) ( )} √ M D M M 2𝜋k + i sin 2𝜋k , (5.3) t D = t M cos D D where the exponent of t is (n + 1)2q − 1 − v − q =

(n + 1)2Nq Dv − Dq Dv − Nv Dq − Nq Dv Dq Dv

=

M , D

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(5.4)

64

5 The Fractional Hyperboletry

and where q = Nq ∕Dq and v = Nv ∕Dv are assumed rational and irreducible, k = 0, 1, · · · Dq Dv − 1, and M/D is rational and in minimal form (i.e., common factors removed). Continuing from Ref. [76] with the permission of ASME: Then, the multivalued, or indexed, definition for the R1 Coshq,v is given by R1 Coshq,v (a, k, t) =

∞ ∑ ]} a2n (t)(n+1)2q−1−v−q { [ cis ((n + 1)2q − 1 − v − q)(2𝜋k) , Γ((n + 1)2q − v − q) n=0

(5.5)

where cis(∘) ≡ cos(∘) + i sin(∘). The R1 Sinh and R1 Tanh are defined similarly: R1 Sinhq,v (a, k, t) ≡

∞ ∑ ]} a2n+1 (t)(n+1)2q−1−v { [ cis ((n + 1)2q − 1 − v)(2𝜋k) , Γ((n + 1)2q − v) n=0

t > 0, (5.6)

with k = 0, 1, … , Dq Dv − 1, and R1 Tanhq,v (a, k, t) ≡

R1 Sinhq,v (a, k, t) R1 Coshq,v (a, k, t)

,

t > 0,

(5.7)

also with k = 0, 1, · · · Dq Dv − 1. It is observed that the definitions given in equations (5.5)–(5.7) are indexed complex functions. The principal R1 -hyperbolic functions are obtained by taking k = 0, in equations (5.5)–(5.7), thus ∞ ∑ (a)2n (t)(n+1)2q−1−v−q R1 Coshq,v (a, 0, t) = R1 Coshq,v (a, t) ≡ , t > 0, (5.8) Γ((n + 1)2q − v − q) n=0 and R1 Sinhq,v (a, 0, t) = R1 Sinhq,v (a, t) ≡

∞ ∑ (a)2n+1 (t)(n+1)2q−1−v n=0

and R1 Tanhq,v (a, 0, t) = R1 Tanhq,v (a, t) ≡

Γ((n + 1)2q − v) R1 Sinhq,v (a, t) R1 Coshq,v (a, t)

,

,

t > 0,

t > 0,

(5.9)

(5.10)

where we have introduced a simplified notation when k = 0, for the principal functions. Also, the subscript “1” is only used to relate to the R1 trigonometry in Chapter 6. In contrast to equations (5.5)–(5.7), these definitions (equations (5.8)–(5.10)) are not indexed and are real functions. The principal functions are used as the basis of the identities and analyses of the following sections. To demonstrate backward compatibility to the classical hyperbolic functions, we substitute q = 1, v = 0, into the defining series (equations (5.5–5.7)) giving R1 Cosh1,0 (a, t) = cosh(a t) ,

t > 0,

(5.11)

R1 Sinh1,0 (a, t) = sinh(a t) ,

t > 0,

(5.12)

R1 Tanh1,0 (a, t) = tanh(at) ,

t > 0.

(5.13)

and Figures 5.1–5.3 show the principal (k = 0) R1 -hyperbolic functions for various values of q. We observe the traditional cosh (t) and sinh (t) when q = 1, v = 0, and k = 0. We also notice that

5.1 The Fractional R1 -Hyperbolic Functions

Figure 5.1 R1 Coshq,0 (1, t) versus t-Time, for q = 0.2–2.0 in steps of 0.2, a = 1, v = 0, and k = 0. Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

10

8 R1Coshq,0(a,t)

q = 0.2

q = 2.0

6

4

2

0

1

0

2 t-Time

3

4

Figure 5.2 R1 Sinhq,0 (1, t) versus t-Time, for q = 0.2–2.0 in steps of 0.2, a = 1, v = 0, and k = 0. Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

10

8 R1sinhq,0(a,t)

q = 0.2

q = 2.0

6

4

2

0

0

1

2 t-Time

3

4

R1 Sinh1∕2,0 (1, 0, t) = et . This is readily verified by appropriate substitution into the defining series (equation (5.6)). The effect of the parameter a on the R1 Coshq,v (a, t), R1 Sinhq,v (a, t), and R1 Tanhq,v (a, t), for values of q = 0.25, 0.50, and 0.75, is shown in Figures 5.4–5.6, respectively. It is seen by comparison with the a = 1 cases shown in Figures 5.1–5.3 that the parameter does not act as a time scaling parameter. The effect of the parameter v on the R1 -fractional hyperbolic functions is presented in Figure 5.7. This is later shown to be the same as a fractional differentiation of order v. In later chapters, it will be seen that for q > 1, special behavior is observed when v = q-1. Figures 5.8–5.10 examine the effect on the fractional hyperbolic functions under these conditions. The most notable changes here occur with overlapping of the R1 Coshq,v (a, t)-functions with t < 2. The behavior of the multivalued functions is shown in Figure 5.11 for R1 Sinh6∕7,0 (1, k, t) with various k values. The order q = 6/7 was chosen for the strongly symmetric behavior observed in the imaginary part of the R1 Sinh6∕7,0 (1, k, t)-function. It is important to observe here that there

65

5 The Fractional Hyperboletry

Figure 5.3 R1 Tanhq,0 (1, t) versus t-Time, for q = 0.2–3.0 in steps of 0.2, a = 1, v = 0, and k = 0. Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

1.4 1.2

R1Tanhq,0(a,t)

1 q = 0.2 0.8 0.6 q = 3.0 0.4 0.2 0

0

2

4

6

8

10

t-Time 30

30

25

25

a = 1.2

a = 2.0

20

R1Coshq,0(a,t)

R1Coshq,0(a,t)

a = 1.0

15 10 a = 0.8

5 0

0

1

2

a = 1.0 15 10

a = 0.8

5

4

3

20

0

0

1

t-Time (a)

2

3

4

t-Time (b)

30 25 a = 2.0 R1Coshq,0(a,t)

66

20 a = 1.0 15 10

a = 0.8

5 0

0

1

2 t-Time (c)

3

4

Figure 5.4 Effect of the a parameter on R1 Coshq,0 (a, t) 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.

5.1 The Fractional R1 -Hyperbolic Functions

30

30

25

a = 1.2

20

R1Sinhq,0(a,t)

R1Sinhq,0(a,t)

25

a = 1.0 15 10

a = 0.8

5 0

0

1

2 t-Time (a)

a = 2.0

20 a = 1.0 15 10 a = 0.8

5

3

4

0 0

1

2 t-Time (b)

3

4

30

R1Sinhq,0(a,t)

25

a = 2.0

a = 1.0

20 15 10

a = 0.8

5 0 0

1

2 t-Time (c)

3

4

Figure 5.5 Effect of the a parameter on R1 Sinhq,0 (a, t) 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.

is only one real R1 Sinh6∕7,0 (1, k, t)-function, that is, k = 0. This behavior does not exist for all q. For example, with q = 7/8, not shown, there are two real functions, they are k = 0 and k = 4. The remainder of this section develops some fundamental relationships between the fractional hyperbolic functions and the R-functions. Continuing from Ref. [76] with the permission of the ASME: It is directly observed that the R1 Cosh-function is an even function (with respect to a), that is, for t > 0 R1 Coshq,v (−a, k, t) = R1 Coshq,v (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1 .

(5.14)

Also, the R1 Sinh-function is odd, that is, R1 Sinhq,v (−a, k, t) = −R1 Sinhq,v (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1 .

(5.15)

67

5 The Fractional Hyperboletry 1.2

1.2

a = 2.0

1 a = 2.0 R1Tanh0.50,0(a,t)

R1Tanh0.25,0(a,t)

1 0.8 0.6

a = 0.2

0.4 0.2 0 0

0.8 0.6

a = 0.2 0.4 0.2

1

2 t-Time (a)

3

0

4

1.2

0

1

2 t-Time (b)

3

4

1.2

1 a = 2.0

1 R1Tanh2.2,0(a,t)

a = 2.0 R1Tanh0.75,0(a,t)

68

0.8 0.6

a = 0.2

0.4

0.8 0.6 0.4

a = 0.2 0.2 0 0

0.2

1

2 t-Time (c)

3

4

0

0

1

2 t-Time (d)

3

4

Figure 5.6 Effect of the a parameter on R1 Tanhq,0 (a, t) 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.

From the definition of the fractional hyperbolic functions based on equation (5.2) with (5.5) and (5.6), it is clear that Rq,v (a, t) = R1 Coshq,v (a, t) + R1 Sinhq,v (a, t) ,

t > 0.

(5.16)

Furthermore, by a similar expansion of Rq,v (−a, t) based on (5.2), we have Rq,v (−a, t) = R1 Coshq,v (a, t) − R1 Sinhq,v (a, t),

t > 0.

(5.17)

Adding and subtracting equations (5.16) and (5.17) gives R1 Coshq,v (a, t) =

] 1[ Rq,v (a, t) + Rq,v (−a, t) , 2

t > 0,

(5.18)

R1 Sinhq,v (a, t) =

] 1[ Rq,v (a, t) − Rq,v (−a, t) , 2

t > 0.

(5.19)

and

30

30

25

25 𝜈= q = 0.2

20

R1Sinhq,𝜈(1,t)

R1Coshq,𝜈(1,t)

5.1 The Fractional R1 -Hyperbolic Functions

15 10 5 0

1

2 t-Time (a)

15 10 5

𝜈= q = 2.0 0

𝜈= q = 0.2

20

3

0

4

𝜈= q = 2.0 0

1

2 t-Time (b)

3

4

30

R1Tanhq,𝜈(1,t)

25 20 𝜈= q = 2.0

15 10 5

0 𝜈= q = 0.2 0 1

2 t-Time (c)

3

4

Figure 5.7 Effect of v on (a) R1 Coshq,0 (a, t), (b) R1 Sinhq,0 (a, t), (c) R1 Tanhq,0 (a, t) 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 R1 Coshq,v (a, t) versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

30

R1Coshq,q–1(a,t)

25 20 q = 1.0

15

q = 2.0

10 5 0 0

1

2

3

t-Time

4

5

69

5 The Fractional Hyperboletry

Figure 5.9 Effect of v on R1 Sinhq,v (a, t) versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

30

R1Sinhq,q–1(a,t)

25 20 q = 1.0

15

q = 2.0

10 5 0 0

1

2

3

4

5

t-Time

Figure 5.10 Effect of v on R1 Tanhq,v (a, t) versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

1.2 1

R1Tanhq,q–1(a,t)

70

0.8

q = 1.0

q = 2.0

0.6 0.4 0.2 0

0

1

2

3

4

5

t-Time

These equations ((5.18) and (5.19)) exactly parallel equations (1.20) and (1.21) and may equally well be taken as the definitions for the principal fractional R1 -hyperbolic functions. Following from these relations and definition (5.10), we have R1 Tanhq,v (a, t) =

Rq,v (a, t) − Rq,v (−a, t) Rq,v (a, t) + Rq,v (−a, t)

,

t > 0.

(5.20)

The right-hand sides of equations (5.5) and (5.6) are recognized as R-functions, that is, R1 Coshq,v (a, t) = R2q,v+q (a2 , t) , R1 Sinhq,v (a, t) = aR2q,v (a , t), 2

t > 0,

(5.21)

t > 0.

(5.22)

Combining the results of equations (5.18) and (5.21), we have the identity R1 Coshq,v (a, t) = R2q,v+q (a2 , t) =

] 1[ Rq,v (a, t) + Rq,v (−a, t) , 2

t > 0.

(5.23)

Imag(R1Sinh6/7,0(1,k,t))

Real(R1Sinh6/7,0(1,k,t))

5.1 The Fractional R1 -Hyperbolic Functions

k=0

5

Figure 5.11 Re{R1 Sinh6∕7,0 (1, k, t)} and Im{R1 Sinh6∕7,0 (1, k, t)} versus t-Time, for k = 0 to 6.

k = 1,6 k = 1,6 k = 2,5

0 k = 3,4 4 t-Time

–5 0

2

6

k=3

5

k=6 k=0

k=2

0

8

k=5

–5

k=4 0

2

k=1 4 t-Time

6

8

Similarly, combining the results of equations (5.19) and (5.22), we have R1 Sinhq,v (a, t) = a R2q,v (a2 , t) =

] 1[ R (a, t) − Rq,v (−a, t) , 2 q,v

t > 0,

(5.24)

which also introduces two new R-function relationships. Taking the ratio of equations (5.23) and (5.24) and combining with the results of equation (5.7) yields ( ) [ ] aR2q,v a2 , t Rq,v (a, t) − Rq,v (−a, t) R1 Tanhq,v (a, t) = =[ (5.25) ] , t > 0. R2q,v+q (a2 , t) Rq,v (a, t) + Rq,v (−a, t) Now, adding equations (5.23) and (5.24) yields Rq,v (a, t) = R2q,v+q (a2 , t) + a R2q,v (a2 , t) ,

t > 0,

(5.26)

while subtraction yields Rq,v (−a, t) = R2q,v+q (a2 , t) − a R2q,v (a2 , t) ,

t > 0.

(5.27)

Subtracting the square of equation (5.19) from the square of equation (5.18) yields the following: R1 Cosh2q,v (a, t) − R1 Sinh2q,v (a, t) = Rq,v (a, t)Rq,v (−a, t) ,

t > 0,

(5.28)

or using the center terms of equations (5.23) and (5.24) R1 Cosh2q,v (a, t) − R1 Sinh2q,v (a, t) = Rq,v (a, t)Rq,v (−a, t) = R22q,v+q (a2 , t) − a2 R22q,v (a2 , t) ,

t > 0.

(5.29)

Taking q = 1, v = 0, equation (5.29) is seen that is a generalization of the well-known equation cosh2 (t) − sinh2 (t) = 1.

(5.30)

Furthermore, adding the square of equation (5.18) to the square of equation (5.19) yields R1 Cosh2q,v (a, k, t) + R1 Sinh2q,v (a, t) =

] 1[ 2 Rq,v (a, t) + R2q,v (−a, t) , 2

t > 0.

(5.31)

71

72

5 The Fractional Hyperboletry

Again, using the center terms of equations (5.23) and (5.24) with (5.31) yields R1 Cosh2q,v (a, t) + R1 Sinh2q,v (a, t)

] 1[ 2 Rq,v (a, t) + R2q,v (−a, t) , t > 0. 2 Taking q = 1, v = 0 shows that this is a generalization of the duplication formula: = R22q,v+q (a2 , t) + a2 R22q,v (a2 , t) =

cosh2 (t) + sinh2 (t) = cosh(2t).

(5.32)

(5.33)

5.2 R1 -Hyperbolic Function Relationship An interrelationship between the R1 -hyperbolic functions is readily determined. Using the definition (5.6), consider a R1 Coshq,v−q (a, k, t) =a

∞ ∑ a2n (t)(n+1)2q−1−v+q−q

n=0 ∞

Γ((n + 1)2q − v)

] [ cis ((n + 1)2q − 1 − v) (2𝜋k)

∑ a2n+1 t (n+1)2q−1−v [ ] = cis ((n + 1)2q − 1 − v)(2𝜋k) , Γ((n + 1)2q − v) n=0

t > 0.

(5.34)

Thus, recognizing the form of equation (5.6), we have a R1 Coshq,v−q (a, k, t) = R1 Sinhq,v (a, k, t) , t > 0, k = 0, 1, … , Dq Dv − 1.

(5.35)

Because of the nature of v this relation may also be considered to be a fractional calculus operation when k = 0. Continuing from Ref. [76] with the permission of ASME:

5.3 Fractional Calculus Operations on the R1 -Hyperbolic Functions The 𝛼-order differintegral of R1 Coshq,v (a, t), is determined as follows: 𝛼 0 dt R1 Coshq,v (a, t)

=

∞ ∑ a2n 0 dt𝛼 (t)(n+1)2q−1−v−q n=0

Γ((n + 1)2q − v − q)

,

t > 0.

(5.36)

Oldham and Spanier [104] p. 67 provide the following differintegration relation: q

p 0 Dx x =

Γ(p + 1)xp−q , Γ(p − q + 1)

p > −1,

(5.37)

which is valid for all q. Then, fractionally differintegrating term-by-term (see Section 3.16) yields =

∞ ∑ n=0

Thus,

a2n (t)(n+1)2q−1−v−q−𝛼 , q > v, Γ((n + 1)2q − v − q − 𝛼)

𝛼 0 dt R1 Coshq,v (a, t)

= R1 Coshq,v+𝛼 (a, t),

t > 0.

q > v,

t > 0.

(5.38)

(5.39)

Applying equation (5.35), this can be written as 𝛼 0 dt R1 Coshq,v (a, t)

=

1 R Sinhq,v+𝛼+q (a, t), a 1

q > v,

t > 0.

(5.40)

5.5 Complexity-Based Hyperbolic Functions

If 𝛼 = −q, we have 1 R Sinhq,v (a, t) , q > v , a 1

t > 0.

(5.41)

= R1 Sinhq,v+𝛼 (a, t), 2q > v,

t > 0,

(5.42)

−q 0 dt R1 Coshq,v (a, t)

=

By similar logic, 𝛼 0 dt R1 Sinhq,v (a, t)

and

𝛼 0 dt R1 Sinhq,v (a, t)

= a R1 Coshq,v+𝛼−q (a, t) , 2q > v,

t > 0.

(5.43)

If 𝛼 = q, we have q 0 dt R1 Sinhq,v (a, t)

= a R1 Coshq,v (a, t), q > v,

t > 0.

(5.44)

5.4 Laplace Transforms of the R1 -Hyperbolic Functions The Laplace transforms for the principal R1 -hyperbolic functions are easily determined from the Laplace transforms of the related R-functions (equation (3.22)). Then, the transform of equation (5.23) is } { } 1 { } { (5.45) L R1 Coshq,v (a, t) = L R2q,v+q (a2 , t) = L Rq,v (a, t) + Rq,v (−a, t) . 2 Then, using equation (3.22), we have ( v ) } { s sv+q 1 sv , q ≥ v. (5.46) = + L R1 Coshq,v (a, t) = 2q s − a2 2 sq − a sq + a In a similar manner, the transform of equation (5.24) is { } { } 1 { } L R1 Sinhq,v (a, t) = a L R2q,v (a2 , t) = L Rq,v (a, t) − Rq,v (−a, t) . (5.47) 2 Again, using equation (3.22) ( v ) } { s asv 1 sv , q ≥ v. (5.48) = − q L R1 Sinhq,v (a, t) = 2q 2 q s −a 2 s −a s +a

5.5

Complexity-Based Hyperbolic Functions

In Section 5.1, fractional hyperbolic functions that parallel the familiar classical hyperbolic functions were derived based on the parity-based series (equation (5.2)). In this section, we shall consider complexity-based functions starting from equation (5.1). Using equation (5.3), (5.1) is written as Rq,v (a, k, t) =

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

cis{((n + 1)q − 1 − v)(2𝜋k)} , t > 0, k = 0, 1, … , Dq Dv − 1.

(5.49)

Now, the real and imaginary parts define the new functions: R1 Cohq,v (a, k, t) ≡

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

cos{((n + 1)q − 1 − v)(2𝜋k)} , t > 0, k = 0, 1, … , Dq Dv − 1.

(5.50)

73

74

5 The Fractional Hyperboletry

and R1 Sihq,v (a, k, t) ≡

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

sin {((n + 1)q − 1 − v)(2𝜋k)} , t > 0.

(5.51)

The related principal functions, k = 0, are R1 Cohq,v (a, 0, t) = R1 Cohq,v (a, t) =

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

= Rq,v (a, t) , t > 0,

(5.52)

and R1 Sihq,v (a, 0, t) = R1 Sihq,v (a, t) =

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

0 = 0,

t > 0. (5.53)

For the case with q = 1 and v = 0, we have R1 Coh1,0 (a, k, t) = eat , and R1 Sih1,0 (a, k, t) = 0 ,

t > 0.

(5.54)

Because the principal functions return to the R-function and zero, further properties are not pursued.

5.6 Fractional Hyperbolic Differential Equations With the availability of the Laplace transforms, various fractional differential equations may be identified that are solvable using R1 -hyperbolic functions. Based on the Laplace transforms, we may easily determine associated fractional differential equations for which they are solutions. For clarity of presentation, we are only concerned with uninitialized fractional differential equations (to include initialization effects, see, e.g., Ref. [71]). Continuing from Ref. [76] with the permission of ASME: Consider the uninitialized fractional differential equation 2q 0 dt x(t)

v+q

− a2 x(t) = 0 dt f (t) ,

q ≥ v,

(5.55)

taking the Laplace transform gives sv+q F(s). − a2 Based on equation (5.46), this has a solution of the form X(s) =

(5.56)

s2q

x(t) = R1 Coshq,v (a, t) ,

t > 0,

q ≥ v,

(5.57)

when f (t) = 𝛿(t), that is, a unit impulse function. For an arbitrary f (t), the solution is given by convolution t

R1 Coshq,v (a, t) ∗ f (t) =

∫

R1 Coshq,v (a, 𝜏)f (t − 𝜏)d𝜏

0 t

=

∫ 0

R1 Coshq,v (a, t − 𝜏)f (𝜏)d𝜏,

t > 0,

q ≥ v.

(5.58)

5.6 Fractional Hyperbolic Differential Equations

Assuming composition, equation (5.55) may also be written as a fractional integrodifferential equation, namely q−v 0 dt x(t)

−v−q

− a2 0 dt

(5.59)

x(t) = g(t).

Taking v = −q in equation (5.46) infers the uninitialized fractional differential equation 2q 0 dt x(t)

− a2 x(t) = f (t),

(5.60)

when f (t) = 𝛿(t), this has a solution of the form x(t) = R1 Coshq,−q (a, t) = R2q,0 (a2 , t) =

1 R Sinhq,0 (a, t) , a 1

t > 0.

(5.61)

In a similar manner, equation (5.48) indicates that the uninitialized fractional differential equation 2q 2 v (5.62) 0 dt x(t) − a x(t) = a0 dt f (t), has a solution of the form x(t) = R1 Sinhq,v (a, t),

t≥0

(5.63)

when f (t) = 𝛿(t), that is, an impulse function. Again for an arbitrary f (t), the solution is given by convolution t

R1 Sinhq,v (a, t) ∗ f (t) =

∫

R1 Sinhq,v (a, 𝜏)f (t − 𝜏)d𝜏

0 t

=

∫

R1 Sinhq,v (a, t − 𝜏)f (𝜏)d𝜏,

t > 0, q ≥ v.

(5.64)

0

Equation (5.62) may also be written as a fractional integro-differential equation, namely 2q−v x(t) 0 dt

− a2 0 dt−v x(t) = af (t).

(5.65)

Linear combinations of fractional hyperbolic functions may also be used to infer fractional differential forms. For example, from the linear combination A R1 Coshq,v (a, t) + B R1 Sinhu,w (b, t),

(5.66)

or using the relationships (5.22) and (5.23) A R2q,v+q (a2 , t) + B b R2u,w (b2 , t),

(5.67)

we may infer the transformed equation

( ) ( ) Asv+q s2u − b2 + B b sw s2q − a2 X(s) A sv+q Bbsw + 2u = = . 2q 2 2 2(q+u) 2 2u 2 2q 2 2 s −a s −b F(s) s −a s −b s +a b

(5.68)

From this, it is observed that the noncommensurate order fractional differential equation 2(q+u) x(t) 0 dt

where

2q

− a2 0 dt2u x(t) − b2 0 dt x(t) + a2 b2 x(t) = g(t),

(5.69)

{ ( ( ) ( ))} , g(t) = L−1 F(s) Asv+q s2u − b2 + B b sw s2q − a2

(5.70)

will have a solution of the form given by equation (5.66) for f (t) = 𝛿(t).

75

76

5 The Fractional Hyperboletry

Figure 5.12 Expanding foam reactor.

𝜈in R 𝜈gen

h

𝜈 out

5.7 Example Consider the generation of an expanding viscous foam in a cylindrical tank. The foam volume in the tank (Figure 5.12) is described by t

v(t) =

∫

(v̇ in (𝜏) + v̇ gen (𝜏) − v̇ out (𝜏))d𝜏,

(5.71)

0

where v̇ in (t) is the volumetric rate of foam entering the tank and v̇ out (t) is the volumetric rate of foam leaving the tank through a long baffled pipeline. The volumetric foam generation rate, v̇ gen (t), which, for short timescales, is proportional to the volume of foam in the tank and is given by v̇ gen (t) = k1 v(t).

(5.72)

The foam volume in the cylindrical tank then is directly proportional to the height of foam in the tank, that is, v(t) = 𝜋R2 h(t). The tank is drained by a baffled diffusive pipeline that we consider to be of semi-infinite length in terms of the time–distance scaling of the problem. The foaming process is photosensitively ended as it enters the pipeline. The volumetric flow into the pipe is proportional to the semiderivative of the pressure (height of foam) at the pipeline inlet. Thus, ( ) 1∕2 1∕2 v̇ out (t) = k 0 Dt h(t) = k 0 dt h(t) + 𝜓(hi , t) , t > 0 , (5.73) or

( ) k k 1∕2 1∕2 D v(t) = d v(t) + 𝜓(v , t) , t > 0, 0 i t 𝜋R2 0 t 𝜋R2 where 𝜓(vi , t) is the initialization function, Lorenzo and Hartley [78] representing the flow history of the long pipeline. The reaction process is started at t = 0 by a unit impulse of the foam volume rate, v̇ in (t) = 𝛿(t). Taking the Laplace transform of the aforementioned equations and solving for L{v(t)} gives 1 − k𝜓(v, s) . (5.74) L {v(t)} = 2 𝜋R s + ks1∕2 − 𝜋R2 k1 v̇ out (t) =

To illustrate the application of the fractional hyperbolic functions, we take the radius of the tank to be small relative to the constants k and k 1 thus ignoring the first term in the denominator.

5.8 Discussions

Then, with 𝛼 2 ≡ 𝜋R2 k1 ∕k, the volume of foam in the tank can be approximated as ( ) 𝜓(v, s) 1 1 − 1∕2 L {v(t)} = . k s1∕2 − 𝛼 2 s − 𝛼2 Note that from equation (5.48) } { 𝛽 . L R1 Sinh1∕4,0 (𝛽, t) = 1∕2 s − 𝛽2

(5.75)

(5.76)

Inverse transforming equation (5.75) and applying the convolution theorem give the solution for the volume of foam in the tank as a function of time to be t

1 1 R Sinh1∕4,0 (𝛼, 𝜏) 𝜓(vi , t − 𝜏) d𝜏. R Sinh1∕4,0 (𝛼, t) − v(t) = k𝛼 1 𝛼∫ 1

(5.77)

0

5.8 Discussions Based on the R-function with real parameters a and t, namely Rq,v (a, t), we have defined the fractional hyperbolic functions. These functions are backward compatible with the classical hyperbolic functions containing them as subfunctions when q = 1 and v = 0. Along with the Laplace transforms for these functions, various properties and identities have been developed. In particular, the fractional differintegrals of the functions have been derived. The fractional hyperbolic functions are applicable to the solutions of certain fractional integro-differential equations. Because of the growing nature of these functions with time (see Figures 5.1–5.10), they will generally be associated with unstable physical phenomena. This is demonstrated in the reactor example provided. In Chapter 6, a similar development based on the R-function is used to develop the R1 -trigonometry; the first of several fractional trigonometries. We will find in Chapter 9 that the R1 -hyperbolic functions and the fractional R1 -trigonometric functions are both contained as special cases of the fractional meta-trigonometry. In other words, the fractional meta-trigonometry will be seen to unify the hyperboletry and the trigonometries.

77

79

6 The R1 -Fractional Trigonometry 6.1 R1 -Trigonometric Functions The R1 -fractional trigonometry [73–75] is defined by taking the parameter a in the argument of the R-function to be imaginary. This is in contrast to Chapter 5 in which both the a and t parameters were taken to be real. Then, the R1 -trigonometry is based on Rq,v (ai, t), where R-function is expanded into even and odd series, that is, series with terms that are even and odd powers of a. This yields Rq,v (ai, t) =

∞ ∑ (ai)n (t)(n+1)q−1−v n=0

Rq,v (ai, t) =

,

∞ ∑ (ai)2n (t)(2n+1)q−1−v n=0

Rq,v (ai, t) =

Γ((n + 1)q − v)

Γ((2n + 1)q − v)

t > 0,

+

∞ ∑ (ai)2n+1 (t)(2n+2)q−1−v

∞ ∑ (−1)n a2n (t)(2n+1)q−1−v n=0

Γ((2n + 1)q − v)

(6.1)

n=0

+i

Γ((2n + 2)q − v)

,

t > 0,

∞ ∑ (−1)n a2n+1 (t)(2n+2)q−1−v n=0

Γ((2n + 2)q − v)

(6.2)

,

t > 0.

(6.3)

We define Eva {Rq,v (ai, t)} to be the even part of Rq,v (ai, t), that is, the terms in equation (6.1) with even powers of a, and Oa {Rq,v (ai, t)} to be the odd part of Rq,v (ai, t), that is, the terms in equation (6.1) with odd powers of a. Then, we have Eva {Rq,v (ai, t)} = Re{Rq,v (ai, t)} and Oa {Rq,v (ai, t)} = Im{Rq,v (ai, t)} from equation (6.3). Therefore, R1 Cos-function is defined as the real part of equation (6.1) and the R1 Sin-function as the imaginary part. Because t is raised to a fractional power in these series, the series are multivalued or indexed. To elucidate the indexed behavior of these functions, we consider the roots of t. Then, from equation (3.122), { ( ) ( )} √ M D M M 2𝜋k + i sin 2𝜋k , t D = t M cos D D where for R1 Cos the exponent of t is (n + 1)2q − 1 − v − q =

(n + 1)2Nq Dv − Dq Dv − Nv Dq − Nq Dv Dq Dv

=

M , D

where q = Nq ∕Dq and v = Nv ∕Dv are assumed rational and irreducible, k = 0, 1, … , Dq Dv − 1, and M/D is rational and in minimal form. From Lorenzo and Hartley [74], with permission of

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

80

6 The R1 -Fractional Trigonometry

Springer: Then, the multivalued, or indexed, definition for the R1 Cosq,v is given by R1 Cosq,v (a, k, t) ≡

∞ ∑ ]} (−1)n a2n (t)(n+1)2q−1−v−q { [ cis ((n + 1)2q − 1 − v − q)(2𝜋k) Γ((n + 1)2q − v − q) n=0

(6.4)

with cis(∘) = cos(∘) + i sin(∘), t > 0, k = 0, 1, … , Dq Dv − 1, with common factors in M/D removed. The R1 Sin and R1 Tan are defined in a similar manner R1 Sinq,v (a, k, t) ∞ ∑ ]} (−1)n a2n+1 (t)(n+1)2q−1−v { [ cis ((n + 1)2q − 1 − v)(2𝜋k) , ≡ Γ((n + 1)2q − v) n=0

(6.5)

with t > 0, k = 0, 1, … , Dq Dv − 1, and R1 Tanq,v (a, k, t) ≡

R1 Sinq,v (a, k, t) R1 Cosq,v (a, k, t)

,

t > 0.

(6.6)

The principal R1 -generalized trigonometric functions are real functions of the real variable t. They are obtained by taking k = 0, in equations (6.4–6.6). Thus, R1 Cosq,v (a, 0, t) = R1 Cosq,v (a, t) ≡

∞ ∑ (−1)n a2n (t)(n+1)2q−1−v−q n=0

R1 Sinq,v (a, 0, t) = R1 Sinq,v (a, t) ≡

Γ((n + 1)2q − v − q)

∞ ∑ (−1)n a2n+1 (t)(n+1)2q−1−v n=0

and R1 Tanq,v (a, 0, t) = R1 Tanq,v (a, t) ≡

Γ((n + 1)2q − v) R1 Sinq,v (a, t) R1 Cosq,v (a, t)

,

,

t > 0,

(6.7)

,

t > 0,

(6.8)

t > 0,

(6.9)

where the simplified notation introduced earlier is used for the principal, k = 0, functions. These functions, equations (6.4–6.9), are generalizations of the circular functions (also known as harmonic functions) of the classical integer-order trigonometry. The indexed behavior seen in equations (6.4–6.6) has no parallel in the classical trigonometry. We note, when q = 1 and v = 0, the R1 -trigonometric functions revert back to the circular functions for t > 0. The principal R1 -trigonometric functions are shown in Figures 6.1a, b and 6.2 for various values of q. As the value of q decreases from 1 to 0.1, the oscillatory nature of the R1 Cos- and R1 Sin-functions is also seen to decrease. For q < 1, it can also be seen that the curves tend to zero as t increases. The functions grow at an increasing rate, as q increases for values of q > 1 (not shown). For smaller values of q in the 𝜋 to 3𝜋∕2 range, the lack of zero crossings of the R1 Cos-function strongly affects the behavior of the R1 Tan-function. Figures 6.3–6.8 show the effects of the a parameter on the R1 Cos-, R1 Sin-, and the R1 Tan-functions, respectively. Increasing the a parameter increases the response rate of the function. In other words, increasing a decreases the apparent period and increases the overshoot. The reader should also consider the effect of the order variable q in these figures. For the R1 Cos and the R1 Tan, it also increases the response rate. For the R1 Sin, it changes the character of the response.

6.1 R1 -Trigonometric Functions

1.5

1.5 q = 1.0

1

q = 1.2

0.5 0

R1Sinq,0(1,t)

R1Cosq,0(1,t)

1

q = 0.2

–0.5

0.5 q = 0.2

0 –0.5

–1 –1.5

q = 1.2

–1

0

1

2

3 4 t-Time (a)

5

6

7

–1.5

q = 1.0 0

1

2

3 4 t-Time (b)

5

6

7

Figure 6.1 (a) R1 Cosq,0 (1, t) versus t-Time, for q = 0.2–1.2 in steps of 0.1, v = 0, a = 1. (b) R1 Sinq,0 (1, t) versus t-Time, for q = 0.2–1.2 in steps of 0.1, v = 0, a = 1. (a, b) Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer. Figure 6.2 R1 Tanq,0 (1, t) versus t-Time, for q = 0.2–1.2 in steps of 0.2, v = 0, a = 1.

20 15

0.2

1 q = 1.2

q = 1.0

q = 1.2

R1Tanq,0(1,t)

10 5

q = 0.8

q = 0.2

0 q = 0.6

–5 –10 –15 –20

0

1

2

4 3 t-Time

5

6

7

Figures 6.9–6.14 show the effects of secondary order parameter v on the R1 Cos-, R1 Sin-, and R1 Tan-functions, respectively. In general, increasing v tends to increase the rate of response of the functions and reduce the period. From the Laplace transforms of the functions (Section 6.5), it is clear that v > 0 fractionally differentiates the v = 0 function, while v < 0 integrates it. The effect of v reduces the period of the R1 Tan-function; this is shown clearly in Figure 6.14. Figure 6.15 shows the phase plane for variations in the order q, with a = 1.0, v = 0.0, and k = 0. Note that for q < 1 the functions are attracted to the origin from infinity, while for q > 1, the functions start at the origin and spiral out to infinity. 6.1.1 R1 -Trigonometric Properties

From the definitions (6.4) and (6.5), we have that R1 Cosq,v (−a, k, t) = R1 Cosq,v (a, k, t) ,

t > 0, k = 0, 1, … , Dq Dv − 1 ,

(6.10)

81

6 The R1 -Fractional Trigonometry

Figure 6.3 Effect of a on R1 Cosq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

1

R1Cos0.25,0(a,t)

0.8 0.6 0.4 a = 0.25

0.2 0

a = 1.0

–0.2 0

1

2

3 4 t-Time

5

6

7

Figure 6.4 Effect of a on R1 Cosq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

1 0.8

a = 0.25

R1Cos0.75,0(a,t)

0.6 0.4 0.2 0 –0.2 –0.4

a = 1.0 0

1

2

3 4 t-Time

5

6

7

Figure 6.5 Effect of a on R1 Sinq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

1

0.8 R1Sin0.25,0(a,t)

82

0.6

0.4

a = 0.75

0.2 a = 0.25 0 0

1

a = 0.50 a = 1.0 2

3 4 t-Time

5

6

7

6.1 R1 -Trigonometric Functions

Figure 6.6 Effect of a on R1 Sinq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

0.8 a = 1.0

R1Sin0.75,0(a,t)

0.6

0.4 a = 0.25

0.2

0

–0.2

0

1

2

3 4 t-Time

5

6

7

Figure 6.7 Effect of a on R1 Tanq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

30 a = 0.75

1.0 20

R1Tan0.25,0(a,t)

a = 0.50 10 a = 0.25

0 –10 –20 –30 0

2

8 6 t-Time

4

10

12

Figure 6.8 Effect of a on R1 Tanq,0 (1, t) for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

30 1.0

0.75

1.0

0.75

1.0

a = 0.75

R1Tan0.75,0(a,t)

20 10

0.50

0.5

0 –10 –20 –30

0

2

4

6 8 t-Time

10

12

83

6 The R1 -Fractional Trigonometry

Figure 6.9 Effect of v on R1 Cosq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

0.8 0.6

R1Cos0.25,𝜈(1,t)

0.4 0.2

𝜈 = –0.20

0 𝜈 = 0.20

–0.2 –0.4 –0.6 –0.8 –1

0

1

2

3

4 t-Time

5

6

7

Figure 6.10 Effect of v on R1 Cosq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

R1Cos0.75,𝜈(1,t)

1

0.5

𝜈 = –0.20 0 𝜈 = 0.20

–0.5

0

1

2

3 4 t-Time

5

6

7

Figure 6.11 Effect of v on R1 Sinq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

1

0.8

R1Sin0.25,𝜈(1,t)

84

0.6

0.4

0.2

𝜈 = –0.20

0

𝜈 = 0.20 0

1

2

3 4 t-Time

5

6

7

6.1 R1 -Trigonometric Functions

Figure 6.12 Effect of v on R1 Sinq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

0.8

R1Sin0.75,𝜈(1,t)

0.6

0.4

𝜈 = –0.20 𝜈 = 0.20

0.2

0 –0.2 0

1

2

3 4 t-Time

5

6

7

Figure 6.13 Effect of v on R1 Tanq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

30 𝜈 = 0.0

R1Tan0.25,𝜈(1,t)

20

𝜈 = –0.20

𝜈 = –0.10

10 0 𝜈 = 0.10 –10

𝜈 = 0.0

–20 –30

𝜈 = –0.10

0

1

2

3 4 t-Time

5

6

7

Figure 6.14 Effect of v on R1 Tanq,0 (1, t) for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

30

R1Tan0.75,𝜈(1,t)

20 𝜈 = 0.2

𝜈 = –0.20 𝜈 = 0.20

𝜈 = –0.20

10 0 –10 –20 –30

0

1

2

3 4 t-Time

5

6

7

85

6 The R1 -Fractional Trigonometry

2

q = 1.50

1.75

Figure 6.15 Phase plane R1 Sinq,v (a, t) versus R1 Cosq,v (a, t) for q = 1.1 and q = 0.25–1.75 in steps of 0.25, a = 1.0, v = 0.0.

0.25

1.5 1 R1Sinq,0(1,t)

86

q = 1.25

0.50

0.5

0.75

0 –0.5 q = 1.0 –1 –1.5 –2 –2

q = 1.1 –1

0 R1Cosq,0(1,t)

1

2

and R1 Sinq,v (−a, k, t) = −R1 Sinq,v (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1.

(6.11)

These results substituted into equation (6.6) yield −R1 Sinq,v (a, k, t) = −R1 Tanq,v (a, k, t), R1 Tanq,v (−a, k, t) = R1 Cosq,v (a, k, t) t > 0, k = 0, 1, … , Dq Dv − 1.

(6.12)

Using the definitions (6.4) and (6.5) with equation (6.3) gives Rq,v (ai, t) = R1 Cosq,v (a, t) + i R1 Sinq,v (a, t),

t > 0,

(6.13)

which is a fractional Euler equation for the R1 -trigonometry. Expanding Rq,v (−ai, t) in the same manner as done for equation (6.3) yields Rq,v (−ai, t) = R1 Cosq,v (a, t) − i R1 Sinq,v (a, t),

t > 0,

(6.14)

a complementary fractional Euler equation. The addition of equations (6.13) and (6.14) gives 1 (6.15) R1 Cosq,v (a, t) = (Rq,v (ai, t) + Rq,v (−ai, t)), t > 0. 2 If we subtract equation (6.15) from (6.14), we have 1 R1 Sinq,v (a, t) = (Rq,v (a i, t) − Rq,v (−a i, t)), t > 0. (6.16) 2i Both equations (6.15) and (6.16) are the fractional generalizations of the classical equations (1.18) and (1.19), respectively. Based on the definitions of the R1 -trigonometric functions, we have ∞ ∑ (−1)n a2n (t)(n+1)2q−1−v−q R1 Cosq,v (a, t) = , t > 0, Γ((n + 1)2q − v − q) n=0 R1 Cosq,v (a, t) =

∞ ∑ ((a i)2 )n (t)(n+1)2q−1−v−q n=0

Γ((n + 1)2q − v − q)

= R2q,v+q (−a2 , t),

t > 0,

(6.17)

6.1 R1 -Trigonometric Functions

another R-function expression. Combining this with equation (6.15) gives R1 Cosq,v (a, t) = R2q,v+q (−a2 , t) =

1 (R (a i, t) + Rq,v (−a i, t)), 2 q,v

t > 0.

(6.18)

Similarly, we have R1 Sinq,v (a, t) =

∞ ∑ (−1)n a2n+1 (t)(n+1)2q−1−v

Γ((n + 1)2q − v) ( 2 )n (n+1)2q−1−v ∑ −a (t)

n=0 ∞

R1 Sinq,v (a, t) = a

n=0

Γ((n + 1)2q − v)

,

t > 0,

( ) = aR2q,v −a2 , t ,

(6.19) t > 0,

(6.20)

taken with equation (6.16) yields ( ) 1 R1 Sinq,v (a, t) = aR2q,v −a2 , t = (Rq,v (a i, t) − Rq,v (−a i, t)), 2i

t > 0.

(6.21)

A similar relation for the R1 Tanq,v (a, t) may be obtained from the ratio of equations (6.18) and (6.21), that is, ( ) aR2q,v −a2 , t Rq,v (ai, t) − Rq,v (−ai, t) R1 Tanq,v (a, t) = = , t > 0. (6.22) R2q,v+q (−a2 , t) i (Rq,v (ai, t) + Rq,v (−ai, t)) The addition of the squares equations (6.15) and (6.16) gives R1 Cos2q,v (a, t) + R1 Sin2q,v (a, t) = Rq,v (a i, t)Rq,v (−a i, t), ( ) = R22q,v+q −a2 , t + a2 R2q,v (−a2 , t),

t > 0,

(6.23)

a fractional-order generalization of the classical identity cos2 (t) + sin2 (t) = 1. Because the right-hand side of this identity is a real number, this complex R product behaves like the multiplication of complex conjugates. In a similar manner, the subtraction of the square of equation (6.16) from the square of equation (6.15) gives 1 2 [R (a i, t) + R2q,v (−a i, t)], 2 q,v ( ( ) ) = R22q,v+q −a2 , t − a2 R2q,v −a2 , t ,

R1 Cos2q,v (a, t) − R1 Sin2q,v (a, t) =

t > 0.

(6.24)

Figure 6.16 shows the value of the product for a range of q and t values. This is a generalization of the identity cos2 (t) − sin2 (t) = cos(2 t). Figure 6.17 shows the value of the conjugate sum for a range of q and t values. The substitution of equations (6.18) and (6.21) into (6.13) gives ( ( ) ) (6.25) Rq,v (a i, t) = R2q,v+q −a2 , t + a i R2q,v −a2 , t , t > 0, the fractional Euler equation in an exponential form. Note that for q = 1/2 and v = 0, ( ) 2 1 R1 Sin 1 ,0 (a, t) = aR1,0 −a2 , t = a e−a t = (R1∕2,0 (a i, t) − R1∕2,0 (−a i, t)), 2 2i Furthermore, with a = 1, we have R1 Sin 1 ,0 (1, t) = e−t , 2

t > 0.

t > 0.

87

6 The R1 -Fractional Trigonometry

Figure 6.16 Conjugate product Rq,0 (i, 0, t)•Rq,0 (−i, 0, t), q = 0.2–1.4 in steps of 0.2, k = 0.

2 1.2

1.4 Rq,0(ia,t)Rq,0(–ia,t)

1.5 q = 1.0

1

0.5 0.8

0.4 0.6 0 0.2 0

1

2

3 4 t-Time

5

6

7

Figure 6.17 Conjugate sum 1 2 (R (i, 0, t) + R2q,0 (−i, 0, t)), q = 0.2–1.2 in 2 q,0 steps of 0.2, k = 0, a = 1.0.

5 4

2

2

1/2[Rq,0[(ia,t)Rq,0(–ia,t)]

88

q = 1.2

3 2 1

q = 1.0

0.8

0 –1 0.4 0.6 0.2 –2 –3

0

1

2

3 4 t-Time

5

6

7

6.2 R1 -Trigonometric Function Interrelationship A relationship between the R1 Cos-function and the R1 Sin-function is easily determined. Based on equation (6.4) aR1 Cosq,v−q (a, k, t) =a

∞ ∑ (−1)n a2n (t)(n+1)2q−1−v+q−q n=0

=

Γ((n + 1)2q − v + q − q)

∞ ∑ (−1)n a2n+1 (t)(n+1)2q−1−v n=0

Γ((n + 1)2q − v)

[ ] cis ((n + 1)2q − 1 − v + q − q)(2𝜋k)

[ ] cis ((n + 1)2q − 1 − v)(2𝜋k) , t > 0, k = 0, 1, … , Dq Dv − 1.

(6.26)

Thus, a R1 Cosq,v−q (a, k, t) = R1 Sinq,v (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1.

Conversely, R1 Sinq,v+q (a, k, t) = a R1 Cosq,v (a, k, t) =,

t > 0, k = 0, 1, … , Dq Dv − 1.

(6.27)

6.4 Fractional Calculus Operations on the R1 -Trigonometric Functions

The substitution q = 1, v = 0 into the defining series for the fractional trigonometric functions gives R1 Cos1,0 (a, k, t) = cos(a t) ,

t > 0, k = 0, 1, … , Dq Dv − 1,

(6.28)

R1 Sin1,0 (a, k, t) = sin(a t),

t > 0, k = 0, 1, … , Dq Dv − 1,

(6.29)

and t > 0, k = 0, 1, … , Dq Dv − 1,

R1 Tan1,0 (a, t) = tan(at),

(6.30)

demonstrating backward compatibility to common trigonometry.

6.3 Relationships to R1 -Hyperbolic Functions The relationship between the R1 -hyperbolic and the R1 -trigonometric functions may be determined by appropriate substitutions into the defining series. It is clear from equations (5.21) and (6.17) that ( ) (6.31) R1 Coshq,v (ai, t) = R1 Cosq,v (a, t) = R2q,v+q −a2 , t , t > 0, and conversely

( ) R1 Cosq,v (ai, t) = R1 Coshq,v (a, t) = R2q,v+q a2 , t , t > 0.

(6.32)

From equations (5.22) and (6.21), we have that

and conversely

( ) R1 Sinhq,v (ai, t) = i R1 Sinq,v (a, t) = i a R2q,v −a2 , t , t > 0,

(6.33)

( ) R1 Sinq,v (ai, t) = iR1 Sinhq,v (a, t) = i a R2q,v a2 , t , t > 0 .

(6.34)

These relationships between R1 -hyperbolic and R1 -trigonometric functions exactly parallel and generalize those between the integer-order trigonometric functions (q = 1, v = 0) and classical hyperbolic functions.

6.4 Fractional Calculus Operations on the R1 -Trigonometric Functions This section determines fractional differintegrals of the principal fractional R-trigonometric functions. This section is from Lorenzo and Hartley [74], with permission of Springer: The 𝛼-order differintegral of R1 Cosq,v (a, t) is determined as follows: Differintegrating term-by-term (see Section 3.16) gives 𝛼 0 dt R1 Cosq,v (a, t)

=

∞ ∑ (−1)n a2n 0 dt𝛼 (t)(n+1)2q−1−v−q n=0

Γ((n + 1)2q − v − q)

,

t > 0.

(6.35)

Then applying equation (5.37), we have 𝛼 0 dt R1 Cosq,v (a, t)

=

∞ ∑ (−1)n a2n (t)(n+1)2q−1−v−q−𝛼 n=0

Thus,

Γ((n + 1)2q − v − q − 𝛼)

𝛼 0 dt R1 Cosq,v (a, t)

, q > v, t > 0.

= R1 Cosq,v+𝛼 (a, t), q > v, t > 0.

(6.36)

89

90

6 The R1 -Fractional Trigonometry

Applying equation (6.27), this may be written as 𝛼 0 dt R1 Cosq,v (a, t)

1 (a, t), R Sin a 1 q,v+𝛼+q

=

q > v, t > 0 .

(6.37)

For 𝛼 = −q, this becomes 1 (6.38) R Sin (a, t), q > v, t > 0. a 1 q,v The 𝛼-order differintegral of R1 Sinq,v (a, t), is determined as follows: We may differintegrate (6.26) term-by-term based on Section 3.16. Then, using equation (5.37) −q 0 dt R1 Cosq,v (a, t)

𝛼 0 dt R1 Sinq,v (a, t) =

=

∞ ∑ (−1)n a2n+1 0 dt𝛼 (t)(n+1)2q−1−v

Γ((n + 1)2q − v)

n=0 ∞

𝛼 0 dt R1 Sinq,v (a, t)

=

∑ (−1)n a2n+1 (t)(n+1)2q−1−v−𝛼 n=0

Thus,

𝛼 0 dt R1 Sinq,v (a, t)

Γ((n + 1)2q − v − 𝛼)

,

t > 0,

(6.39)

, 2q > v, t > 0 .

= R1 Sinq,v+𝛼 (a, t), 2q > v, t > 0.

(6.40)

Applying equation (6.27), this may be written as 𝛼 0 dt R1 Sinq,v (a, t)

= a R1 Cosq,v+𝛼−q (a, t) , 2q > v, t > 0,

(6.41)

and for 𝛼 = q, we have the important case q 0 dt R1 Sinq,v (a, t)

= a R1 Cosq,v (a, t), 2q > v, t > 0.

(6.42)

6.5 Laplace Transforms of the R1 -Trigonometric Functions Because the series are uniformly convergent (see Chapter 14), the Laplace transforms for the principal R1 -trigonometric functions are readily determined by transforming the defining infinite series term-by-term. 6.5.1

Laplace Transform of R1 Cosq. v (a, k, t)

Then, the transform of equation (6.4) is {∞ } ∑ (−1)n a2n t (n+1)2q−1−v−q [ ] cis ((n + 1)2q − 1 − v − q)(2𝜋 k) L{R1 Cosq,v (a, k, t)} = L , Γ((n + 1)2q − v − q) n=0 k = 0, 1, 2 · · · Dq Dv − 1.

(6.43)

Inverse transforming term-by-term (see Section 3.16), ( ) ∞ ∑ ] [ t (n+1)2q−1−v−q n 2n (−1) a L cis n𝜆k + 𝜉k , L{R1 Cosq,v (a, k, t)} = Γ((n + 1)2q − v − q) n=0 k = 0, 1, 2 · · · Dq Dv − 1, where 𝜆k = 4q𝜋k and 𝜉k = (q − 1 − v)(2𝜋k). Thus, ∞ ] } ∑ [ { (−1)n a2n s−[(n+1)2q−v−q] cis n𝜆k + 𝜉k , L R1 Cosq,v (a, k, t) = n=0

q > v, k = 0, 1, 2, … , Dq Dv − 1,

(6.44)

6.5 Laplace Transforms of the R1 -Trigonometric Functions ∞ ] { } ∑ [ L R1 Cosq,v (a, k, t) = (−1)n a2n s−[(n+1)2q−v−q] cos n𝜆k + 𝜉k n=0

+i

∞ ∑

] [ (−1)n a2n s−[(n+1)2q−v] sin n𝜆k + 𝜉k ,

n=0

q > v, k = 0, 1, 2, … , Dq Dv − 1. Replacing cos and sin with exponential forms,

) ei[n𝜆+𝜉] + e−i[n𝜆+𝜉] (−1) a s L R1 Cosq,v (a, k, t) = 2 n=0 ) ( i[n𝜆+𝜉] ∞ ∑ − e−i[n𝜆+𝜉] e + i (−1)n a2n s−[(n+1)2q−v−q] , 2i n=0 {

}

∞ ∑

(

n 2n −[(n+1)2q−v−q]

k = 0, 1, 2, … , Dq Dv − 1.

(6.45)

∞ { } ∑ L R1 Cosq,v (a, k, t) = (−1)n a2n s−[(n+1)2q−v−q] ei[n𝜆+𝜉] , k = 0, 1, 2, … , Dq Dv − 1, n=0 ∞ { } ∑ L R1 Cosq,v (a, k, t) = (−1)n a2n n=0

1 ei[n𝜆+𝜉] , k = 0, 1, 2, … , Dq Dv − 1, s[(n+1)2q−v−q]

∞ { } ei𝜉k sv+q ∑ (−a2 ei𝜆k )n L R1 Cosq,v (a, k, t) = , q > v, k = 0, 1, 2, … , Dq Dv − 1 . 2q s (s2q )n n=0

Now,

1 1 ∑ (−b)n ; = sq + b sq n=0 sqn ∞

therefore, we have } { L R1 Cosq,v (a, k, t) =

ei𝜉k sv+q , q > v, k = 0, 1, 2, … , Dq Dv − 1, + a2 ei𝜆k where 𝜆k = 4q𝜋k and 𝜉k = (q − 1 − v)(2𝜋k). For the principal function, k = 0 { } sv+q L R1 Cosq,v (a, t) = 2q , q > v. s + a2

6.5.2

(6.46)

s2q

(6.47)

(6.48)

(6.49)

Laplace Transform of R1 Sinq. v (a, k, t)

Determination of this transform proceeds in a similar manner as that for the R1 Cosq.v (a, k, t); thus, {∞ } ∑ (−1)n a2n+1 t (n+1)2q−1−v [ { } ] L R1 Sinq,v (a, k, t) = L cis ((n + 1)2q − 1 − v)(2𝜋 k) , Γ((n + 1)2q − v) n=0 k = 0, 1, 2, … , Dq Dv − 1,

(6.50)

From Section 3.16, we may transform term-by-term; thus, ( ) ∞ ] } ∑ [ { t (n+1)2q−1−v n 2n+1 (−1) a L cis n𝜆k + 𝜉k , L R1 Sinq,v (a, k, t) = Γ((n + 1)2q − v) n=0 k = 0, 1, 2, … , Dq Dv − 1,

(6.51)

91

92

6 The R1 -Fractional Trigonometry

where 𝜆k = 4q𝜋k and 𝜉k = (2q − 1 − v)(2𝜋k). Thus, ∞ } ∑ { (−1)n a2n+1 s−[(n+1)2q−v] cis[n𝜆k + 𝜉k ] , L R1 Sinq,v (a, k, t) = n=0

2q > v, k = 0, 1, 2, … , Dq Dv − 1, ∞ { } ∑ L R1 Sinq,v (a, k, t) = (−1)n a2n+1 s−[(n+1)2q−v] cos[n𝜆k + 𝜉k ] n=0

+i

∞ ∑

(−1)n a2n+1 s−[(n+1)2q−v] sin[n𝜆k + 𝜉k ], 2q > v,

n=0

k = 0, 1, 2, … , Dq Dv − 1. Replacing cos and sin with exponential forms, ) ei[n𝜆+𝜉] + e−i[n𝜆+𝜉] L R1 Sinq,v (a, k, t) = (−1) a s 2 n=0 ) ( ∞ ∑ ei[n𝜆+𝜉] − e−i[n𝜆+𝜉] n 2n+1 −[(n+1)2q−v] + i (−1) a s , 2i n=0 {

}

∞ ∑

(

n 2n+1 −[(n+1)2q−v]

2q > v, k = 0, 1, 2, … , Dq Dv − 1,

(6.52)

∞ { } ∑ L R1 Sinq,v (a, k, t) = (−1)n a2n+1 s−[(n+1)2q−v] ei[n𝜆+𝜉] , k = 0, 1, 2, … , Dq Dv − 1, n=0 ∞ { } ∑ L R1 Sinq,v (a, k, t) = (−1)n a2n+1

1 ei[n𝜆+𝜉] , k = 0, 1, 2, … , Dq Dv − 1, [(n+1)2q−v] s n=0 ( 2 i𝜆 )n ∞ −a e k } aei𝜉k sv ∑ { , k = 0, 1, 2, … , Dq Dv − 1. L R1 Sinq,v (a, k, t) = 2q s (s2q )n n=0 Now applying equation (6.47), we have { } L R1 Sinq,v (a, k, t) =

aei𝜉k sv , 2q > v, k = 0, 1, 2, … , Dq Dv − 1, + a2 ei𝜆k

s2q

(6.53)

where 𝜆k = 4q𝜋k and 𝜉k = (2q − 1 − v)(2𝜋k). For the principal function, k = 0 } { L R1 Sinq,v (a, t) =

asv , + a2

s2q

2q > v.

(6.54)

The principal R1 -trigonometric and R1 -hyperbolic functions, their defining series, Laplace transforms, and R-function relationships are presented in Table 6.1 along with those of the classical circular functions.

6.6 Complexity-Based R1 -Trigonometric Functions In Section 6.1, fractional trigonometric functions that parallel the familiar classical trigonometric functions were derived based on the parity-based series (equation (6.2)). In this section, we

6.6 Complexity-Based R1 -Trigonometric Functions

Table 6.1 Summary of the principal R1 -trigonometric, R1 -hyperbolic, and the traditional functions. Function

Defining Series ∞ ∑

eat

0

(at)n Γ(n + 1)

Laplace Transform

R-Function

1 s−a

R1,0 (a, t)

sin(at)

∞ ∑ (−1)n (at)2n+1 (2n + 1) ! n=0

a s2 + a2

aR2,0 (−a2 , t)

cos(at)

∞ ∑ (−1)n t 2n (2n) ! n=0

s s2 + a2

R2,1 (−a2 , t)

sinh(at)

∞ ∑ (at)2n+1 (2n + 1) ! n=0

a s2 − a2

aR2,0 (a2 , t)

cosh(at)

∞ ∑ (at)2n (2n) ! n=0

s s2 − a2

R2,1 (a2 , t)

Rq,v (a, t), t > 0

∞ ∑ an t (n+1)q−1−v , Γ((n + 1)q − v) n=0

R1 Sinq,v (a, t), t > 0

∞ ∑ (−1)n a2n+1 (t)(n+1)2q−1−v , t>0 Γ((n + 1)2q − v) n=0

a sv , 2q > v s2q + a2

aR2q,v (−a2 , t)

∞ ∑ (−1)n a2n (t)(n+1)2q−1−v−q , t>0 Γ((n + 1)2q − v − q) n=0

s2q

sv+q , q>v + a2

R2q,v+q (−a2 , t)

∞ ∑ (a)2n+1 (t)(n+1)2q−1−v , t>0 Γ((n + 1)2q − v) n=0

a sv , 2q > v s2q − a2

aR2q,v (a2 , t)

∞ ∑ (a)2n (t)(n+1)2q−1−v−q , t>0 Γ((n + 1)2q − v − q) n=0

s2q

sv+q , q>v − a2

R2q,v+q (a2 , t)

R1 Cosq,v (a, t), R1 Sinhq,v (a, t),

t>0 t>0

R1 Coshq,v (a, t), t > 0

t>0

sq

sv , q>v −a

–

Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

shall consider complexity-based functions starting from equation (6.1). Equation (6.1) can be written as Rq,v (ai, t) =

∞ ∑ (ai)n (t)(n+1)q−1−v n=0 ∞

Rq,v (ai, t) =

=

n=0

∑ (−1)n (a)n (t)(n+1)q−1−v n=0 ∞

Rq,v (ai, t) =

Γ((n + 1)q − v)

∞ ∑ (a)n (t)(n+1)q−1−v

Γ((n + 1)q − v)

Γ((n + 1)q − v)

+i

in , t > 0,

∞ ∑ (−1)n (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

(6.55) , t > 0,

∑ (−a)n (t)(n+1)q−1−v n=0

∞ ∑ (−a)n (t)(n+1)q−1−v +i , t > 0. Γ((n + 1)q − v) Γ((n + 1)q − v) n=0

(6.56)

Furthermore, we have that Rq,v (ai, t) = Rq,v (−a, t) + iRq,v (−a, t) = (1 + i)Rq,v (−a, t),

t > 0,

(6.57)

a complex plane version of the R-function. Because the results are R-functions which have been studied earlier, we do not pursue this area further.

93

94

6 The R1 -Fractional Trigonometry

6.7 Fractional Differential Equations Continuing from Ref. [74], with permission of Springer: The R1 -trigonometric functions may also be solutions to fractional differential equations. This transform (equation (6.49)) indicates that the uninitialized fractional differential equation 2q v+q 2 q>v (6.58) 0 dt x(t) + a x(t) = 0 dt f (t), has a solution of the form x(t) = R1 Cosq,v (a, t),

t ≥ 0,

(6.59)

when f (t) = 𝛿(t), that is, an impulse function. As with the R1 -hyperbolic functions for arbitrary f (t), the solution is given by the convolution t

x(t) = R1 Cosq,v (a, t) ∗ f (t) =

∫

R1 Cosq,v (a, 𝜏)f (t − 𝜏)d𝜏

0 t

=

∫

R1 Cosq,v (a, t − 𝜏)f (𝜏)d𝜏,

t > 0,

q ≥ v.

(6.60)

0

Assuming composition equation (6.58) may also be written in the form of a fractional integro-differential equation, namely q−v 0 dt x(t)

−v−q

+ a2 0 dt

x(t) = g(t).

In a similar manner, equation (6.54) indicates that the uninitialized fractional differential equation 2q 2 v (6.61) 0 dt x(t) + a x(t) = a0 dt f (t), has a solution of the form x(t) = R1 Sinq,v (a, t), 2q > v, t > 0,

(6.62)

when f (t) = 𝛿(t), that is, an impulse function. For arbitrary f (t), the solution is obtained by the convolution similar to equation (6.60). Equation (6.61) may also be written as a fractional integro-differential equation, namely 2q−v x(t) 0 dt

+ a2 0 dt−v x(t) = af (t).

(6.63)

Linear combinations of fractional trigonometric functions may also be used to infer solutions for fractional differential forms of noncommensurate order. For example, from the linear combination A R1 Cosq,v (a, t) + B R1 Sinu,w (b, t),

t > 0,

(6.64)

or using the relationships (6.17) and (6.20) A R2q,v+q (−a2 , t) + B b R2u,w (−b2 , t),

t > 0,

we may infer the Laplace transformed equation X(s) A sv+q Bbsw + 2u = , 2 2 +a s +b F(s)

s2q

(6.65)

6.7 Fractional Differential Equations

or

X(s) A sv+q+2u + Ab2 sv+q + B b sw+2q + B b a2 sw = . F(s) s2(q+u) + a2 s2u + b2 s2q + a2 b2

From this, it is seen that the uninitialized fractional differential equation 2(q+u) x(t) 0 dt

where

2q

+ a2 0 dt2u x(t) + b2 0 dt x(t) + a2 b2 x(t) = g(t),

(6.66)

{ ( ( ) ( ))} , g(t) = L−1 F(s) Asv+q s2u + b2 + B b sw s2q + a2

will have a solution of the form given by equation (6.64). Various specializations of the orders and parameters lead to other useful results. Still more varied fractional differential equations may be obtained by considering linear combinations of the both the R1 -trigonometric functions and the R1 -hyperbolic functions, using, for example, forms such as AR1 Coshq,v (a, t) + BR1 Sinhu,w (b, t) + CR1 Cosx,y + DR1 Sinp,r (a, t), t > 0.

(6.7.14)

Chapter 7 explores the effect of imaginary t in the definition of a fractional trigonometry.

95

97

7 The R2 -Fractional Trigonometry The basis for the R1 -trigonomtry (developed in Chapter 6) was to replace the parameter a by ia in the R-function. For the R2 -trigonometry, we consider a as a real parameter and let t be imaginary, that is, t → i t in the R-function. Specifically, the basis for the R2 -trigonometry is Rq,v (a, it) [73, 74, 75]. In the traditional trigonometry (Section 1.3), the common exponential function is expanded in integer powers of t, and we found that the real part (equation (1.18)) was the summation of even powers of t and the imaginary part (equation (1.19)) was the summation of odd powers of t. In this chapter, we find that this is no longer the case when it is raised to fractional powers. This introduces families of new functions not found in the traditional trigonometry. We find that these new functions occur in a pairwise manner similar to the classical sine and cosine functions. The development of the R2 -trigonometry follows from Lorenzo and Hartley [74] with permission of Springer:

7.1 R2 -Trigonometric Functions: Based on Real and Imaginary Parts We start by considering the real and imaginary parts of Rq,v (a, i t) =

∞ ∑ an (i t)(n+1) q−1−v , Γ((n + 1) q − v) n=0

t > 0.

(7.1)

Now, for a complex number z = x + i y = r(cos 𝜃 + i sin 𝜃), we have from equation (3.122) { [ ] [ ]} m m zm∕n = rm∕n cos (𝜃 + 2𝜋k) + i sin (𝜃 + 2𝜋k) , n n k = 0, 1, 2, … , n − 1 . (7.2) Therefore, let us write for the exponent of (it) in equation (7.1) (n + 1) q − 1 − v = (n + 1) =

Nq Dq

−1−

Nv Dv

(n + 1)Nq Dv − Dq Dv − Nv Dq Dq Dv

=

M , D

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(7.3)

98

7 The R2 -Fractional Trigonometry

where q = Nq ∕Dq and v = Nv ∕Dv are assumed to be rational and irreducible and M/D is in minimal form. Then, equation (7.1) becomes Rq,v (a, i t) =

∞ ∑ n=0

an (i t)M∕D , Γ((n + 1) q − v)

t>0

(7.4)

and by (7.2) Rq,v (a, k, i t) =

∞ ∑

an Γ((n + 1) q − v) n=0 { ( { M[ )} { ( )}]} M 𝜋 M 𝜋 × t D cos + 2𝜋k + i sin + 2𝜋k , D 2 D 2 t > 0, k = 0, 1, … Dq Dv − 1 .

(7.5)

where k has been introduced into the R argument since it is now explicit on the right-hand side. We propose the following definitions: R2 Cosq,v (a, k, t) ≡

∞ ∑

an t (n+1) q−1−v Γ((n + 1) q − v) n=0 { ( )} 𝜋 × cos ((n + 1) q − 1 − v) + 2𝜋k , 2 t > 0, k = 0, 1, … , Dq Dv − 1,

(7.6)

and R2 Sinq,v (a, k, t) ≡

∞ ∑

an t (n+1) q−1−v Γ((n + 1) q − v) n=0 { ( )} 𝜋 × sin ((n + 1) q − 1 − v) + 2𝜋k , 2 t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.7)

Thus, we see that the fractional character of q and v gives rise to a family of Dq Dv -many fractional trigonometric functions for rational values of q and v and Dq Dv in minimal form. This situation is unparalleled in the ordinary trigonometry. The principal functions, k = 0, are defined as R2 Cosq,v (a, 0, t) =R2 Cosq,v (a, t) ≡

∞ ∑ n=0

{ ( )} an t (n+1) q−1−v 𝜋 , cos ((n + 1) q − 1 − v) Γ((n + 1) q − v) 2

t > 0,

(7.8)

t > 0.

(7.9)

and R2 Sinq,v (a, 0, t) =R2 Sinq,v (a, t) ≡

∞ ∑ n=0

{ ( )} an t (n+1) q−1−v 𝜋 , sin ((n + 1) q − 1 − v) Γ((n + 1) q − v) 2

The R2 Tan-function is defined as the ratio R2 Tanq,v (a, k, t) ≡

R2 Sinq,v (a, k, t) R2 Cosq,v (a, k, t)

,

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.10)

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

Figure 7.1 R2 Cosq,0 (1, t) versus t-Time for q = 0.2–2.0 in steps of 0.1, with a = 1.0, v = 0, k = 0. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

6 q = 0.2 4

R2Cosq,0(1,t)

2

0

q = 2.0

–2 –4 –6

0

1

2

3 4 t-Time

5

6

7

Figure 7.2 R2 Sinq,0 (1, t) versus t-Time for q = 0.2–2.0 in steps of 0.1, with a = 1.0, v = 0, k = 0. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

6 4

R2Sinq,0(1,t)

2

0

q = 2.0

–2 –4 q = 0.2 –6

0

1

2

3 4 t-Time

5

6

7

By substituting q = 1, v = 0 in equations (7.6) and (7.7), it can be seen that R2 -trigonometric functions generalize the classical trigonometric functions, that is, R2 Cos1,0 (a, k, t) = cos(a t),

t > 0,

and R2 Sin1,0 (a, k, t) = sin(a t),

t > 0.

(7.11)

Figures 7.1 and 7.2 show the principal R2 Cos- and R2 Sin- and R2 Tan-functions for 0.2 ≤ q ≤ 2. Note that there is a reversal in the amplitude of the R2 Sin-function for q > 1.9 (Figure 7.3). Furthermore, because R2 Cos2,0 (1, t) = 0 the tangent plot is limited to q = 1.9. The effect of the a parameter on the R2 Cos-function is shown in Figures 7.4 and 7.5 for order variable q = 0.25 and 0.75, respectively. It can be seen that as a increases, the functions become increasingly oscillatory, also the amplitudes grow, and the frequency of oscillation increases. Similar effects are observed for the R2 Sin-functions in Figures 7.6 and 7.7. The effect of the change in frequency is clearly seen for the tangent functions in Figures 7.8 and 7.9. Note, when q = 1 and v = 0, from

99

7 The R2 -Fractional Trigonometry

Figure 7.3 R2 Tanq,0 (1, t) versus t-Time for q = 0.2–1.9 in steps of 0.1, with a = 1.0, v = 0, k = 0.

30

20 q = 1.9

R2Tanq,0(1,t)

10

0

q = 0.2

–10

q = 1.9

–20 –30

0

1

2

3

4 t-Time

5

6

7

Figure 7.4 Effect of a for R2 Cosq,0 (1, t) versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.25, k = 0, v = 0.0.

6 4

R2Cos0.25,0(a,t)

100

a = 1.1

a = 1.0

2

0

a = 0.2

–2 –4 –6 0

1

2

3

4 t-Time

5

6

7

equations (7.11) the R2 Cos-functions revert to the classical circular functions and a becomes exactly the frequency parameter. Figures 7.10–7.15 examine the effects of the differintegration of order v on the principal R2 Cos-, R2 Sin-, and R2 Tan-functions for v = −0.6–0.6 in steps of 0.6, with q = 0.25 and q = 0.75. It is interesting to observe that in all cases, after a short transient period of less than one cycle, the v = 0 case and the v = 0.6 case are nearly indistinguishable. The effect of the k index on the R2 Cos6∕7,0 (1, t) is shown in Figure 7.16. Note the strong symmetric behavior for this particular case. This is not always found for general values of q. Figure 7.17 shows a phase plane plot for R2 Cosq,0 versus R2 Sinq,0 with order, q, varying between 0.40 and 2.0. For 0 < q < 2, the amplitudes of both the R2 Cosq,0 - and R2 Sinq,0 -functions are asymptotic to 1∕q. In other words, the principal functions are attracted to circles of radius 1∕q, for 0 < q < 2. We observe that for values of q < 1, the functions originate at infinity while for value of q > 1, they begin at the origin.

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

Figure 7.5 Effect of a for R2 Cosq,0 (1, t) versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.75, k = 0, v = 0.0.

2 1.5

R2Cos0.75,0(a,t)

1

a = 0.2

0.5 0 –0.5 –1

a = 1.1

–1.5 –2

0

1

2

3 4 t-Time

5

6

7

6 a = 1.1

R2Sin0.25,0(a,t)

4

a = 1.0

2

0

a = 0.2

–2 –4 –6

0

1

2

3

4

5

6

7

t-Time

Figure 7.6 Effect of a for R2 Sinq,0 (1, t) 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.

Continuing from Ref. [74] with permission of Springer: We also observe from equations (7.5)–(7.7) that Rq,v (a, k, it) = R2 Cosq,v (a, k, t) + i R2 Sinq,v (a, k, t) , t > 0, k = 0, 1, 2, … , Dq Dv − 1 ,

(7.12)

paralleling the form of equation (6.13) for the R1 -trigonometry. Furthermore, Rq,v (a, k, −it) = R2 Cosq,v (a, k, t) − i R2 Sinq,v (a, k, t) , t > 0, k = 0, 1, 2, … , Dq Dv − 1 is also validated in Section 7.4 (equation (7.70)).

(7.13)

101

7 The R2 -Fractional Trigonometry

Figure 7.7 Effect of a for R2 Sinq,0 (1, t) 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.

1.5 a = 1.1

R2Sin0.75,0(a,t)

1 0.5

0

a = 0.2

–0.5 –1 –1.5

0

1

2

3 4 t-Time

5

6

7

20 a = 1.0

15

a = 0.9

a = 0.8 a = 1.0

0.7

Figure 7.8 Effect of a for R2 Tanq,0 (1, t) 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.

10 R2Tan0.25,0(a,t)

102

5 0 –5 –10 –15 –20

0

1

2

3 4 t-Time

5

6

7

7.2 R2 -Trigonometric Functions: Based on Parity Because the real part of Rq,v (a, it) is no longer the same as the even part of Rq,v (a, it) and the imaginary part is no longer the same as the odd part of Rq,v (a, it), we now consider the terms of the series containing only even or odd powers of a, that is, the parity of the Rq,v (a, it) series. Expanding Rq,v (a, it) Rq,v (a, i t) =

∞ ∞ ∑ ∑ an (i t)(n+1) q−1−v an (it)nq = (it)q−1−v , Γ((n + 1) q − v) Γ((n + 1) q − v) n=0 n=0

Rq,v (a, i t)

{ = (it)

q−1−v

t > 0,

(7.14)

} a(it)q a2 (it)2q a3 (it)3q a4 (it)4q 1 + + + + +··· . Γ(q − v) Γ(2q − v) Γ(3q − v) Γ(4q − v) Γ(5q − v) (7.15)

7.2 R2 -Trigonometric Functions: Based on Parity

Figure 7.9 Effect of a for R2 Tanq,0 (1, t) 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.

20 a = 1.0

15

a = 0.7 a = 1.0

0.7

R2Tan0.75,0(a,t)

10 5 0 –5 –10 –15 –20

0

1

2

3 4 t-Time

5

6

7

Figure 7.10 Effect of v for R2 Cosq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.

10

R2Cos0.25,𝜈(1,t)

𝜈 = 0.6 5 0.0

0 𝜈 = –0.6

–5

0

1

2

3 4 t-Time

5

6

7

Separating this into summations of even and odd powers of a gives } {∞ ∞ ∑ ∑ (a2 )n (it)2nq a (a2 )n (it)(2n+1) q q−1−v Rq,v (a, i t) = (it) + , Γ((n + 1) 2q − v − q) n=0 Γ((n + 1) 2q − v) n=0 Rq,v (a, it) =

∞ ∞ ∑ ∑ a2n (it)(n+1) 2q−1−v−q a2n (it)(n+1) 2q−1−v +a , Γ((n + 1) 2q − v − q) Γ((n + 1) 2q − v) n=0 n=0

t > 0,

(7.16)

from which we observe Rq,v (a, it) = R2q,v+q (a2 , it) + aR2q,v (a2 , it) , t > 0.

(7.17)

Equation (7.16) then separates Rq,v (a, it) into series with even and odd powers of a. These complex series are of theoretical importance and are named the Corotation and the Rotation

103

7 The R2 -Fractional Trigonometry

Figure 7.11 Effect of v for R2 Cosq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.

2 1.5 𝜈 = 0.0, 0.6

R2Cos0.75,𝜈(1,t)

1 0.5 0 –0.5

𝜈 = –0.6

–1 –1.5 –2

5

1

0

2

3 4 t-Time

5

6

7

Figure 7.12 Effect of v for R2 Sinq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.

𝜈 = 0.6 𝜈 = –0.6

R2Sin0.25,𝜈(1,t)

104

0 𝜈 = 0.0

–5

0

1

2

3 4 t-Time

5

6

7

functions and are defined as R2 Corq,v (a, t) ≡ R2 Rotq,v (a, t) ≡

∞ ∑ a2n (it)(n+1) 2q−1−v−q , Γ((n + 1) 2q − v − q) n=0 ∞ ∑ a2n+1 (it)(n+1) 2q−1−v n=0

Γ((n + 1) 2q − v)

,

t > 0, t > 0,

(7.18)

(7.19)

where R2 Corq,v (a, it) = Ea {Rq,v (a, it)} and R2 Rotq,v (a, it) = Oa {Rq,v (a, it)}, both with respect to powers of a. Now, the real and imaginary parts of equations (7.18) and (7.19) will be used to define four new real fractional R2 -trigonometric functions. Continuing from Ref. [74] with permission of Springer:

7.2 R2 -Trigonometric Functions: Based on Parity

Figure 7.13 Effect of v for R2 Sinq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.

2 𝜈 = –0.6

1.5

R2Sin0.75,𝜈(1,t)

1 0.5 0 –0.5

𝜈 = 0.0

–1

𝜈 = 0.6

–1.5 –2

0

1

2

3 4 t-Time

5

6

7

Figure 7.14 Effect of v for R2 Tanq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.

5

R2Tan0.25,𝜈(1,t)

𝜈 = –0.6

𝜈 = 0,0.6

𝜈 = 0,0.6

𝜈 = 0,0.6

0.6 𝜈 = –0.6

0 𝜈 = 0.0

𝜈 = –0.6

–5 0

1

2

3

4

5

6

7

t-Time

From the R2 Rotq,v (a, t)-function, we have R2 Rotq,v (a, t) =

∞ ∑ a2n+1 (it)(n+1) 2q−1−v n=0

=

Γ((n + 1) 2q − v)

∞ ∑ a2n+1 (t)(n+1) 2q−1−v n=0

Γ((n + 1) 2q − v)

i(n+1)

2q−1−v

,

t > 0.

We now write (n + 1) 2q − 1 − v =

(n + 1)2Nq Dv − Dq Dv − Dq Nv Dq Dv

=

M , D

(7.20)

105

7 The R2 -Fractional Trigonometry

Figure 7.15 Effect of v for R2 Tanq,0 (1, t) versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.

5

R2Tan0.75,𝜈(1,t)

–0.6

𝜈 = 0,0.6

𝜈 = 0,0.6

𝜈 = –0.6

0 𝜈 = 0.0 𝜈 = 0.6 𝜈 = –0.6

–5

0

1

2

3

4

5

6

7

t-Time

Figure 7.16 Effect of k for R2 Cosq,0 (1, t) versus t-Time for k = 0–6 in steps of 1, with q = 6/7, a = 1.0, k = 0, v = 0.

5 k=1

R2Cos6 / 7,0(1,t)

106

k=2

k=3

k=6

k=5

0 k=4

k=0

k=2 –5

0

1

2

k=1 3

4

5

6

7

t-Time

where q = Nq ∕Dq and v = Nv ∕Dv are assumed rational, and M/D is in minimal form. Then, equation (7.20) may be written as R2 Rotq,v (a, k, t) =

∞ ∑ a2n+1 (t)(n+1) 2q−1−v n=0

Γ((n + 1) 2q − v)

cos(𝜆)

∞ ∑ a2n+1 (t)(n+1) 2q−1−v

sin(𝜆), Γ((n + 1) 2q − v) )) ( ( where 𝜆 = ((n + 1) 2q − 1 − v) 𝜋2 + 2𝜋 k with t > 0, k = 0, 1, · · · Dq Dv − 1. The Coflutter function is defined as the real part of the R2 Rot-function, that is, +i

(7.21)

n=0

R2 Cof lq,v (a, k, t) ≡

( ( )) 𝜋 cos ((n + 1) 2q − 1 − v) + 2𝜋 k , Γ((n + 1) 2q − v) 2

∞ ∑ a2n+1 (t)(n+1) 2q−1−v n=0

t > 0, k = 0, 1, … , Dq Dv − 1 ,

(7.22)

7.2 R2 -Trigonometric Functions: Based on Parity

Figure 7.17 Phase plane plot R2 Sinq,0 (1, t) versus R2 Cosq,0 (1, t) for q = 0.4–2.0 in steps of 0.2, a = 1.0, v = 0, k = 0. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

2.5 2

q = 0.4

1.5

0.6 0.8

R2Sinq,0(1,t)

1 0.5 0 –0.5 –1 –1.5

2.0 1.2

–2 –2.5 –2

–1

0

1

2

R2Cosq,0(1,t)

and the Flutter function is defined as the imaginary part of the R2 Rot-function, namely R2 Flutq,v (a, k, t) ≡

( ( )) 𝜋 sin ((n + 1) 2q − 1 − v) + 2𝜋 k , Γ((n + 1) 2q − v) 2

∞ ∑ a2n+1 (t)(n+1) 2q−1−v n=0

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.23)

In a similar manner, from R2 Corq,v (a, t), we have for t > 0 R2 Corq,v (a, t) =

∞ ∑ a2n (it)(n+1) 2q−1−v−q Γ((n + 1) 2q − v − q) n=0

=

∞ ∑ n=0

a2n (t)(n+1) 2q−1−v−q (n+1) i Γ((n + 1) 2q − v − q)

2q−1−v−q

.

(7.24)

We now write (n + 1) 2q − 1 − v − q =

(n + 1) 2Nq Dv − Dq Dv − Dq Nv − Nq Dv Dq Dv

=

M , D

where q = Nq ∕Dq and v = Nv ∕Dv are assumed rational, and M/D is in minimal form. Then, equation (7.24) may be written as R2 Corq,v (a, t) =

∞ ∑ n=0

∞ ∑ a2n (t)(n+1) 2q−1−v−q a2n (t)(n+1) 2q−1−v−q cos(𝜉) + i sin(𝜉), Γ((n + 1) 2q − v − q) Γ((n + 1) 2q − v − q) n=0

(7.25)

( ( )) where 𝜉 = ((n + 1) 2q − 1 − v − q) 𝜋2 + 2𝜋 k with t > 0, and k = 0, 1, … , Dq Dv − 1. From equation (7.25), the Covibration function is defined as, Re{R2 Corq,v }, that is,

107

108

7 The R2 -Fractional Trigonometry

R2 Covibq,v (a, k, t) ≡

∞ ∑ n=0

( ( )) a2n (t)(n+1) 2q−1−v−q 𝜋 cos ((n + 1) 2q − 1 − v − q) + 2𝜋 k Γ((n + 1) 2q − v − q) 2

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.26)

Similarly, the vibration function is defined as, Im{R2 Corq,v (a, it)}, namely R2 Vibq,v (a, k, t) ≡

∞ ∑ n=0

( ( )) a2n (t)(n+1) 2q−1−v−q 𝜋 sin ((n + 1) 2q − 1 − v − q) + 2𝜋 k , Γ((n + 1) 2q − v − q) 2

t > 0 k = 0, 1, … , Dq Dv − 1.

(7.27)

By substituting q = 1, and v = 0, into the parity series definitions, we can determine the backward compatibility of these functions. Therefore, we have continuing from Ref. [74] with permission of Springer: (7.28) R2 Cos1,0 (a, t) = cos(a t), t > 0, R2 Sin1,0 (a, t) = sin(a t),

t > 0,

(7.29)

R2 Cor1,0 (a, t) = cosh(i at) = cos(at),

t > 0,

(7.30)

R2 Rot1,0 (a, t) = sinh(i at) = i sin(at),

t > 0,

(7.31)

R2 Cof l1,0 (a, k, t) = 0,

t > 0,

(7.32)

R2 Flut1,0 (a, k, t) = sin(at),

t > 0,

(7.33)

R2 Covib1,0 (a, k, t) = cos(at),

t > 0,

(7.34)

R2 Vib1,0 (a, k, t) = 0,

t > 0.

(7.35)

Thus, we see that complex series – the Rotation and the Corotation – revert to hyperbolic functions of complex arguments. The real functions defined from the Rotation and Corotation are observed to either degenerate to zero or in the cases of R2 Flut1,0 (a, k, t) and R2 Covib1,0 (a, k, t), to be backward compatible with the circular functions. Thus, clearly, we could have defined the R2 Flutq,v (a, k, t)- and R2 Covibq,v (a, k, t)-functions to be the fractional generalizations of the traditional sine and cosine functions; however, we have given precedence to complexity over parity in our naming conventions. We again use the notations as follows: Ea (∘) to mean the terms of (∘) having even powers of a, and Oa (∘) to mean the terms of (∘) having odd powers of a. Then, the following observations are readily made: R2 Cosq,v (a, k, t) = Re(Rq,v (a, k, it)),

(7.36)

R2 Sinq,v (a, k, t) = Im(Rq,v (a, k, it)),

(7.37)

R2 Covibq,v (a, k, t) = Re(Ea (Rq,v (a, k, it))) = Ea (Re(Rq,v (a, k, it))),

(7.38)

7.2 R2 -Trigonometric Functions: Based on Parity

R2 Vibq,v (a, k, t) = Im(Ea (Rq,v (a, k, it))) = Ea (Im(Rq,v (a, k, it))),

(7.39)

R2 Cof lq,v (a, k, t) = Re(Oa (Rq,v (a, k, it))) = Oa (Re(Rq,v (a, k, it))),

(7.40)

R2 Flutq,v (a, k, t) = Im(Oa (Rq,v (a, k, it))) = Oa (Im(Rq,v (a, k, it))),

(7.41)

R2 Rotq,v (a, t) = Oa (Rq,v (a, it)),

(7.42)

R2 Corq,v (a, t) = Ea (Rq,v (a, it)).

(7.43)

The commutative property of equations (7.38)–(7.41) are readily proved by taking the even and odd parts of equations (7.6) and (7.7) and relating them to the real and imaginary parts of equations (7.18) and (7.19), which are the functions on the left-hand side of equations (7.38)–(7.41). It is of interest to also write the equivalent relationships for the R1 -functions: R1 Coshq,v (a, t) = Ea (Rq,v (a, t)),

(7.44)

R1 Sinhq,v (a, t) = Oa (Rq,v (a, t)),

(7.45)

R1 Cosq,v (a, t) = Re(Rq,v (ia, t)) = Ea (Rq,v (ia, t)),

(7.46)

R1 Sinq,v (a, t) = Im(Rq,v (a, it)) = Oa (Rq,v (ia, t)).

(7.47)

It is noted that the R1 -functions, that is, equations (7.44)–(7.47), exactly parallel the traditional circular and hyperbolic functions with regard to complexity and parity. The principal, k = 0, functions, R2 Cof lq,v (a, 0, t), R2 Flutq,v (a, 0, t), R2 Covibq,v (a, 0, t), and R2 Vibq,v (a, 0, t), are presented in Figures 7.18 and 7.19 for selected q values. Observe that R2 Cof l1,0 (1, 0, t) = R2 Vib1,0 (1, 0, t) = 0 as indicated by equations (7.32) and (7.35). We also see

R2 Coflq,0(1,t)

4 2

q = 0.5 dashed q = 1.0

0

q = 0.2

–2

R2 Flutq,0(1,t)

0

1

2

3 4 t-Time

5

6

7

2 q = 0.5 and 1.0 dashed 0

q = 0.8

–2

q = 0.2 0

1

2

3 4 t-Time

5

6

7

Figure 7.18 R2 Cof lq,0 (1, t) and R2 Flutq,0 (1, t) 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. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

109

7 The R2 -Fractional Trigonometry

R2 Covibq,0(1,t)

4 2

q = 0.5 and q = 1.0 dashed

0

q = 0.8

–2

q = 0.4

R2 Covibq,0(1,t)

0

1

2

q = 0.2

3 4 t-Time

2

5

6

7

6

7

q = 0.5 dashed q = 1.0

0

q = 0.2

–2 0

1

2

3

4

5

t-Time

Figure 7.19 R2 Covibq,0 (1, t) and R2 Vibq,0 (1, t) 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. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

that R2 Cof l1∕2,0 (1, 0, t) = cos(t) and that R2 Flut1∕2,0 (1, 0, t) = R2 Flut1,0 (1, 0, t) = sin(t), t > 0. Figures 7.20 and 7.21 show phase plane plots for R2 Cof lq,v (a, k, t) versus R2 Flutq,v (a, k, t) and R2 Covibq,v (a, k, t) versus R2 Vibq,v (a, k, t), for various values of q, compare these to Figure 7.17. For the parity functions we observe that for values of q < 0.5 the functions originate at infinity, while for values of q > 0.5 they begin at the origin. 3

2

q = 0.2 q = 0.5 dashed

1 R2 Flutq,0(1,t)

110

0 q = 0.4 –1

q = 1.0

q = 0.8

–2

–3 –3

q = 0.6

–2

–1

1 0 R2 Coflq,0(1,t)

2

3

Figure 7.20 Phase plane plot R2 Flutq,0 (1, t) versus R2 Cof lq,0 (1, t) 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.

7.3 Laplace Transforms of the R2 -Trigonometric Functions

3

2

q = 0.2 q = 0.5 dashed

R2 Vibq,0(1,t)

1

q = 0.4

0

–1

q = 0.6 q = 1.0

q = 0.8

–2

–3 –3

–2

–1

1 0 R2 Covibq,0(1,t)

2

3

Figure 7.21 Phase plane plot R2 Vibq,0 (1, t) versus R2 Covibq,0 (1, t) 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.

7.3 Laplace Transforms of the R2 -Trigonometric Functions 7.3.1 R2 Cosq,v (a, k, t)

Using definition (7.6), we first consider {∞ } { ( )} ∑ an t (n+1) q−1−v 𝜋 cos ((n + 1)q − 1 − v) + 2𝜋k L{R2 Cosq,v (a, k, t)} = L , Γ((n + 1)q − v) 2 n=0 t > 0, k = 0, 1, … , Dq Dv − 1.

(7.48)

Because the series converges uniformly (see Sections 3.16 and 14.3), we may integrate term-by-term; thus, { } ∞ ∑ t (n+1) q−1−v n L{R2 Cosq,v (a, k, t)} = a L cos{n𝛼q + 𝛽q,v } , Γ((n + 1)q − v) n=0 t > 0, k = 0, 1, … , Dq Dv − 1 ,

(7.49)

where 𝛼q = q(𝜋∕2 + 2𝜋k) and 𝛽q,v = (q − 1 − v)(𝜋∕2 + 2𝜋k). Then, L{R2 Cosq,v (a, k, t)} =

∞ ∑

an s−((n+1)q−v) cos{n𝛼q + 𝛽q,v } ,

n=0

k = 0, 1, … , Dq Dv − 1, q > v. Applying the cosine definition (equation (1.18)), this yields L{R2 Cosq,v (a, k, t)} =

∞ ∞ ei𝛽q,v ∑ an ein𝛼q e−i𝛽q,v ∑ an e−in𝛼q + , 2 n=0 s((n+1)q−v) 2 n=0 s((n+1)q−v)

q > v, k = 0, 1, … , Dq Dv − 1.

(7.50)

111

112

7 The R2 -Fractional Trigonometry

This may be written as L{R2 Cosq,v (a, k, t)} =

∞ ∞ ei𝛽q,v ∑ (aei 𝛼q )n e−i𝛽q,v ∑ (ae−i 𝛼q )n + , 2 sq−v n=0 snq 2 sq−v n=0 snq

q > v, k = 0, 1, … , Dq Dv − 1 .

(7.51)

Based on equation (2.5), we write ∞ ∑ (−b)n

snq

n=0

=

sq

sq . +b

(7.52)

Applying this relation, equation (7.51) may be written as L{R2 Cosq,v (a, k, t)} =

ei 𝛽q,v sv e−i 𝛽q,v sv + , 2 (sq − aei 𝛼q ) 2 (sq − a e−i 𝛼q ) q > v, k = 0, 1, … , Dq Dv − 1 ,

(7.53)

or inverse transforming 1 q−1−v Rq,v (aiq , t) + i−(q−1−v) Rq,v (ai−q , t)), q > v. (i 2 After some algebraic manipulation of equation (7.53), we obtain [ ] sv (ei 𝛽q,v + e−i 𝛽q,v )sq − a(e(𝛽q,v −𝛼q ) i + e−(𝛽q,v −𝛼q ) i ) L{R2 Cosq,v (a, k, t)} = , 2 s2q − a(e𝛼q i + e−𝛼q i )sq + a2 R2 Cosq,v (a, k, t) =

(7.54)

q > v, k = 0, 1, … , Dq Dv − 1 . The bracketed terms are recognized as cosine terms, giving L{R2 Cosq,v (a, k, t)} =

sv (cos(𝛽q,v )sq − a cos(𝛽q,v − 𝛼q )) s2q − 2 a cos(𝛼q )sq + a2

,

(7.55) q > v, k = 0, 1, … , Dq Dv − 1, ( ) where 𝛽q,v = (q − 1 − v) 𝜋2 + 2𝜋k , 𝛼q = q 𝜋2 + 2𝜋k , q = Nq ∕Dq , v = Nv ∕Dv , q > v, and t > 0. The following is an alternate statement of equations (7.55) and (7.53): (

)

L{R2 Cosq,v (a, k, t)} =

sv (cos(𝛽q,v )sq − a cos(𝛽q,v − 𝛼q )) (sq − a ei𝛼q ) (sq − a e−i𝛼q )

,

q > v, k = 0, 1, … , Dq Dv − 1.

(7.56)

The development of the Laplace transforms of the remaining functions follows the same pattern used here; the key results are therefore presented without detailed derivation. For the transforms that follow, the general notations 𝛼q = q(𝜋∕2 + 2𝜋k) and 𝛽q,v = (q − 1 − v)(𝜋∕2 + 2𝜋k) apply. Furthermore, q = Nq ∕Dq , v = Nv ∕Dv ,t ≥ 0, k = 0, 1, … Dq Dv − 1, and q ≥ v unless otherwise noted. The Laplace transforms are provided in the normal format and in a factored format. 7.3.2 R2 Sinq,v (a, k, t)

L{R2 Sinq,v (a, k, t)} =

ei 𝛽q,v sv e−i 𝛽q,v sv − , i 𝛼 2 i (sq − a e q ) 2 i (sq − a e−i 𝛼q ) q > v, k = 0, 1, … , Dq Dv − 1 ,

(7.57)

7.4 R2 -Trigonometric Function Relationships

L{R2 Sinq,v (a, k, t)} =

sv (sin(𝛽q,v )sq − a sin(𝛽q,v − 𝛼q )) s2q − 2 a cos(𝛼q )sq + a2

,

q > v, k = 0, 1, … , Dq Dv − 1 .

(7.58)

7.3.3 L{R2 Coflq,v (a, k, t)}

L{R2 Cof lq,v (a, k, t)} = L{R2 Cof lq,v (a, k, t)} = where 𝛼2q = 2q

(

𝜋 2

a e𝛽2q,v i sv a e−𝛽2q,v i sv + , 2(s2q − a2 e𝛼2q i ) 2(s2q − a2 e−𝛼2q i )

a sv (cos(𝛽2q,v )s2q − a2 cos(𝛽2q,v − 𝛼2q ))

) + 2𝜋 k and 𝛽2q,v

s4q − 2a2 cos(𝛼2q )s2q + a4 ( ) = (2q − 1 − v) 𝜋2 + 2𝜋 k .

7.3.4 L{R2 Flutq,v (a, k, t)}

1 L{R2 Flutq,v (a, k, t)} = 2i L{R2 Flutq,v (a, k, t)} = where 𝛼2q = 2q

(

𝜋 2

2q > v,

(

a e𝛽2q,v i sv 2q s − a2 e𝛼2q i

)

1 − 2i

(

2q > v,

a e−𝛽2q,v i sv 2q s − a2 e−𝛼2q i

a sv (sin(𝛽2q,v )s2q − a2 sin(𝛽2q,v − 𝛼2q ))

) + 2𝜋k and 𝛽2q,v

,

s4q − 2a2 cos(𝛼2q )s2q + a4 ( ) = (2q − 1 − v) 𝜋2 + 2𝜋 k .

(7.59)

,

(7.60)

) ,

(7.61)

2q > v,

(7.62)

7.3.5 L{R2 Covibq,v (a, k, t)}

L{R2 Covibq,v (a, k, t)} = L{R2 Covibq,v (a, k, t)} = where 𝛼2q = 2q

(

𝜋 2

) + 2𝜋 k and 𝛽q,v

7.3.6 L{R2 Vibq,v (a, k, t)}

L{R2 Vibq,v (a, k, t)} = L{R2 Vibq,v (a, k, t)} = where 𝛼2q

e𝛽q,v i sv+q 2i

e𝛽q,v i sv+q e−𝛽q,v i sv+q + , 𝛼2q i 2 2q − a e ) 2(s − a2 e−𝛼2q i )

(7.63)

2(s2q

sv+q (cos(𝛽q,v )s2q − a2 cos(𝛽q,v − 𝛼2q )) s4q − 2a2 cos(𝛼2q )s2q + a4 ( ) = (q − 1 − v) 𝜋2 + 2𝜋 k . (

1 2q s − a2 e𝛼2q i

) −

(

e−𝛽q,v i sv+q 2i

sv+q (sin(𝛽q,v )s2q − a2 sin(𝛽q,v − 𝛼2q ))

s4q − 2a2 cos(𝛼2q )s2q + a4 ( ) ( ) = 2q 𝜋2 + 2𝜋k and 𝛽q,v = (q − 1 − v) 𝜋2 + 2𝜋 k .

,

,

q > v,

1 2q s − a2 e−𝛼2q i q > v,

(7.64)

) ,

(7.65)

(7.66)

7.4 R2 -Trigonometric Function Relationships Various properties of the R2 -trigonometric functions in terms of the fractional exponential functions are derived here. Among these results are identities that parallel the basic equations (1.18) and (1.19). Some special R-function relationships also result. The results derived in the following are valid only for t > 0.

113

114

7 The R2 -Fractional Trigonometry

7.4.1 R2 Cosq,v (a, k, t) and R2 Sinq,v (a, k, t) Relationships and Fractional Euler Equation

We observe from equations (7.5)–(7.7) that Rq,v (a, k, it) = R2 Cosq,v (a, k, t) + iR2 Sinq,v (a, k, t),

k = 0, 1, 2, … , Dq Dv − 1.

This is similar to the form of equation (6.13) for the R1 -trigonometry. Consider ∞ ∞ ∑ an (−it)(n+1) q−1−v ∑ an (t)(n+1) q−1−v Rq,v (a, −it) = = (−i)(n+1)q−1−v . Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0

(7.67)

(7.68)

Now,

( )𝜙 [ ( )] [ ( )] 1 𝜋 𝜋 = i−𝜙 = cos −𝜙 + 2k𝜋 + i sin −𝜙 + 2k𝜋 i 2 2 [ ( )] [ ( )] 𝜋 𝜋 = cos 𝜙 + 2k𝜋 − i sin 𝜙 + 2k𝜋 . 2 2 Thus, equation (7.68) becomes ∞ ∑ an (t)(n+1)q−1−v Rq,v (a, k ∗ , −it) = Γ((n + 1)q − v) n=0 ) [ ( 𝜋 × cos[(n + 1)q − 1 − v] + 2𝜋k ∗ 2 )] ( 𝜋 (7.69) + 2𝜋k ∗ , − sin[(n + 1)q − 1 − v] 2 where k ∗ = 0, 1, 2, … , Dq Dv − 1. Although k and k ∗ have the same range, they may not be enumerated in the same direction; however, it is clear that when k = 0 and k ∗ = 0, the principal values are equal. Therefore, it may be seen that (−i)𝜙 =

Rq,v (a, −it) = R2 Cosq,v (a, t) − i R2 Sinq,v (a, t) ,

t > 0.

(7.70)

Consider now the R2 Cosq,v (a, k, t)-function as defined in equation (7.6), that is, ∞ { ( )} ∑ an t (n+1) q−1−v 𝜋 cos ((n + 1)q − 1 − v) + 2𝜋k , R2 Cosq,v (a, k, t) ≡ Γ((n + 1)q − v) 2 n=0 t > 0, k = 0, 1, 2, … , Dq Dv − 1 . Continuing from Ref. [74] with permission of Springer: Applying definition (1.18), we may write R2 Cosq,v (a, k, t) =

∞ ∑ an t (n+1) q−1−v Γ((n + 1)q − v) n=0 ) ) ( ( 𝜋 ⎛ i[(n+1)q−1−v] 𝜋 + 2𝜋k −i[(n+1)q−1−v] + 2𝜋k ⎞ ⎟ ⎜e 2 2 +e ×⎜ ⎟, 2 ⎟ ⎜ ⎠ ⎝

t > 0, k = 0, 1, 2, … , Dq Dv − 1 . (

i[(n+1)q−1−v]

𝜋 2

)

+2𝜋k

= i [(n+1)q−1−v] , and (3.127), we have ( [(n+1)q−1−v] −[(n+1)q−1−v] ) ∞ ∑ an t (n+1) q−1−v +i i . R2 Cosq,v (a, t) = Γ((n + 1)q − v) 2 n=0

Now, by equation (3.124), e

(7.71)

7.4 R2 -Trigonometric Function Relationships

This may be written as ∞ ∑ an t (n+1)q−1−v R2 Cosq,v (a, t) = Γ((n + 1)q − v) n=0

(

Then, combining terms R2 Cosq,v (a, t) =

∞ ∑ n=0

an Γ((n + 1)q − v)

i[(n+1)q−1−v] + (−i)[(n+1)q−1−v] 2 (

) ,

t > 0.

(i t)[(n+1)q−1−v] + (−i t)[(n+1)q−1−v] 2

) ,

t > 0, or

{ 1 R2 Cosq,v (a, t) = 2

∞ ∑ an (i t)[(n+1)q−1−v] n=0

Γ((n + 1)q − v)

+

∞ ∑ an (−i t)[(n+1)q−1−v] n=0

} ,

Γ((n + 1)q − v)

t > 0.

(7.72)

Recognizing the summations as R-functions, we write R2 Cosq,v (a, t) =

Rq,v (a, it) + Rq,v (a, −it) 2

,

t > 0.

(7.73)

This equation exactly parallels equation (1.18) in the ordinary trigonometry. It could also serve as a definition of R2 Cosq,v (a, t). The development for the R2 Sinq,v (a, t)-function follows in a similar manner, and thus, will not be detailed. The result for the R2 Sinq,v (a, t)-function is Rq,v (a, it) − Rq,v (a, −it)

, t > 0. (7.74) 2i This equation exactly parallels equation (1.19) in the classical trigonometry. It could also serve as a definition of R2 Sinq,v (a, t). Continuing from Ref. [74] with permission of Springer: R2 Sinq,v (a, t) =

Now, squaring equations (7.73) and (7.74) yields for t > 0 R2 Cos2q,v (a, t) = R2 Sin2q,v (a, t) =

R2q,v (a, it) + R2q,v (a, −it) + 2Rq,v (a, it)Rq,v (a, −it) 4

,

−R2q,v (a, it) − R2q,v (a, −it) + 2Rq,v (a, it)Rq,v (a, −it)

4 Adding and subtracting equations (7.75) and (7.76) gives R2 Cos2q,v (a, t) + R2 Sin2q,v (a, t) = Rq,v (a, it)Rq,v (a, −it),

t > 0,

(7.75) .

(7.76)

(7.77)

and 1 2 (7.78) (R (a, it) + R2q,v (a, −it)), t > 0 . 2 q,v These equations parallel equations (6.23) and (6.24) and generalize the identities cos2 (t) + sin2 (t) = 1 and cos2 (t) − sin2 (t) = cos(2t). We rewrite equation (7.73) and (7.74) as (7.79) Rq,v (a, it) + Rq,v (a, −it) = 2R2 Cosq,v (a, t), t > 0, R2 Cos2q,v (a, t) − R2 Sin2q,v (a, t) =

and Rq,v (a, it) − Rq,v (a, −it) = 2 i R2 Sinq,v (a, t),

t > 0.

(7.80)

115

116

7 The R2 -Fractional Trigonometry

Adding equations (7.79) and (7.80) returns the fractional Euler equations for the R2 -trigonometry Rq,v (a, it) = R2 Cosq,v (a, t) + i R2 Sinq,v (a, t),

t > 0.

(7.81)

This equation is the fractional generalization of equation (1.29) and contains it as a special case, that is, when q = 1 and v = 0. Subtracting equation (7.80) from (7.79) gives the complementary equation Rq,v (a, −it) = R2 Cosq,v (a, t) − i R2 Sinq,v (a, t),

t > 0.

(7.82)

7.4.2 R2 Rotq,v (a, t) and R2 Corq,v (a, t) Relationships

As previously mentioned, R2 Rotq,v (a, t) and R2 Corq,v (a, t) are complex functions. From the definitions (equations (7.17)–(7.19)), we have Rq,v (a, it) = R2 Corq,v (a, t) + R2 Rotq,v (a, t),

t > 0.

(7.83)

The summations of equations (7.18) and (7.19) are R-functions of complex arguments, that is, R2 Corq,v (a, t) = R2q,v+q (a2 , it),

t > 0,

(7.84)

R2 Rotq,v (a, t) = aR2q,v (a2 , it) ,

t > 0.

(7.85)

Combining the results of equations (7.84) and (7.85) with equation (7.83) gives the following identity: (7.86) Rq,v (a, it) = R2q,v+q (a2 , it) + aR2q,v (a2 , it), t > 0. 7.4.3 R2 Coflq,v (a, t) and R2 Flutq,v (a, t) Relationships

The analysis starts from the Laplace transform of R2 Cof lq,v (a, k, t), that is, equation (7.63) a e𝛽2q,v i a e−𝛽2q,v i sv sv L{R2 Cof lq,v (a, k, t)} = + , (7.87) 𝛼 i 2 s2q − a2 e 2q 2 s2q − a2 e−𝛼2q i ( ) ( ) where 𝛼2q = 2q 𝜋2 + 2𝜋 k and 𝛽2q,v = (2q − 1 − v) 𝜋2 + 2𝜋 k . Continuing from Ref. [74] with permission of Springer: Performing the inverse transform using equation (3.22), we have ( ( ) a e−𝛽2q,v i ) a e𝛽2q,v i R2 Cof lq,v (a, t) = R2q,v a2 e𝛼2q i , t + R2q,v a2 e−𝛼2q i , t , 2 2 t > 0.

(7.88)

Applying equation (3.118), we write R2q,v (a2 e𝛼2q i , t) = (e𝛼2q i )

− (2q−1−v) 2q

and R2q,v (a2 e−𝛼2q i , t) = (e−𝛼2q i ) Now ( i

𝜋 2

+2𝜋k

)e

(𝛽2q,v −𝛼2q (2q−1−v)∕2q)i

=

− (2q−1−v) 2q

e0(= 1 and ) −i 𝜋2 +2𝜋k

e = i and e−𝛼2q i∕2q = e written as

R2q,v (a2 , e𝛼2q i∕2q t) ,

e

t > 0,

R2q,v (a2 , e−𝛼2q i∕2q t) ,

(−𝛽2q,v +𝛼2q (2q−1−v)∕2q)i

t > 0. 0

= e = 1,

also

e𝛼2q i∕2q =

= −i. With these results, equation (7.88) may be

7.4 R2 -Trigonometric Function Relationships

( R2 Coflq,v (a, t) = a

R2q,v (a2 , it) + R2q,v (a2 , −it)

)

2

,

t > 0.

(7.89)

The R2 Flutq,v (a, t)-function is derived in a similar manner. We give the final result: ( ) R2q,v (a2 , it) − R2q,v (a2 , −it) , t > 0. (7.90) R2 Flutq,v (a, t) = a 2i We now derive some additional properties of these functions. Rewrite equations (7.89) and (7.90) as 2 (7.91) R2q,v (a2 , it) + R2q,v (a2 , −it) = R2 Cof lq,v (a, t), t > 0, a R2q,v (a2 , it) − R2q,v (a2 , −it) =

2i R Flutq,v (a, t), a 2

t > 0.

(7.92)

Adding equations (7.91) and (7.92) gives a R2q,v (a2 , it) = R2 Coflq,v (a, t) + i R2 Flutq,v (a, t) = R2 Rotq,v (a, t),

t > 0.

(7.93)

This equation might be considered as a “fractional semi-Euler” equation as discussed later. Subtracting equation (7.92) from equation (7.91) yields the complimentary equation a R2q,v (a2 , −it) = R2 Coflq,v (a, t) − i R2 Flutq,v (a, t) = R2 Rotq,v (a2 , −it),

t > 0.

(7.94)

2

Letting 2q = u and a = b in equation (7.93) gives √ √ √ ± b Ru,v (b, it) = R2 Cofl u ,v (± b, t) + i R2 Flut u ,v (± b, t), 2

t > 0.

2

Alternatively, we may write ( ) √ √ 1 Rq,v (a, it) = √ R2 Cofl q ,v (± a, t) + i R2 Flut q ,v (± a, t) , 2 2 ± a

t > 0.

(7.95)

(7.96)

Equating the real parts to real parts and imaginary parts to imaginary parts with the fractional Euler equation (7.67), namely Rq,v (a, it) = R2 Cosq,v (a, t) + i R2 Sinq,v (a, t), we have the relationships

t > 0,

(7.97)

R2 Cosq,v (a, t) =

√ 1 √ R2 Cof l q2 ,v (± a, t) , ± a

t > 0,

(7.98)

R2 Sinq,v (a, t) =

√ 1 √ R2 Flut q2 ,v (± a, t) , ± a

t > 0.

(7.99)

and

Forms paralleling equations (7.77) and (7.78) have also been derived for the R2 Cof lq,v (a, t)and R2 Flutq,v (a, t)-functions [74]. These are 2 2 (a, t) + R2 Flutq,v (a, t) = a2 [R2q,v (a2 , it)R2q,v (a2 , −it)], R2 Coflq,v

t > 0,

(7.100)

and 2 2 (a, t) − R2 Flutq,v (a, t) = R2 Coflq,v

a2 2 [R (a2 , it) + R22q,v (a2 , −it)], 2 2q,v

t > 0.

(7.101)

117

118

7 The R2 -Fractional Trigonometry

7.4.4 R2 Covibq,v (a, t) and R2 Vibq,v (a, t) Relationships

This analysis also starts with the Laplace transform of the R2 Covibq,v (a, k, t)-function. From equation (7.63) e𝛽q,v i e−𝛽q,v i sv+q sv+q + , q > v, (7.102) 𝛼 i 2 s2q − a2 e 2q 2 s2q − a2 e−𝛼2q i ( ) ( ) where 𝛼2q = 2q 𝜋2 + 2𝜋 k and 𝛽q,v = (q − 1 − v) 𝜋2 + 2𝜋 k . Performing the inverse transform using equation (3.22), we have L{R2 Covibq,v (a, k, t)} =

e𝛽q,v i e−𝛽q,v i (a2 e−𝛼2q i , t), R2q,v+q (a2 e𝛼2q i , t) + R 2 2 2q,v+q Applying equation (3.118), we write

t > 0.

R2 Covibq,v (a, k, t) =

R2q,v+q (a2 e𝛼2q i , t) = (e𝛼2q i )

− (q−1−v) 2q

and R2q,v+q (a2 e−𝛼2q i , t) = (e−𝛼2q i )

R2q,v+q (a2 , e𝛼2q i∕2q t) ,

− (q−1−v) 2q

t > 0,

R2q,v+q (a2 , e−𝛼2q i∕2q t) ,

t > 0. (

Now e(𝛽q,v −𝛼2q (q−1−v)∕2q)i ( ) −i 𝜋2 +2𝜋k −𝛼2q i∕2q

e

0

= e = 1 and e

(−𝛽q,v +𝛼2q (q−1−v)∕2q)i

0

(7.103)

= e = 1, also e

𝛼2q i∕2q

=e

i

𝜋 2

= −i. With these results, equation (7.103) may be written as ( ) R2q,v+q (a2 , it) + R2q,v+q (a2 , −it) R2 Covibq,v (a, t) = , q > v, t > 0. 2

) +2𝜋k

= i and

=e

The derivation for the R2 Vibq,v (a, t) follows in a similar manner, giving ( ) R2q,v+q (a2 , it) − R2q,v+q (a2 , −it) , q > v, t > 0. R2 Vibq,v (a, t) = 2i

(7.104)

(7.105)

We now derive some additional properties of these functions. Rewrite equations (7.104) and (7.105) as R2q,v+q (a2 , it) + R2q,v+q (a2 , −it) = 2 R2 Covibq,v (a, t), R2q,v+q (a2 , it) − R2q,v+q (a2 , −it) = 2 i R2 Vibq,v (a, t),

t > 0, t > 0.

(7.106) (7.107)

Adding equations (7.106) and (7.107) gives an additional Euler-like equation: R2q,v+q (a2 , it) = R2 Covibq,v (a, t) + i R2 Vibq,v (a, t),

t > 0.

(7.108)

This equation may also be considered as a “fractional semi-Euler” equation as discussed later. Subtracting equation (7.107) from equation (7.106) yields the complimentary equation R2q,v+q (a2 , −it) = R2 Covibq,v (a, t) − i R2 Vibq,v (a, t),

t > 0.

Let 2q = u and a2 = b in equation (7.108), then √ √ Ru,v+ u (b, it) = R2 Covib u ,v (± b, t) + i R2 Vib u ,v (± b, t), 2

2

2

(7.109)

t > 0.

(7.110)

t > 0,

(7.111)

This may also be written as ( ) √ √ Rq,v+ q (a, it) = R2 Covib q ,v (± a, t) + i R2 Vib q ,v (± a, t) , 2

(

2

2

) √ √ Rq,v (a, it) = R2 Covib q ,v− q (± a, t) + i R2 Vib q ,v− q (± a, t) , 2

2

2

2

7.5 Fractional Calculus Operations on the R2 -Trigonometric Functions

another fractional semi-Euler equation. Comparing the real and imaginary parts with the fractional Euler equation (7.67) Rq,v (a, it) = R2 Cosq,v (a, t) + i R2 Sinq,v (a, t),

t > 0,

(7.112)

we have the relationships

√ R2 Cosq,v (a, t) = R2 Covib q ,v− q (± a, t) , 2

and

2

√ R2 Sinq,v (a, t) = R2 Vib q ,v− q (± a, t) , 2

t>0

(7.113)

t > 0.

2

(7.114)

Then, using equations (7.67), (7.96) together with equation (7.111), we obtain 1 R2 Cosq,v (a2 , t) = R2 Cof l q ,v (a, t) = R2 Covib q ,v− q (a, t) , t > 0, 2 2 2 a

(7.115)

1 (7.116) R Flut q ,v (a, t) = R2 Vib q ,v− q (a, t) , t > 0. 2 2 2 a 2 Forms paralleling equations (7.115) and (7.116) have also been derived for the R2 Covibq,v (a, t)and R2 Vibq,v (a, t)-functions [74]. These are R2 Sinq,v (a2 , t) =

R2 Covib2q,v (a, t) + R2 Vib2q,v (a, t) = R2q,v+q (a2 , it)R2q,v+q (a2 , −it),

t > 0,

(7.117)

and 1 2 (a2 , it) + R22q,v+q (a2 , −it)], t > 0 . (7.118) [R 2 2q,v+q Many more such relationships are possible for the R2 -trigonometric functions. Relationships such as the multiple-angle and fractional-angle formulas from the integer trigonometry and more are yet to be developed. The following section derives some fractional calculus relationships. R2 Covib2q,v (a, t) − R2 Vib2q,v (a, t) =

7.5 Fractional Calculus Operations on the R2 -Trigonometric Functions 7.5.1 R2 Cosq,v (a, k, t)

The 𝛼-order differintegral of R2 Cosq,v (a, k, t) is determined as 𝛼 𝛼 0 dt R2 Cosq,v (a, k, t) = 0 dt

∞ ∑ n=0

{ ( )} an t (n+1) q−1−v 𝜋 cos ((n + 1)q − 1 − v) + 2𝜋 k , Γ((n + 1) q − v) 2 t > 0,

k = 0, 1, … Dq Dv − 1. (7.119)

Based on Section 3.16, we may term-by-term differintegrate ∞ { ( )} ∑ an 0 dt𝛼 t (n+1) q−1−v 𝜋 𝛼 d R Cos (a, k, t) = cos ((n + 1)q − 1 − v) + 2𝜋 k , 0 t 2 q,v Γ((n + 1) q − v) 2 n=0 t > 0,

k = 0, 1, … Dq Dv − 1.

(7.120)

Applying equation (5.37), that is, 𝛼 p 0 dt x

=

Γ(p + 1)xp−𝛼 , Γ(p − 𝛼 + 1)

p > −1 .

(7.121)

119

120

7 The R2 -Fractional Trigonometry

Continuing from Ref. [74], with permission of Springer: Thus, equation (7.120) becomes 𝛼 0 dt R2 Cosq,v (a, k, t)

=

∞ ∑ n=0

{ ( )} an t (n+1) q−1−v−𝛼 𝜋 cos ((n + 1) q − 1 − v) + 2𝜋 k , Γ((n + 1) q − v − 𝛼) 2

q > v, t > 0, k = 0, 1, ...., Dq Dv − 1 .

(7.122)

From the sum and difference formulas for the integer-order trigonometry, the following identities may be derived: cos(x) = cos(y) cos(x − y) − sin(y) sin(x − y),

(7.123)

sin(x) = cos(y) sin(x − y) + sin(y) cos(x − y).

(7.124)

and Now, in equation (7.122), let ) ( 𝜋 + 2𝜋 k , x = ((n + 1) q − 1 − v) 2 ( ) 𝜋 y=𝛼 + 2𝜋k . 2 Then, applying equation (7.123), 𝛼 0 dt R2 Cosq,v (a, k, t)

= cos(y)

∞ ∑ n=0

− sin(y)

k = 0, 1, … , Dq Dv − 1, also, let

an t (n+1) q−1−v−𝛼 cos{x − y} Γ((n + 1) q − v − 𝛼)

∞ ∑ n=0

an t (n+1) q−1−v−𝛼 sin{x − y} , Γ((n + 1) q − v − 𝛼)

t > 0, q > v, k = 0, 1, … , Dq Dv − 1 .

(7.125)

The summations are recognized as R2 Cosq,v+𝛼 (a, k, t) and R2 Sinq,v+𝛼 (a, k, t), respectively, yielding the final result 𝛼 0 dt R2 Cosq,v (a, k, t)

= cos(y)R2 Cosq,v+𝛼 (a, k, t) − sin(y)R2 Sinq.v+𝛼 (a, k, t)

(7.126) t > 0, q > v, k = 0, 1, … , Dq Dv − 1 , ) 𝜋 where y = 𝛼 + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving 2 (

0 dt R2 Cosq,v (a, k, t)

= −R2 Sinq.v+1 (a, k, t), t > 0, q > v, k = 0, 1, … , Dq Dv − 1.

(7.127)

We may check backward compatibility by taking q = 1 and v = 0 0 dt R2 Cos1,0 (a, k, t)

= −R2 Sin1,1 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1 ,

which evaluates to 0 dt

as expected.

cos(at) = −a sin(at),

t > 0,

(7.128)

7.5 Fractional Calculus Operations on the R2 -Trigonometric Functions

An interesting alternative development of the result of equation (7.127) is obtained as follows: ] [ Rq,v (a, it) + Rq,v (a, −it) 𝛼 𝛼 d R Cos (a, t) = d , t > 0. (7.129) 0 t 2 q,v 0 t 2 By the differentiation equation (3.114), 𝛼 0 dt R2 Cosq,v (a, t) =

i𝛼 Rq,v+𝛼 (a, it) + (−i)𝛼 Rq,v+𝛼 (a, −it) 2

,

t > 0.

(7.130)

When 𝛼 = 1, we have 0 dt R2 Cosq,v (a, t)

=

iRq,v+1 (a, it) − iRq,v+1 (a, −it)

=

2 −Rq,v+1 (a, it) + Rq,v+1 (a, −it) 2i

, t > 0,

which is recognized as 0 dt R2 Cosq,v (a, t)

(7.131)

= −R2 Sinq,v+1 (a, t).

7.5.2 R2 Sinq,v (a, k, t)

Determination of the derivative for the R2 Sinq,v (a, k, t)-function proceeds in a manner similar to that of R2 Cosq,v (a, k, t). The key results for the 𝛼-order differintegral of R2 Sinq,v (a, k, t) ( ) follow. Using equation (7.124) with x = ((n + 1)q − 1 − v) 𝜋2 + 2𝜋 k , k = 0, 1, … , Dq Dv − 1, ) ( q > v, and y = 𝛼 𝜋2 + 2𝜋k , we have 𝛼 0 dt R2 Sinq,v (a, k, t)

= cos(y)

∞ ∑ n=0

+ sin(y)

an t (n+1) q−1−v−𝛼 sin{x − y} Γ((n + 1)q − v − 𝛼)

∞ ∑ n=0

an t (n+1) q−1−v−𝛼 cos{x − y} , Γ((n + 1)q − v − 𝛼)

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.132)

The summations are recognized as R2 Sinq,v+𝛼 (a, k, t) and R2 Cosq,v+𝛼 (a, k, t), respectively, yielding the final result 𝛼 0 dt R2 Sinq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R2 Sinq,v+𝛼 (a, k, t) + sin(y)R2 Cosq.v+𝛼 (a, k, t) q > v, t > 0, k = 0, 1, … Dq Dv − 1 ,

)

(7.133)

+ 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R2 Sinq,v (a, k, t)

= R2 Cosq.v+1 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.134)

Backward compatibility is obtained by taking q = 1 and v = 0; thus, 0 dt R2 Sin1,0 (a, k, t)

= R2 Cos1,1 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1 ,

(7.135)

which evaluates to 0 dt

sin(at) = a cos(at), t > 0,

(7.136)

again the expected result. An alternative R-function-based development using equation (7.74) yields 𝛼 0 dt R2 Sinq,v (a, t) =

i𝛼−1 Rq,v+1 (a, it) − i−𝛼−1 Rq,v+1 (a, −it) 2

, t > 0,

(7.137)

121

122

7 The R2 -Fractional Trigonometry

when 𝛼 = 1, we have 0 dt R2 Sinq,v (a, t)

=

Rq,v+1 (a, it) + Rq,v+1 (a, −it) 2

,

t > 0,

(7.138)

which is recognized as 0 dt R2 Sinq,v (a, t)

= R2 Cosq,v+1 (a, t),

t > 0.

(7.139)

7.5.3 R2 Corq,v (a, t)

The 𝛼-order differintegral of R2 Corq,v (a, t) is determined as 𝛼 𝛼 0 dt R2 Corq,v (a, t) = 0 dt

𝛼 0 dt R2 Corq,v (a,

t) =

∞ ∑ a2n (it)(n+1) 2q−1−v−q , Γ((n + 1)2q − v − q) n=0

t > 0,

∞ ∑ a2n i(n+1)2q−1−v−q 0 dt𝛼 t (n+1)2q−1−v−q n=0

Γ((n + 1)2q − v − q)

,

(7.140)

t > 0.

Application of equation (7.121) to this equation gives 𝛼 0 dt R2 Corq,v (a, t)

=

∞ ∑ a2n i(n+1)2q−1−v−q t (n+1)2q−1−v−q−𝛼 n=0

Γ((n + 1)2q − v − q − 𝛼)

,

t > 0,

(7.141)

with q > v. The summation is recognized as both an R-function and as an R2 Cor-function, yielding the final result 𝛼 0 dt R2 Corq,v (a, t)

= i𝛼 R2q,v+q+𝛼 (a2 , i t) = i𝛼 R2 Corq,v+𝛼 (a, t),

q > v,

t > 0.

(7.142)

2q > v,

t > 0.

(7.143)

7.5.4 R2 Rotq,v (a, t)

Similarly, the 𝛼-order differintegral of R2 Rotq,v (a, t) is 𝛼 0 dt R2 Rotq,v (a, t)

= a i𝛼 R2q,v+𝛼 (a2 , it) = i𝛼 R2 Rotq,v+𝛼 (a, t),

Only key results are given for the remaining functions. 7.5.5 R2 Coflutq,v (a, t)

The 𝛼-order differintegral of R2 Coflutq,v (a, k, t) is determined as 𝛼 0 dt R2 Coflutq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R2 Coflutq,v+𝛼 (a, k, t) − sin(y)R2 Flutq,v+𝛼 (a, k, t)

(7.144) t > 0, 2q > v, 𝛼 ≥ 0, k = 0, 1, … Dq Dv − 1 , ) + 2𝜋k . Taking 𝛼 = 1, 2q > v, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R2 Coflutq,v (a, k, t)

= −R2 Flutq,v+1 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1.

(7.145)

Taking q = 1 and v = 0 gives 𝛼 0 dt R2 Coflut1,0 (a, k, t)

= cos(y)R2 Coflut1,𝛼 (a, k, t) − sin(y)R2 Flut1,𝛼 (a, k, t) , k = 0, 1, … , Dq Dv − 1 ,

(7.146)

and with 𝛼 = 1, we have 0 dt R2 Coflut1,0 (a, k, t)

= −R2 Flut1,1 (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.147)

7.5 Fractional Calculus Operations on the R2 -Trigonometric Functions

However, R2 Coflut1,0 (a, k, t) = 0; thus, R2 Flut1,1 (a, k, t) = 0,

k = 0, 1, … , Dq Dv − 1,

t > 0.

(7.148)

The alternative R-function-based development gives 𝛼 0 dt R2 Coflutq,v (a, t) =

i𝛼 R2q,v+1 (a2 , it) + i−𝛼 R2q,v+1 (a2 , −it) 2

,

t > 0.

(7.149)

7.5.6 R2 Flutq,v (a, k, t)

The 𝛼-order differintegral of R2 Flutq,v (a, k, t) is given as 𝛼 0 dt R2 Flutq,v (a, k, t)

(

= cos(y)R2 Flutq,v+𝛼 (a, k, t) + sin(y)R2 Coflutq,v+𝛼 (a, k, t) t > 0, 2q > v, k = 0, 1, … , Dq Dv − 1 ,

)

(7.150)

𝜋 + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, with 2q > v, gives 2 t > 0, k = 0, 1, … , Dq Dv − 1 . (7.151) 0 dt R2 Flutq,v (a, k, t) = R2 Coflutq,v+1 (a, k, t),

where y = 𝛼

Taking q = 1 and v = 0, in equation (7.150), yields 𝛼 0 dt R2 Flut1,0 (a, k, t)

= cos(y)R2 Flut1,𝛼 (a, k, t) + sin(y)R2 Coflut1,𝛼 (a, k, t) t > 0, k = 0, 1, … Dq Dv − 1 .

(7.152)

When we also have 𝛼 = 1, this becomes 0 dt R2 Flut1,0 (a, k, t)

t > 0, k = 0, 1, … , Dq Dv − 1 .

= R2 Coflut1,1 (a, k, t),

(7.153)

However, from equation (7.33),R2 Flut1,0 (a, k, t) = sin(a t), thus indicating that t > 0, k = 0, 1, … , Dq Dv − 1 .

R2 Coflut1,1 (a, k, t) = a cos(a t),

The alternative R-function-based development yields [ 𝛼 ] i R2q,v+𝛼 (a2 , it) − i−𝛼 R2q,v+𝛼 (a2 , −it) 𝛼 , 0 dt R2 Flutq,v (a, t) = a 2i

t > 0.

(7.154)

(7.155)

7.5.7 R2 Covibq,v (a, k, t)

The 𝛼-order differintegral of R2 Covibq,v (a, k, t) is 𝛼 0 dt R2 Covibq,v (a, k, t)

where y = 𝛼

(

𝜋 2

)

= cos(y)R2 Covibq,v+𝛼 (a, k, t) − sin(y)R2 Vibq,v+𝛼 (a, k, t) q > v, t > 0, k = 0, 1, … , Dq Dv − 1 ,

(7.156)

+ 2𝜋k . With 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R2 Covibq,v (a, k, t)

= −R2 Vibq,v+1 (a, k, t),

q > v, t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.157)

In addition, with q = 1 and v = 0, we obtain 0 dt R2 Covib1,0 (a, k, t)

= −R2 Vib1,1 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1 .

From equation (7.34), R2 Covib1,0 (a, k, t) = cos(a t); thus, we have R2 Vib1,1 (a, k, t) = a sin(a t),

t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.158)

123

124

7 The R2 -Fractional Trigonometry

Alternatively, 𝛼 0 dt R2 Covibq,v (a, t)

=

i𝛼 R2q,v+q+𝛼 (a2 , it) + i−𝛼 R2q,v+q+𝛼 (a2 , −it) 2

,

t > 0.

(7.159)

When 𝛼 = 1, we have 0 dt R2 Covibq,v (a, t)

=−

R2q,v+q+1 (a2 , it) − R2q,v+q+1 (a2 , −it)

= −R2 Vibq,v+1 (a, t),

2i t > 0.

(7.160)

7.5.8 R2 Vibq,v (a, k, t)

The 𝛼-order differintegral of R2 Vibq,v (a, k, t) is determined as 𝛼 0 dt R2 Vibq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R2 Vibq,v+𝛼 (a, k, t) + sin(y)R2 Covibq,v+𝛼 (a, k, t)

q > v, t > 0, k = 0, 1, … , Dq Dv − 1 , ) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R2 Vibq,v (a, k, t)

= R2 Covibq,v+1 (a, k, t), q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

(7.161)

(7.162)

Now, taking q = 1 and v = 0, in equation (7.161), gives 𝛼 0 dt R2 Vib1,0 (a, k, t)

= cos(y)R2 Vib1,𝛼 (a, k, t) + sin(y)R2 Covib1,𝛼 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1 .

(7.163)

With both 𝛼 = 1 and with q = 1 and v = 0, we have 0 dt R2 Vib1,0 (a, k, t)

t > 0, k = 0, 1, … , Dq Dv − 1 .

= R2 Covib1,1 (a, k, t),

However, R2 Vib1,0 (a, k, t) = 0; therefore, we have R2 Covib1,1 (a, k, t) = 0.

(7.164)

The alternative R-function-based development of the results of equation (7.105) is obtained as i𝛼 R2q,v+q+𝛼 (a2 , it) − (−i)𝛼 R2q,v+q+𝛼 (a2 , −it) 𝛼 d R Vib (a, t) = , t > 0. (7.165) 0 t 2 q,v 2i When 𝛼 = 1, we have 0 dt R2 Vibq,v (a, t) =

R2q,v+q+1 (a2 , it) + R2q,v+q+1 (a2 , −it) 2

,

t > 0,

(7.166)

which is recognized as 0 dt R2 Vibq,v (a, t)

= R2 Covibq,v+1 (a, t), t > 0.

(7.167)

Tables 7.1 and 7.2 summarize the various properties of the R2 -trigonometric functions. Continuing from Ref. [74], with permission of Springer: 7.5.9

Summary of Fractional Calculus Operations on the R2 -Trigonometric Functions

For ease of reference, the fractional calculus operations are ( ) summarized here. The derivations are for t > 0, k = 0, 1, … , Dq Dv − 1, and y = 𝛼 𝜋2 + 2𝜋k . For all cases, except the Flut and

For this table 𝛼q = q

( ) ) 𝜋 𝜋 + 2𝜋k , 𝛽q,v = (q − 1 − v) + 2𝜋 k , t > 0, and k = 0, 1, 2 · · · Dq Dv − 1. 2 2 Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

_

∞ ∑ a2n (it)(n+1) 2q−1−v−q Γ((n + 1)2q − v − q) n=0

R2 Corq,v (a, t)

(

_

s4q − 2a2 cos(𝛼2q )s2q + a4

q>v

, 2q > v

, 2q > v a sv (cos(𝛽2q,v )s2q − a2 cos(𝛽2q,v − 𝛼2q ))

s4q − 2a2 cos(𝛼2q )s2q + a4

a sv (sin(𝛽2q,v )s2q − a2 sin(𝛽2q,v − 𝛼2q ))

s4q − 2a2 cos(𝛼2q )s2q + a4

,

,q > v

q>v

sv+q (cos(𝛽q,v )s2q − a2 cos(𝛽q,v − 𝛼2q ))

s4q − 2a2 cos(𝛼2q )s2q + a4

sv+q (sin(𝛽q,v )s2q − a2 sin(𝛽q,v − 𝛼2q ))

s2q − 2 a cos(𝛼q )sq + a2

,

, q>v

sv (cos(𝛽q,v )sq − a cos(𝛽q,v − 𝛼q ))

∞ ∑ a2n+1 (it)(n+1) 2q−1−v n=0 Γ((n + 1)2q − v)

( ( )) ∞ ∑ a2n+1 (t)(n+1) 2q−1−v 𝜋 cos ((n + 1)2q − 1 − v) + 2𝜋 k 2 n=0 Γ((n + 1)2q − v)

( ( )) ∞ ∑ a2n+1 (t)(n+1) 2q−1−v 𝜋 sin ((n + 1)2q − 1 − v) + 2𝜋 k 2 n=0 Γ((n + 1)2q − v)

( ( )) ∞ ∑ a2n (t)(n+1) 2q−1−v−q 𝜋 cos ((n + 1)2q − 1 − v − q) + 2𝜋 k 2 n=0 Γ((n + 1)2q − v − q)

( ( )) ∞ ∑ a2n (t)(n+1) 2q−1−v−q 𝜋 sin ((n + 1)2q − 1 − v − q) + 2𝜋 k 2 n=0 Γ((n + 1)2q − v − q)

sv (sin(𝛽q,v )sq − a sin(𝛽q,v − 𝛼q ))

{ ( )} ∞ ∑ an t (n+1) q−1−v 𝜋 sin ((n + 1)q − 1 − v) + 2𝜋k 2 n=0 Γ((n + 1)q − v) { ( )} ∞ n (n+1) q−1−v ∑ a t 𝜋 cos ((n + 1)q − 1 − v) + 2𝜋k 2 n=0 Γ((n + 1)q − v) s2q − 2 a cos(𝛼q )sq + a2

Laplace transform

Series definition

R2 Rotq,v (a, t)

R2 Cof lq,v (a, k, t)

R2 Flutq,v (a, k, t)

R2 Covibq,v (a, k, t)

R2 Vibq,v (a, k, t)

R2 Cosq,v (a, k, t)

R2 Sinq,v (a, k, t)

Function

Table 7.1 Summary of R2 -functions.

Oa (Rq,v (ia, it)) Ea (Rq,v (ia, it))

a R2q,v (a2 , i t)

R2q,v+q (a2 , i t)

R2 Corq,v (a, t)

For this table t > 0, k = 0, 1, 2, … , Dq Dv − 1. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

2

R2q,v (a2 , it) + R2q,v (a2 , −it)

Re(Oa (Rq,v (ia, k, it))) = Oa (Re(Rq,v (ia, k, it)))

R2 Rotq,v (a, t)

Im(Oa (Rq,v (ia, k, it))) = Oa (Im(Rq,v (ia, k, it)))

a

2i )

)

R2 Cof lq,v (a, k, t)

(

R2q,v (a2 , it) − R2q,v (a2 , −it)

Re(Ea (Rq,v (ia, k, it))) = Ea (Re(Rq,v (ia, k, it)))

Im(Ea (Rq,v (ia, k, it))) = Ea (Im(Rq,v (ia, k, it)))

Re(Rq,v (ia, k, it))

Im(Rq,v (ia, k, it))

Relation to Rq,v (a, it)

a

(

2

R2q,v+q (a2 , it) + R2q,v+q (a2 , −it)

2i

R2q,v+q (a2 , it) − R2q,v+q (a2 , −it)

2

Rq,v (a, it) + Rq,v (−a, −it)

2i

Rq,v (a, it) − Rq,v (−a, −it)

R-Function equivalent

R2 Flutq,v (a, k, t)

R2 Covibq,v (a, k, t)

R2 Vibq,v (a, k, t)

R2 Cosq,v (a, k, t)

R2 Sinq,v (a, k, t)

Function

Table 7.2 Summary of R2 -functions.

cosh(iat) = cos(at)

sinh(iat) = i sin(at)

0

sin(at)

cos(at)

0

cos(at)

sin(at)

q = 1, v = 0, k = 0

7.6 Inferred Fractional Differential Equations

Coflut, q > v, for the Flut and Coflut, we have 2q > v. 𝛼 0 dt R2 Cosq,v (a, k, t)

= cos(y)R2 Cosq,v+𝛼 (a, k, t) − sin(y)R2 Sinq.v+𝛼 (a, k, t),

(7.168)

𝛼 0 dt R2 Sinq,v (a, k, t)

= cos(y)R2 Sinq,v+𝛼 (a, k, t) + sin(y)R2 Cosq.v+𝛼 (a, k, t),

(7.169)

𝛼 0 dt R2 Corq,v (a, t)

= i𝛼 R2q,v+q+𝛼 (a2 , i t) = i𝛼 R2 Corq,v+𝛼 (a, t),

(7.170)

𝛼 0 dt R2 Rotq,v (a, t)

= a i𝛼 R2q,v+𝛼 (a2 , i t) = i𝛼 R2 Rotq,v+𝛼 (a, t),

(7.171)

𝛼 0 dt R2 Coflutq,v (a, k, t)

= cos(y)R2 Coflutq,v+𝛼 (a, k, t) − sin(y)R2 Flutq,v+𝛼 (a, k, t),

(7.172)

𝛼 0 dt R2 Flutq,v (a, k, t)

= cos(y)R2 Flutq,v+𝛼 (a, k, t) + sin(y)R2 Coflutq,v+𝛼 (a, k, t),

(7.173)

𝛼 0 dt R2 Covibq,v (a, k, t)

= cos(y)R2 Covibq,v+𝛼 (a, k, t) − sin(y)R2 Vibq,v+𝛼 (a, k, t),

(7.174)

𝛼 0 dt R2 Vibq,v (a, k, t)

= cos(y)R2 Vibq,v+𝛼 (a, k, t) + sin(y)R2 Covibq,v+𝛼 (a, k, t).

(7.175)

7.6 Inferred Fractional Differential Equations Because the approach to obtaining special classes of FDEs for the R2 -trigonometry is so similar to that for the R1 -trigonometry, one only needs to change the functions in that analysis. We only point out that combinations of R1 - and R2 -functions may also be used.

127

129

8 The R3 -Trigonometric Functions In the integer-order trigonometry, replacing the frequency parameter a by the imaginary parameter ia toggles the functions back and forth between the trigonometric and the hyperbolic functions. For example, sin(i y) ⇒ i sinh(y)

and

sinh(i y) ⇒ i sin(y),

and

cosh(iy) ⇒ cos(y).

and for the cosine function cos(i y) ⇒ cosh(y)

Parallels to this were found for the R1 -trigonometry with the R1 -hyperboletry. However, this is not the case relative to the R2 -trigonometry. This chapter develops the effect of an imaginary frequency parameter and an imaginary time parameter in the R-function.

8.1 The R3 -Trigonometric Functions: Based on Complexity The basis for definition of the R1 -trignometric functions is to allow a → i a. The R2 -trigonometric functions are based on t → i t. The obvious question is: What is the nature of the results when both a → i a and t → i t? This assumption is the basis of the R3 -trignometric functions. We start by separating Rq,v ( i a, i t) into real and imaginary parts. Thus, we consider Rq,v ( i a, i t) =

∞ ∑ (ia)n (it)(n+1)q−1−v n=0 ∞

Rq,v ( i a, i t) =

Γ((n + 1)q − v)

,

t > 0,

∑ an t (n+1)q−1−v i(n+1)q−1−v+n n=0

Γ((n + 1)q − v)

,

(8.1) t > 0.

(8.2)

Now, for rational q and v, we may write nq + q − 1 − v + n =

nNq Dv + Nq Dv − Dq Dv − Nv Dq + nDq Dv Dq Dv

=

M . D

Thus, i(n+1)q−1−v+n = cos(𝜉) + i sin(𝜉) = cis(𝜉),

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(8.3)

130

8 The R3 -Trigonometric Functions

( ) ( ) 𝜋 𝜋 where 𝜉 = M + 2𝜋k , 𝜉 = (n(q + 1) + q − 1 − v) + 2𝜋k , and k = 0, 1, 2, … , Dq Dv − 1. D 2 2 Therefore, equation (8.2) may be written as Rq,v ( i a, k, i t) =

∞ ∑ an t (n+1)q−1−v cis(𝜉), Γ((n + 1)q − v) n=0

Rq,v ( i a, k, i t) =

∞ ∞ ∑ ∑ an t (n+1)q−1−v an t (n+1)q−1−v cos(𝜉) + i sin(𝜉), Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0

t > 0,

k = 0, 1, 2, … , D − 1,

(8.4)

where k is included in the R-function argument to reflect its presence in 𝜉 on the right-hand side. Then, by similarity to equations (7.6) and (7.7), we define ∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 cos (n(q + 1) + q − 1 − v) + 2𝜋k , R3 Cosq,v (a, k, t) ≡ Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, 2, … , D − 1,

(8.5)

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 sin (n(q + 1) + q − 1 − v) + 2𝜋k , R3 Sinq,v (a, k, t) ≡ Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, 2, … , D − 1.

(8.6)

Thus, these definitions are quite similar to those for R2 Cosq,v ( a, t) and R2 Sinq,v ( a, t), differing from their R2 counterparts only by the presence of an extra “n” in the arguments of the circular functions. As with the R2 -trigonometric functions, we define the principal functions for t > 0, as ∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , (8.7) cos (n(q + 1) + q − 1 − v) R3 Cosq,v (a, t) ≡ R3 Cosq,v (a, 0, t) ≡ Γ((n + 1)q − v) 2 n=0 R3 Sinq,v (a, t) ≡ R3 Sinq,v (a, 0, t) ≡

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 . sin (n(q + 1) + q − 1 − v) Γ((n + 1)q − v) 2 n=0

(8.8)

Combining equations (8.4)–(8.6) gives Rq,v ( i a, i t) = R3 Cosq,v (a, k, t) + iR3 Sinq,v (a, k, t),

t > 0, k = 0, 1, 2, … , D − 1.

(8.9)

Figure 8.1 shows the principal R3 Cosq,v - and the R3 Sinq,v -functions for values of q between q = 0.2 and q = 1.0, with a = 1.0 and v = 0. We observe from Figure 8.1 that for q = 1 and v = 0 we have R3 Cos1,0 (a, t) = e−a t , and we can see that R3 Sin1,0 (1, t) = 0. These results are readily verified by appropriate substitution into equations (8.7) and (8.8). The character of these functions changes considerably for q > 1, which is illustrated in Figure 8.2. For q ≈ 1.495 and greater, the principal R3 Sinq,v - and R3 Cosq,v -functions also grow in oscillation amplitude. In Figures 8.3 and 8.4, we study the effect of changes in the a parameter on the principal R3 Cosq,v - and the R3 Sinq,v -functions for values of q = 0.25 and 0.75, respectively. As seen in the R1 and R2 trigonometries, the effect of increasing a leads to faster transient responses of the trigonometric variables. The effect of the differintegration order variable, v, is presented in Figures 8.5 and 8.6 for values of q = 0.25 and 0.75, respectively. Here, we see that v not only changes the transient rate of response of the trigonometric functions, but also the nature of the response, in particular when the sign of v changes.

R3Cosq,0(1,t)

8.1 The R3 -Trigonometric Functions: Based on Complexity

1 q = 1.0

0.5

R3Sinq,0(1,t)

q = 0.2 0 0

1

0 0.2

2 t-Time

0.8

3

4

3

4

q = 1.0

0.6 0.4

–0.2 –0.4 0

1

2 t-Time

Figure 8.1 R3 Cosq,0 (1, t) and R3 Sinq,0 (1, t) versus t-Time for a = 1, k = 0, v = 0, q = 0.2–1.0 in steps of 0.2.

R3Cosq,0(1,t)

1 q = 1.5

0.5 q = 1.0

0 –0.5 –1 0

1

2

3

4

5

6

7

5

6

7

t-Time

R3Sinq,0(1,t)

1 0.5 q = 1.0

0

q = 1.5

–0.5 –1 0

1

2

3

4 t-Time

Figure 8.2 R3 Cosq,0 (1, t) and R3 Sinq,0 (1, t) versus t-Time for a = 1, k = 0, v = 0, q = 1.0–1.5 in steps of 0.1.

131

8 The R3 -Trigonometric Functions

R3Cos0.25,0(a,t)

1

0.5 a = 0.25, 0.50 0

a = 1.0 0

1

2 t-Time

3

4

2 t-Time

3

4

R3Sin0.25,0(a,t)

0 1.0 a = 0.25 –0.5

–1

0

1

R3Cos0.75,0(a,t)

Figure 8.3 Effect of a, R3 Cosq,0 (1, t), and R3 Sinq,0 (1, t) versus t-Time for a = 0.25–1.0 in steps of 0.25, with q = 0.25, k = 0, v = 0.

1 a = 0.25 0.5 a = 1.0 0

0

1

2

3

4

3

4

t-Time 0 a = 1.0 R3Sin0.75,0(a,t)

132

a = 0.25 –0.5

–1 0

1

2 t-Time

Figure 8.4 Effect of a, R3 Cosq,0 (1, t), and R3 Sinq,0 (1, t) versus t-Time for a = 0.25–1.0 in steps of 0.25, with q = 0.75, k = 0, v = 0.

8.1 The R3 -Trigonometric Functions: Based on Complexity

R3Cos0.25,𝜈(1,t)

1

0.5

0

𝜈 = 0.0

𝜈 = 0.6

0.3 0

1

𝜈 = –0.3

2 t-Time

𝜈 = –0.6

3

4

R3Sin0.25,𝜈(a,t)

1 𝜈 = 0.6

0.3 0

𝜈 = –0.6

𝜈 = –0.3 and 0.0

–1 0

1

2 t-Time

3

4

Figure 8.5 Effect of v, R3 Cosq,0 (1, t), and R3 Sinq,0 (1, t) versus t-Time for v = −0.6–0.6 in steps of 0.3, with q = 0.25, k = 0, a = 1.

R3Cos0.75,𝜈(1,t)

1

0.5 𝜈 = –0.6 0.6 0 0

𝜈 = .3 1

𝜈 = 0.0 2 t-Time

𝜈 = –0.3 3

4

R3Sin0.75,𝜈(a,t)

1 𝜈 = 0.6

𝜈 = –0.6 𝜈 = –0.3

0 𝜈 = 0.3 𝜈 = 0.0

–1 0

1

2 t-Time

3

4

Figure 8.6 Effect of v, R3 Cosq,0 (1, t), and R3 Sinq,0 (1, t) versus t-Time for v = −0.6–0.6 in steps of 0.3, with q = 0.75, k = 0, a = 1.

133

8 The R3 -Trigonometric Functions

5

R3Sin5/7,0(1,t)

k=1

k=3

k=4

0

k=6 k=0

k=5 k=2 –5 0

1

2 t-Time

3

4

Figure 8.7 Effect of k, for R3 Sin5∕7,0 (1, k, t), with k = 0–6 in steps of 1, q = 5/7, a = 1.0, v = 0. Figure 8.8 Phase plane R3 Sinq,0 (1, t) versus R3 Cosq,0 (1, t) for q = 1.0–2.0 in steps of 0.2, with a = 1.0, v = 0, and t = 0–7.0.

2 1.5

q = 2.0

q = 1.6

q = 1.8

1 R3Sinq,0(1,t)

134

0.5 q = 1.4

0

q = 1.0

–0.5 –1 –1.5 –2

–1

0 R3Cosq,0(1,t)

1

2

The effect of k, for R3 Sin5∕7,0 (1, k, t), with k = 0 to 6 in steps of 1, and q = 5/7, a = 1.0, v = 0 is shown in Figure 8.7. In this case, strong symmetry is again seen. It has been observed that the R3 Sin-function maintains this symmetry for q = 1∕n with n an odd integer. Figure 8.8 shows a phase plane for R3 Sinq,0 (1, t) versus R3 Cosq,0 (1, t) for q = 1.0 to 2.0. In this figure, all cases start from the origin. The cases with 1 ≤ q ≤ 1.4 return to the origin, while those for q ≥ 1.6 diverge with increasing rates.

8.2 The R3 -Trigonometric Functions: Based on Parity We now consider Rq,v ( i a, i t) based on parity of the exponent of a. Then, equation (8.1) is written as ∞ ∑ (iq+1 a)n ( t q )n Rq,v ( i a, i t) = (i t)q−1−v , t > 0. (8.10) Γ((n + 1)q − v) n=0

8.2 The R3 -Trigonometric Functions: Based on Parity

The summation becomes (iq+1 a) t q (iq+1 a)2 t 2q (iq+1 a)3 t 3q (iq+1 a)4 t 4q (iq+1 a)5 t 5q 1 + + + + + + · · ·. Γ(q − v) Γ(2q − v) Γ(3q − v) Γ(4q − v) Γ(5q − v) Γ(6q − v)

(8.11)

Forming two summations by separating terms with even and odd powers of a, we may write } {∞ ∞ ∑ (iq+1 a )2n (t q )2n+1 ∑ (iq+1 a )2n+1 (t q )2n+2 q−1−v −1−v t + , t > 0, Rq,v ( i a, i t) = i Γ((2n + 1)q − v) n=0 Γ((2n + 2)q − v) n=0 (8.12)

Rq,v ( i a, i t) = (i )

q−1−v

{∞ } ∞ ∑ (i2(q+1) a2 )n t (n+1)2q−1−v−q ∑ (i2(q+1) a2 )n t (n+1)2q−1−v q+1 + (i a ) . Γ((n + 1)2q − v − q) Γ((n + 1)2q − v) n=0 n=0 (8.13)

We note in passing the identity Rq,v ( i a, i t) = (i )q−1−v {R2q,v+q (i2(q+1) a2 , t) + (iq+1 a )R2q,v (i2(q+1) a2 , t)},

t > 0,

(8.14)

which after application of equation (3.119) becomes Rq,v ( i a, i t) = R2q,v+q (−a2 , i t) + i aR2q,v (−a2 , i t),

t > 0.

For t > 0, equation (8.13) is rewritten as Rq,v ( i a, i t) =

∞ ∑

a2n t (n+1)2q−1−v−q 2n(q+1)+q−1−v i Γ((n + 1)2q − v − q)

n=0 ∞

+

∑ a2n+1 t (n+1)2q−1−v i2n(q+1)+2q−v , Γ((n + 1)2q − v) n=0

(8.15)

where the summations are separated into even and odd powers of a in (8.12). In parallel with the development of the R2 -trigonometric functions, and based on equations (8.13) and (8.14), we define the R3 -Corotation and R3 -Rotation functions as the summations with even and odd powered a terms, respectively: R3 Corq,v (a, t) ≡ i

q−1−v

∞ ∑ (i2(q+1) a2 )n t (n+1)2q−1−v−q

Γ((n + 1)2q − v − q)

n=0

= iq−1−v t −1−v R3 Rotq,v (a, t) ≡ i2q−v a

= iq−1−v R2q,v+q (i2(q+1) a2 , t),

∞ ∑ (i2(q+1) a2 )n (t q )2n+1 , Γ((n + 1)2q − v − q) n=0

∞ ∑ (i2(q+1) a2 )n t (n+1)2q−1−v

Γ((n + 1)2q − v)

n=0

= i2q−v t −1−v a

n=0

Γ((n + 1)2q − v)

t > 0,

= i2q−v a R2q,v (i2(q+1) a2 , t),

∞ ∑ (i2(q+1) a2 )n (t q )2n+2

(8.16)

= a i R2q,v (−a2 , i t),

(8.17)

t > 0.

We now use the real and imaginary parts of these functions to define the four new real fractional trigonometric functions that parallel those defined for the R2 -trigonometry. The R3 Corq,v (a, t)-function is rewritten as R3 Corq,v (a, t) =

∞ ∑ (a2 )n t (n+1)2q−1−v−q 2n(q+1)+q−1−v , i Γ((n + 1)2q − v − q) n=0

t > 0.

(8.18)

135

136

8 The R3 -Trigonometric Functions

Applying equation (3.123) to i2n(q+1)+q−1−v with q and v rational, we have 2nq + 2n + q − 1 − v =

2nNq Dv + 2nDq Dv + Nq Dv − Dq Dv − Nv Dq Dq Dv

=

M , D

where q = Nq ∕Dq and v = Nv ∕Dv are assumed to be rational and irreducible and M/D is in minimal form. Thus, ( ( )) 𝜋 i2n(q+1)+q−1−v = cos (2n(q + 1) + q − 1 − v) + 2𝜋k 2( )) ( 𝜋 + 2𝜋k + i sin (2n(q + 1) + q − 1 − v) 2 where k = 0, 1, 2, … , Dq Dv − 1. Then R3 Corq,v (a, k, t) may be written as R3 Corq,v (a, k, t) =

∞ ∑ n=0

+i

( ( )) a2n t (n+1)2q−1−v−q 𝜋 cos (2n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)2q − v − q) 2

∞ ∑ n=0

( ( )) a2n t (n+1)2q−1−v−q 𝜋 sin (2n(q + 1) + q − 1 − v) + 2𝜋k , Γ((n + 1)2q − v − q) 2

t > 0.

(8.19)

The real part of the R3 Corq,v (a, k, t) is defined as the R3 -Covibration function R3 Covibq,v (a, k, t) ≡

∞ ∑ n=0

( ( )) a2n t (n+1)2q−1−v−q 𝜋 cos (2n(q + 1) + q − 1 − v) + 2𝜋k , Γ((n + 1)2q − v − q) 2 t > 0,

k = 0, 1, 2, … , Dq Dv − 1. (8.20)

The imaginary part of equation (8.19) defines the R3 -Vibration function ∞ ( ( )) ∑ a2n t (n+1)2q−1−v−q 𝜋 R3 Vibq,v (a, k, t) ≡ sin (2n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 t > 0,

k = 0, 1, 2, … , Dq Dv − 1.

(8.21)

We now examine the nature of these functions when q = 1 and v = 0: ∞ ( ( )) ∑ a2n t 2n 𝜋 cos (4n) + 2𝜋k , R3 Covib1,0 (a, k, t) = Γ(2n + 1) 2 n=0 ∞ ∑ a2n t 2n = = cosh(at), (2n) ! n=0

t > 0, k = 0, 1, 2, … , Dq Dv − 1,

(8.22)

and R3 Vib1,0 (a, k, t) =

∞ ∑ n=0

( ( )) a2n t 2n 𝜋 sin (4n) + 2𝜋k = 0, Γ(2n + 1) 2 t > 0, k = 0, 1, 2, … , Dq Dv − 1.

(8.23)

The R3 Rot-function (equation (8.17)) is now written as R3 Rotq,v (a, t) = a

∞ ∑ (a2 )n t (n+1)2q−1−v 2n(q+1)+2q−v , i Γ((n + 1)2q − v) n=0

t > 0.

(8.24)

8.2 The R3 -Trigonometric Functions: Based on Parity

Applying equation (3.123) to i2n(q+1)+2q−v with q and v rational, we have 2nq + 2n + 2q − v =

2nNq Dv + 2nDq Dv + 2Nq Dv − Nv Dq Dq Dv

=

M , D

where q = Nq ∕Dq and v = Nv ∕Dv are assumed to be rational and irreducible and M/D is in minimal form. Thus ( ( )) 𝜋 i2n(q+1)+2q−v = cos (2n(q + 1) + 2q − v) + 2𝜋k 2( )) ( 𝜋 + 2𝜋k + i sin (2n(q + 1) + 2q − v) 2 where k = 0, 1, 2, … , Dq Dv − 1. Then, R3 Rotq,v (a, k, t) may be written as R3 Rotq,v (a, k, t) =

∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 cos (2n(q + 1) + 2q − v) + 2𝜋 k Γ((n + 1)2q − v) 2 n=0 ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 sin (2n(q + 1) + 2q − v) + 2𝜋 k , +i Γ((n + 1)2q − v) 2 n=0

t > 0, k = 0, 1, … , Dq Dv − 1.

(8.25)

Thus, we define the real part as the R3 -Coflutter function ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 cos (2n(q + 1) + 2q − v) + 2𝜋 k , R3 Cof lq,v (a, k, t) ≡ Γ((n + 1)2q − v) 2 n=0 t > 0, k = 0, 1, … , Dq Dv − 1,

(8.26)

and the imaginary part as the R3 -Flutter function ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 R3 Flutq,v (a, k, t) ≡ sin (2n(q + 1) + 2q − v) + 2𝜋 k , Γ((n + 1)2q − v) 2 n=0 t > 0, k = 0, 1, … , Dq Dv − 1.

(8.27)

Again, we examine the nature of these functions when q = 1, and v = 0, then ∞ ( ( )) ∑ a2n+1 t 2n+1 𝜋 R3 Cof l1,0 (a, k, t) = cos (4n + 2) + 2𝜋 k , Γ(2n + 2) 2 n=0 =−

∞ ∑ a2n+1 t 2n+1 = − sinh(at), (2n + 1) ! n=0

t > 0, k = 0, 1, … , Dq Dv − 1.

and R3 Flut1,0 (a, k, t) =

∞ ( ( )) ∑ a2n+1 t 2n+1 𝜋 sin (4n + 2) + 2𝜋 k = 0, Γ(2n + 2) 2 n=0

t > 0, k = 0, 1, … , Dq Dv − 1. The backward compatibility, that is, the q = 1, and v = 0 evaluation, of the R3 -functions for t > 0, is summarized as R1,0 ( i a, i t) = e−at ,

(8.28)

,

(8.29)

R3 Cos1,0 (a, t) = e

−a t

R3 Sin1,0 (1, t) = 0,

(8.30)

137

8 The R3 -Trigonometric Functions

R3 Cor1,0 (a, t) = cosh(at),

(8.31)

R3 Rot1,0 (a, t) = − sinh(at),

(8.32)

R3 Covib1,0 (a, t) = cosh(at), R3 Vib1,0 (a, t) = 0,

(8.33) (8.34)

R3 Cof l1,0 (a, t) = − sinh(at),

(8.35)

R3 Flut1,0 (a, t) = 0.

(8.36)

Thus, we see that these functions are backward compatible with the common exponential and the hyperbolic functions and indeed might be considered as the R2 -hyperbolic functions. Here, the complexity and parity properties of Rq,v ( i a, i t) in terms of the R3 -trigonometric functions are summarized; for t > 0 R3 Cosq,v (a, k, t) = Re(Rq,v (ia, k, it)),

(8.37)

R3 Sinq,v (a, k, t) = Im(Rq,v (ia, k, it)),

(8.38)

R3 Covibq,v (a, k, t) = Re(Ea (Rq,v (ia, k, it))) = Ea (Re(Rq,v (ia, k, it))),

(8.39)

R3 Vibq,v (a, k, t) = Im(Ea (Rq,v (ia, k, it))) = Ea (Im(Rq,v (ia, k, it))),

(8.40)

R3 Cof lq,v (a, k, t) = Re(Oa (Rq,v (ia, k, it))) = Oa (Re(Rq,v (ia, k, it))),

(8.41)

R3 Flutq,v (a, k, t) = Im(Oa (Rq,v (ia, k, it))) = Oa (Im(Rq,v (ia, k, it))),

(8.42)

R3 Rotq,v (a, t) = Oa (Rq,v (ia, it)),

(8.43)

R3 Corq,v (a, t) = Ea (Rq,v (ia, it)),

(8.44)

where Ea and Oa refer to the forms with terms containing the even and odd powers of a. Also we see that, by design, these forms exactly parallel those of the related R2 -functions. Similar to the R2 case, the commutative property of equations (8.39)–(8.42) are readily proved by taking the real and imaginary parts of equations (8.16) and (8.17) and relating them to the even and odd parts of equations (8.7) and (8.8). The R3 Cof lq,v (a, k, t)-, R3 Flutq,v (a, k, t)-, R3 Covq,v (a, k, t)-, and R3 Vibq,v (a, k, t)-functions are presented in Figures 8.9–8.12 for 0.1 ≤ q ≤ 0.5, with a = 1, and v = 0. In the figures, and validated by substitution in the appropriate series, we observe R3 Cof l1∕2,0 (a, k, t) = a sin(a2 t) and R3 Flut1∕2,0 (a, k, t) = a cos(a2 t). Figure 8.13 presents a phase plane R3 Flutq,0 (1, t) versus R3 Cof lq,0 (1, t) for q = 0.5–1.5. For q > 0.5, the functions spiral out from the origin with increasing rate as q increases. Figure 8.9 R3 Cof lq,0 (1, t) versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.

1.5 1 R3Coflq,0(1,t)

138

q = 0.50

0.5 0

q = 0.10

–0.5 –1 –1.5 0

1

2

3 4 t-Time

5

6

7

8.2 The R3 -Trigonometric Functions: Based on Parity

Figure 8.10 R3 Flutq,0 (1, t) versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.

1.5

R3Flutq,0(1,t)

1

q = 0.50

0.5 0 q = 0.10 –0.5 –1 –1.5 0

1

2

3 4 t-Time

5

6

7

Figure 8.11 R3 Covibq,0 (1, t) versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.

R3Covibq,0(1,t)

1

0.5 q = 0.10

0

–0.5 q = 0.50 –1 0

1

2

4 3 t-Time

5

6

7

Figure 8.12 R3 Vibq,0 (1, t) versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.

1.5

R3Vibq,0(1,t)

1 0.5

q = 0.10

0 –0.5

q = 0.50

–1 –1.5 0

1

2

4 3 t-Time

5

6

7

The phase plane R3 Vibq,0 (1, t) versus R3 Covibq,0 (1, t) for q = 0.5–1.5 is presented in Figure 8.14. For 0.5 ≤ q < 1 the curves approach from infinity and are repelled before reaching the unit circle at q = 0.5 outward back toward infinity. For q > 1, the curves leave the origin and spiral out to infinity with increasing rate. The q = 1 case starts at the unit circle and proceeds to infinity. Also, it may be shown that R3 Flut1,0 (1, 0, t) = R3 Vib1,0 (1, 0, t) = 0.

139

8 The R3 -Trigonometric Functions

Figure 8.13 Phase plane R3 Flutq,0 (1, t) versus R3 Cof lq,0 (1, t) for q = 0.5–1.5 in steps of 0.1, a = 1.0, k = 0, v = 0, t = 0–10.

5 4 3

q = 0.9

q = 0.8

q = 0.7

R3Flutq,0(1,t)

2

q = 0.6

1 q = 1.0

0

q = 0.5

q = 1.1

–1 –2 –3

q = 1.2

q = 1.5

1.3 1.4

–4 –5 –5

0

5

R3Coflq,0(1,t)

Figure 8.14 Phase plane R3 Vibq,0 (1, t) versus R3 Covibq,0 (1, t) for q = 0.5–1.5 in steps of 0.1, a = 1.0, k = 0, v = 0, t = 0–12.

5 q = 0.60

4 3

1.4 1.3 q = 1.2

1.5

2 R3Vibq,0(1,t)

140

q = 1.1

1

q = 1.0

0 –1 –2 –3

q = 0.70

–4

q = 0.80

–5 –5

0

0.60 q = 0.50 5

R3Covibq,0(1,t)

8.3 Laplace Transforms of the R3 -Trigonometric Functions The development of the Laplace transform for the R3 -trigonometric functions parallels that of the R2 -functions. Thus, we state the results in normal and factored form without development.

L{R3 Cosq,v (a, k, t)} = =

sv (cos(𝜉q,v )sq − a cos(𝜉q,v − 𝜇q+1 )) s2q − 2a cos(𝜇q+1 )sq + a2 ei𝜉q,v sv e−i𝜉q,v sv + , 2(sq − aei𝜇q+1 ) 2(sq − ae−i𝜇q+1 )

,

q > v, q > v.

(8.45)

8.4 R3 -Trigonometric Function Relationships

L{R3 Sinq,v (a, k, t)} = =

L{R3 Cof lq,v (a, k, t)} = =

L{R3 Flutq,v (a, k, t)} = =

L{R3 Covibq,v (a, k, t)} = =

L{R3 Vibq,v (a, k, t)} =

sv (sin(𝜉q,v )sq − a sin(𝜉q,v − 𝜇q+1 ))

,

q > v,

ei𝜉q,v sv e−i𝜉q,v sv + i𝜇 2 i (sq − ae q+1 ) 2 i (sq − ae−i𝜇q+1 )

q > v.

s2q − 2a cos(𝜇q+1 )sq + a2

a sv (cos(𝜉2q,v−1 )s2q − a2 cos(𝜉2q,v−1 − 𝜇2q+2 )) s4q − 2a2 cos(𝜇2q+2 )s2q + a4 aei𝜉2q,v−1 sv ae−i𝜉2q,v−1 sv + , 2(s2q − a2 ei𝜇2q+2 ) 2(s2q − a2 e−i𝜇2q+2 )

s4q − 2a2 cos(𝜇2q+2 )s2q + a4 a ei𝜉2q,v−1 sv a e−i𝜉2q,v−1 sv + , 2 i (s2q − a2 ei𝜇2q+21 ) 2 i (s2q − a2 e−i𝜇2q+2 )

s4q − 2a2 cos(𝜇2q+2 )s2q + a4 ei𝜉q,v sq+v e−i𝜉q,v sq+v + , 2(s2q − a2 ei𝜇q+1 ) 2(s2q − a2 e−i𝜇q+1 ) sv+q (sin(𝜉q,v )s2q − a2 sin(𝜉q,v − 𝜇2q+2 )) s4q − 2a2 cos(𝜇2q+2 )s2q + a4

,

2q > v,

(8.47)

2q > v.

a sv (sin(𝜉2q,v−1 )s2q − a2 sin(𝜉2q,v−1 − 𝜇2q+2 ))

sv+q (cos(𝜉q,v )s2q − a2 cos(𝜉q,v − 𝜇2q+2 ))

(8.46)

,

,

2q > v,

(8.48)

2q > v. q > v,

(8.49)

q > v. ,

q > v,

(8.50)

ei𝜉q,v sq+v e−i𝜉q,v sq+v + , q > v, i𝜇q+1 2 2q − a e ) 2 i (s − a2 e−i𝜇q+1 ) ( ) ( ) where 𝜉q,v = (q − 1 − v) 𝜋2 + 2𝜋k and 𝜇q = q 𝜋2 + 2𝜋k , with k = 0, 1, 2, … , Dq Dv − 1. Comparison of these transforms with the parallel results for the R2 -trigonometric functions shows that the transforms are structurally the same and vary from each other primarily in the coefficient of the s2q term in the denominator. We will later see that this influences the damping behavior of the transform. Note that most of the figures presented in this and previous chapters are of results in the stable domain that is of trigonometric order, 0 < q ≤ 1 for the complexity functions. The reader is encouraged to numerically explore values of q > 1 as many important applications are found there; see, for example, Chapters 17–20. For the parity functions, explore q > 1/2. =

2 i (s2q

8.4 R3 -Trigonometric Function Relationships This section determines expressions for the R3 -trigonometric functions in terms of the fractional exponential functions. These expressions parallel the basic equations (1.18) and (1.19). Some special R-function relationships also result. The organization of the subsections of this section has been set by the ease of development of relationships.

141

142

8 The R3 -Trigonometric Functions

8.4.1 R3 Cosq,v (a, t) and R3 Sinq,v (a, t) Relationships and Fractional Euler Equation

This section develops some important properties of the R3 -trigonometric functions. We observe from equations (8.4)–(8.6) that Rq,v (ia, k, it) =R3 Cosq,v (a, k, t) + i R3 Sinq,v (a, k, t), t > 0, k = 0, 1, 2, … , Dq Dv − 1.

(8.51)

This equation parallels the form of equations (6.15) and (7.12) for the R1 and R2 -trigonometries, respectively. Consider now the R3 Cosq,v (a, k, t)-function as defined in equation (8.5), that is, R3 Cosq,v (a, k, t) ≡

∞ { ( )} ∑ an t (n+1)q−1−v 𝜋 cos (n(q + 1) + q − 1 − v) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, 2, … , Dq Dv − 1.

(8.52)

Applying definition (1.18), we may write R3 Cosq,v (a, k, t)

) ) ( ( 𝜋 ⎛ i[n(q+1)+q−1−v] 𝜋 + 2𝜋k −i[n(q+1)+q−1−v] + 2𝜋k ⎞ ∞ n (n+1)q−1−v ∑ ⎜e ⎟ 2 2 a t +e = ⎜ ⎟, Γ((n + 1)q − v) 2 n=0 ⎜ ⎟ ⎝ ⎠ t > 0, k = 0, 1, 2, … , Dq Dv − 1. ( i[n(q+1)+q−1−v]

𝜋 2

+2𝜋k

= i [n(q+1)+q−1−v] , and (3.127), we have ( [n(q+1)+q−1−v] −[n(q+1)+q−1−v] ) ∞ ∑ an t (n+1)q−1−v +i i , t > 0. R3 Cosq,v (a, t) = Γ((n + 1)q − v) 2 n=0

Now, by equation (3.124), e

(8.53)

)

(8.54)

This may be written as ∞ ⎧∑ an i(n(q+1)+q−1−v) ( t)[(n+1)q−1−v] ⎪ Γ((n + 1)q − v) 1 ⎪ n=0 R3 Cosq,v (a, t) = ⎨ ∞ 2⎪ ∑ an i−(n(q+1)+q−1−v) ( t)[(n+1)q−1−v] + ⎪ Γ((n + 1)q − v) ⎩ n=0

⎫ ⎪ ⎪ ⎬ , ⎪ ⎪ ⎭

∞ ∑ ⎧ (aiq+1 )n ( t)[(n+1)q−1−v] q−1−v i ⎪ Γ((n + 1)q − v) n=0 1⎪ R3 Cosq,v (a, t) = ⎨ ∞ 2⎪ ∑ (ai−(q+1) )n ( t)[(n+1)q−1−v] −(q−1−v) +i ⎪ Γ((n + 1)q − v) ⎩ n=0

Recognizing the summations as R-functions, we have } 1 { q−1−v i Rq,v (aiq+1 , t) + i−(q−1−v) Rq,v (ai−(q+1) , t) , R3 Cosq,v (a, t) = 2

(8.55)

⎫ ⎪ ⎪ ⎬ . ⎪ ⎪ ⎭

(8.56)

t > 0,

(8.57)

or applying equation (3.119) R3 Cosq,v (a, t) =

Rq,v (ia, it) + Rq,v (−ia, −it) 2

,

t > 0.

(8.58)

8.4 R3 -Trigonometric Function Relationships

This equation generalizes equation (1.18) in the ordinary trigonometry. It could also serve as a definition of R3 Cosq,v (a, t). The development for the R3 Sinq,v (a, t)-function follows in similar manner, yielding R3 Sinq,v (a, t) =

Rq,v (ia, it) − Rq,v (−ia, −it) 2i

,

t > 0.

(8.59)

This equation exactly parallels equation (1.19) in the ordinary trigonometry. It could also serve as a definition of R3 Sinq,v (a, t). Solving for Rq,v (ia, it) using both equations (8.58) and (8.59) yields equation (8.51) and subtracting (8.59) from (8.58) gives, for t > 0 (8.60)

Rq,v (−ia, −it) = R3 Cosq,v (a, t) − i R3 Sinq,v (a, t), the companion to equation (8.51). Now, squaring equations (8.58) and (8.59) yields R3 Cos2q,v (a, t) = R3 Sin2q,v (a, t) =

R2q,v (ia, it) + R2q,v (−ia, −it) + 2Rq,v (ia, it)Rq,v (−ia, −it) 4 R2q,v (ia, it) + R2q,v (−ia, −it) − 2Rq,v (ia, it)Rq,v (−ia, −it) −4

,

t > 0,

(8.61)

,

t > 0.

(8.62)

Adding and subtracting equations (8.61) and (8.62) gives t > 0,

R3 Cos2q,v (a, t) + R3 Sin2q,v (a, t) = Rq,v (ia, it)Rq,v (−ia, −it),

(8.63)

and R3 Cos2q,v (a, t) − R3 Sin2q,v (a, t) =

R2q,v (ia, it) + R2q,v (−ia, −it) 2

,

t > 0.

(8.64)

These equations parallel equations (6.23) and (6.24) and provide additional generalizations of the equations cos2 (t) + sin2 (t) = 1 and cos2 (t) − sin2 (t) = cos(2t). 8.4.2 R3 Rotq,v (a, t) and R3 Corq,v (a, t) Relationships

As previously mentioned, R3 Rotq,v (a, t) and R3 Corq,v (a, t) are complex functions. From the definitions (equations (8.15)–(8.17)), we have for t > 0 (8.65)

Rq,v (ia, it) = R3 Corq,v (a, t) + R3 Rotq,v (a, t).

We observe that the summations of equations (8.16) and (8.17) are R-functions of complex arguments, and using equation (3.118), we may write R3 Corq,v (a, t) = iq−1−v R2q,v+q (i2(q+1) a2 , t) = iq−1−v R2q,v+q (−i2q a2 , t) = R2q,v+q (−a2 , i t),

t > 0,

(8.66)

R3 Rotq,v (a, t) = a i2q−v R2q,v (i2(q+1) a2 , t) = a i2q−v R2q,v (−i2q a2 , t) = a iR2q,v (−a2 , i t) ,

t > 0.

(8.67)

Combining the results of equations (8.66) and (8.67) with equation (8.65) gives the following identity: Rq,v (ia, it) = R2q,v+q (−a2 , i t) + a i R2q,v (−a2 , i t) ,

t > 0.

(8.68)

143

144

8 The R3 -Trigonometric Functions

8.4.3 R3 Coflq,v (a, t) and R3 Flutq,v (a, t) Relationships

The analysis starts by replacing the cos function in the definition of R3 Cof lq,v (a, k, t) in equation (8.26). Then, using logic similar to the process used for the R3 Cos, that is, equations (8.53) to (8.57), we have for t > 0 a e𝜉2q,v−1 i a e−𝜉2q,v−1 i (8.69) R2q,v (a2 e𝜇2q+2 i , t) + R2q,v (a2 e−𝜇2q+2 i , t), 2 2 ( ) ( ) where 𝜇2q+2 = (2q + 2) 𝜋2 + 2𝜋 k , 𝜉2q,v−1 = (2q − v) 𝜋2 + 2𝜋 k with t > 0, and k = 0, 1, 2 · · · Dq Dv − 1. Applying equation (3.124) to the exponential functions, we have ( ) R3 Cof lq,v (a, k, t) =

e𝜇2q+2 i = i2q+2 = −i2q and e𝜉2q,v−1 i = e R3 Cof lq,v (a, t) =

(2q−v)

𝜋 2

+2𝜋k i

= i2q−v . Thus, we have

a i2q−v a i−(2q−v) R2q,v (−a2 i2q , t) + R2q,v (−a2 i−2q , t), 2 2

t > 0.

Applying equation (3.119) to these R-functions, we write the final result as ( ) R2q,v (−a2 , −it) − R2q,v (−a2 , it) , t > 0. R3 Cof lq,v (a, t) = a 2i The R3 Flutq,v (a, t)-function is derived in a similar manner. We give the final result ( ) R2q,v (−a2 , it) + R2q,v (−a2 , −it) R3 Flutq,v (a, t) = a , t > 0. 2

(8.70)

(8.71)

We now derive some additional properties of these functions. Rewrite equations (8.70) and (8.71) as R2q,v (−a2 , −it) − R2q,v (−a2 , it) =

2i R Cof lq,v (a, t), a 3

R2q,v (−a2 , −it) + R2q,v (−a2 , it) =

2 R Flutq,v (a, t), a 3

t > 0, t > 0.

(8.72) (8.73)

Adding equations (8.72) and (8.73) gives a R2q,v (−a2 , −it) = R3 Flutq,v (a, t) + iR3 Cof lq,v (a, t),

t > 0.

(8.74)

This equation may be considered as a fractional “semi-Euler” equation. Subtracting equation (8.72) from equation (8.73) and using (8.17) yields the complimentary equation a R2q,v (−a2 , it) =R3 Flutq,v (a, t) − iR3 Cof lq,v (a, t) = −i R3 Rotq,v (a, t),

t > 0.

(8.75)

Now, in equations (8.74) and (8.75), let 2q = u; then, a Ru,v (−a2 , −it) = R3 Flut u ,v (a, t) + i R3 Cofl u ,v (a, t), 2

2

and a Ru,v (−a2 , it) = R3 Flut u ,v (a, t) − iR3 Cofl u ,v (a, t), 2

2

t > 0,

(8.76)

which may be considered as fractional “semi-Euler” equations. Replacing a → ia2 in equation (8.9) gives Rq,v (−a2 , i t) = R3 Cosq,v (i a2 , t) + iR3 Sinq,v (i a2 , t),

t > 0.

(8.77)

8.4 R3 -Trigonometric Function Relationships

Comparing the real and imaginary parts of this equation with equation (8.76), we observe that 1 (8.78) R3 Cosq,v (i a2 , t) = R3 Flut q ,v (a, t) , t > 0, 2 a and

1 R3 Sinq,v (i a2 , t) = − R3 Cof l q ,v (a, t) , 2 a

t > 0.

(8.79)

8.4.4 R3 Covibq,v (a, t) and R3 Vibq,v (a, t) Relationships

This analysis starts by rewriting equation (8.20), the R3 Covibq,v (a, t)-function, as ( i(n𝛼 +𝛽 ) ) ∞ ∑ a2n t (n+1)2q−1−v−q e 2q+2 q,v + e−i(n𝛼2q+2 +𝛽q,v ) R3 Covibq,v (a, t) = , t > 0, Γ((n + 1)2q − v − q) 2 n=0

(8.80)

where 𝛼2q+2 = (2q + 2)𝜋∕2 and 𝛽q,v = (q − 1 − v)𝜋∕2. This becomes R3 Covibq,v (a, t) = (

Since e

i𝛽q,v

=e

i(q−1−v)

𝜋 2

e𝛽q,v i e−𝛽q,v i (a2 e−𝛼2q+2 i , t), R2q,v+q (a2 e𝛼2q+2 i , t) + R 2 2 2q,v+q

t > 0.

)

+2𝜋k

R3 Covibq,v (a, t) =

= iq−1−v and ei𝛼2q+2 = −i2q , we have

iq−1−v i−(q−1−v) R2q,v+q (−a2 i2q , t) + R2q,v+q (−a2 i−2q , t), 2 2

Applying equation (3.119) to both terms gives ( ) R2q,v+q (−a2 , it) + R2q,v+q (−a2 , −it) , R3 Covibq,v (a, t) = 2

t > 0.

The derivation for the R3 Vibq,v (a, t) follows in a similar manner; thus, ( ) R2q,v+q (−a2 , it) − R2q,v+q (−a2 , −it) R3 Vibq,v (a, t) = , t > 0. 2i

t > 0.

(8.81)

(8.82)

Additional properties of these functions are now determined. Rewrite equations (8.81) and (8.82) as R2q,v+q (−a2 , it) + R2q,v+q (−a2 , −it) = 2 R3 Covibq,v (a, t), R2q,v+q (−a2 , it) − R2q,v+q (−a2 , −it) = 2 i R3 Vibq,v (a, t),

t > 0,

(8.83)

t > 0.

(8.84)

t > 0.

(8.85)

Adding equations (8.83) and (8.84) gives R2q,v+q (−a2 , it) = R3 Covibq,v (a, t) + i R3 Vibq,v (a, t) ,

This equation may also be considered as a fractional “semi-Euler” equation. Subtracting equation (8.84) from equation (8.83) yields the complimentary equation R2q,v+q (−a2 , −it) = R3 Covibq,v (a, t) − i R3 Vibq,v (a, t),

t > 0.

(8.86)

t > 0.

(8.87)

Now, in equation (8.85), let 2q = u; then, Ru,v+ u (−a2 , it) = R3 Covib u ,v (a, t) + i R3 Vib u ,v (a, t), 2

2

2

145

146

8 The R3 -Trigonometric Functions

Alternatively, we may write

( ) Rq,v+ q (−a2 , it) = R3 Covib q ,v (a, t) + i R3 Vib q ,v (a, t) , t > 0, 2 2 2 ( ) 2 Rq,v (−a , it) = R3 Covib q ,v− q (a, t) + i R3 Vib q ,v− q (a, t) , t > 0. 2

2

2

(8.88)

2

Replacing a in equation (8.51) by i a2 gives Rq,v (−a2 , it) = R3 Cosq,v (i a2 , t) + i R3 Sinq,v (i a2 , t) ,

t > 0.

(8.89)

Comparing this result with equation (8.88), we observe that R3 Cosq,v (ia2 , t) = R3 Covib q ,v− q (a, t), 2

t > 0,

2

(8.90)

and R3 Sinq,v (ia2 , t) = R3 Vib q ,v− q (a, t), 2

t > 0.

2

(8.91)

Rewrite equation (8.85) as R2q,v (−a2 , it) = R3 Covibq,v−q (a, t) + i R3 Vibq,v−q (a, t) ,

t > 0.

(8.92)

Relating the real and imaginary parts of equations (8.92) and (8.75), we obtain t > 0,

R3 Flutq,v (a, t) = aR3 Covibq,v−q (a, t), R3 Cof lq,v (a, t) = a R3 Vibq,v−q (a, t),

t >0,

(8.93) (8.94)

exposing relationships between the parity functions. It should be remembered, because of the role of the v as an order variable, that these relationships also represent fractional differintegration relationships. These are but a few of the many relationships possible for the R3 -trigonometric functions. Many other relations paralleling the multiple-angle and fractional-angle formulas from the integer-order trigonometry and more are yet to be derived.

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions 8.5.1 R3 Cosq,v (a, k, t)

The 𝛼-order differintegral of R3 Cosq,v (a, k, t) is determined for t > 0 as 𝛼 𝛼 0 dt R3 Cosq,v (a, k, t) = 0 dt

∞ { ( )} ∑ an t (n+1)q−1−v 𝜋 cos (n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)q − v) 2 n=0

t > 0,

k = 0, 1, … , Dq Dv − 1. (8.95)

Based on Section 3.16, we may differintegrate term-by-term: ∞ { ( )} ∑ an 0 dt𝛼 t (n+1)q−1−v 𝜋 𝛼 d R Cos (a, k, t) = cos (n(q + 1) + q − 1 − v) + 2𝜋 k , 0 t 3 q,v Γ((n + 1)q − v) 2 n=0 t > 0,

k = 0, 1, … , Dq Dv − 1. (8.96)

Applying equation (5.37), that is, 𝛼 p 0 dx x

=

Γ(p + 1)xp−𝛼 , Γ(p − 𝛼 + 1)

p > −1,

(8.97)

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

valid for all 𝛼. Differintegrating equation (8.96) term-by-term yields 𝛼 0 dt R3 Cosq,v (a, k, t) =

∞ ∑ n=0

{ ( )} an t (n+1)q−1−v−𝛼 𝜋 cos (n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)q − v − 𝛼) 2 q > v,

t > 0, k = 0, 1, … , Dq Dv − 1. (8.98)

From the sum and difference formulas for the integer-order trigonometry, the following identities are derived: cos(x) = cos(y) cos(x − y) − sin(y) sin(x − y), (8.99) and sin(x) = cos(y) sin(x − y) + sin(y) cos(x − y). (8.100) ) ( Now, in equation (8.98), let x = (n(q + 1) + q − 1 − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, also, ) ( let y = 𝛼 𝜋2 + 2𝜋k , then applying equation (8.99) 𝛼 0 dt R3 Cosq,v (a, k, t) ∞

= cos(y)

∑

an t (n+1)q−1−v−𝛼 cos{x − y} Γ((n + 1)q − v − 𝛼)

n=0 ∞

− sin(y)

∑ n=0

an t (n+1)q−1−v−𝛼 sin{x − y}, q > v, t > 0 k = 0, 1, … , Dq Dv − 1. (8.101) Γ((n + 1)q − v − 𝛼)

The summations are recognized as R3 Cosq,v+𝛼 (a, k, t) and R3 Sinq,v+𝛼 (a, k, t), respectively, yielding the final result 𝛼 0 dt R3 Cosq,v (a, k, t)

= cos(y)R3 Cosq,v+𝛼 (a, k, t) − sin(y)R3 Sinq.v+𝛼 (a, k, t), q > v, t > 0, k = 0, 1, … , Dq Dv − 1,

where y = 𝛼

(

𝜋 2

(8.102)

) + 2𝜋k . Taking 𝛼 = 1 we have cos(y) = 0 and sin(y) = 1 giving

0 dt R3 Cosq,v (a, k, t)

q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

= −R3 Sinq.v+1 (a, k, t),

(8.103)

Now, taking q = 1 and v = 0 gives 0 dt R3 Cos1,0 (a, k, t)

= −R3 Sin1,1 (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1,

(8.104)

which evaluates to 0 dt e

−at

= −a e−at ,

t > 0.

(8.105)

An alternative development of the result of equation (8.103) is obtained as follows: ] [ Rq,v (ia, it) + Rq,v (−ia, −it) 𝛼 𝛼 d R Cos (a, t) = d , t > 0, 0 t 3 q,v 0 t 2

(8.106)

by the differintegration equation (3.114) 𝛼 0 dt R3 Cosq,v (a, t) =

i𝛼 Rq,v+𝛼 (ia, it) + (−i)𝛼 Rq,v+𝛼 (−ia, −it) 2

,

q > v, t > 0.

(8.107)

147

148

8 The R3 -Trigonometric Functions

When 𝛼 = 1, we have 0 dt R3 Cosq,v (a, t)

= =

iRq,v+1 (ia, it) − iRq,v+1 (−ia, −it) 2 −Rq,v+1 (ia, it) + Rq,v+1 (−ia, −it) 2i

,

q > v, t > 0,

(8.108)

which is recognized as 0 dt R3 Cosq,v (a, t)

= −R3 Sinq,v+1 (a, t),

q > v,

t > 0.

(8.109)

8.5.2 R3 Sinq,v (a, k, t)

Determination of the differintegral for the R3 Sinq,v (a, k, t)-function proceeds in a similar manner to the R3 Cosq,v (a, k, t). Then, the 𝛼-order differintegral of R3 Sinq,v (a, k, t) is determined as 𝛼 0 dt R3 Sinq,v (a, k, t)

=

𝛼 0 dt

∞ { ( )} ∑ an t (n+1)q−1−v 𝜋 sin (n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, … , Dq Dv − 1, (8.110) 𝛼 0 dt R3 Sinq,v (a, k, t) =

{ ( )} 𝜋 sin (n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)q − v) 2

∞ ∑ an 0 dt𝛼 t (n+1)q−1−v n=0

t > 0, k = 0, 1, … , Dq Dv − 1.

(8.111)

Application of equation (8.97) to this equation gives ∞ { ( )} ∑ an t (n+1)q−1−v−𝛼 𝜋 𝛼 sin (n(q + 1) + q − 1 − v) + 2𝜋 k , 0 dt R3 Sinq,v (a, k, t) = Γ((n + 1)q − v − 𝛼) 2 n=0 q > v,

t > 0, k = 0, 1, … , Dq Dv − 1. (8.112) ) ( Now, in equation (8.112), let x = (n(q + 1) + q − 1 − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, ) ( also, let y = 𝛼 𝜋2 + 2𝜋k ; then applying equation (8.100), we have 𝛼 0 dt R3 Sinq,v (a, k, t) ∞

= cos(y)

∑

an t (n+1)q−1−v−𝛼 sin{x − y} Γ((n + 1)q − v − 𝛼)

n=0 ∞

+ sin(y)

∑ n=0

an t (n+1)q−1−v−𝛼 cos{x − y}, Γ((n + 1)q − v − 𝛼)

q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

The summations are recognized as R3 Sinq,v+𝛼 (a, k, t) and R3 Cosq,v+𝛼 (a, k, t), respectively, yielding the final result 𝛼 0 dt R3 Sinq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R3 Sinq,v+𝛼 (a, k, t) + sin(y)R3 Cosq.v+𝛼 (a, k, t),

q > v, t > 0, k = 0, 1, … , Dq Dv − 1, ) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R3 Sinq,v (a, k, t)

= R3 Cosq.v+1 (a, k, t),

q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

(8.113)

(8.114)

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

Taking q = 1 and v = 0 yields 0 dt R3 Sin1,0 (a, k, t)

t > 0, k = 0, 1, … , Dq Dv − 1,

= R3 Cos1,1 (a, k, t),

(8.115)

which evaluates to 0 dt 0

= a• 0

again the expected result. The alternative R-function-based development of the result of equation (8.114) is obtained based on equation (8.59), as ] [ Rq,v (ia, it) − Rq,v (−ia, −it) 𝛼 𝛼 d R Sin (a, t) = d , t > 0, (8.116) 0 t 3 q,v 0 t 2i by the differintegration equation (3.114) 𝛼 0 dt R3 Sinq,v (a, t)

i𝛼 Rq,v+𝛼 (ia, it) − (−i)𝛼 Rq,v+𝛼 (−ia, −it)

, q > v, t > 0, 2i i𝛼−1 Rq,v+1 (ia, it) − i−𝛼−1 Rq,v+1 (−ia, −it) 𝛼 d R Sin (a, t) = , q > v, t > 0. 0 t 3 q,v 2 =

When 𝛼 = 1, we have 0 dt R3 Sinq,v (a, t)

=

Rq,v+1 (ia, it) + Rq,v+1 (−ia, −it) 2

,

q > v, t > 0,

(8.117)

which is recognized as 0 dt R3 Sinq,v (a, t)

= R3 Cosq,v+1 (a, t),

q > v, t > 0.

(8.118)

8.5.3 R3 Corq,v (a, t)

Determination of the derivative for the R3 Corq,v (a, t)-function proceeds in similar manner to R3 Sinq,v (a, k, t). Then, the 𝛼-order differintegral of R3 Corq,v (a, t) is determined, using definition (8.16), as 𝛼 0 dt R3 Corq,v (a, t)

𝛼 0 dt R3 Corq,v (a,

=

t) =

𝛼 0 dt

∞ ∑ a2n (t)(n+1)2q−1−v−q i2n(q+1)+q−1−v

i a i

n=0

,

t > 0,

,

t > 0.

q > v,

t > 0.

Γ((n + 1)2q − v − q)

n=0 ∞ 2n 2n (n+1)2q−1−v−q ∑

𝛼 (n+1)2q−1−v−q 0 dt t

Γ((n + 1)2q − v − q)

(8.119)

Application of equation (8.97) to this equation gives 𝛼 0 dt R3 Corq,v (a, t) =

∞ ∑ i2n a2n i(n+1)2q−1−v−q t (n+1)2q−1−v−q−𝛼 n=0

Γ((n + 1)2q − v − q − 𝛼)

,

(8.120)

The summation is recognized as both an R-function and as an R3 Rot-function, yielding the final result 𝛼 0 dt R3 Corq,v (a, t)

= i𝛼 R2q,v+q+𝛼 (−a2 , i t) =

i𝛼−1 (a, t), R Rot a 3 q,v+q+𝛼

q > v, t > 0.

(8.121)

149

150

8 The R3 -Trigonometric Functions

8.5.4 R3 Rotq,v (a, t)

Determination of the derivative for the R3 Rotq,v (a, t)-function proceeds in a manner that is similar to R3 Corq,v (a, t). Then, the 𝛼-order differintegral of R3 Rotq,v (a, t) is determined as 𝛼 𝛼 2q−v a 0 dt R3 Rotq,v (a, t) = 0 dt i

∞ ∑ (i2(q+1) a2 )n (t)(n+1)2q−1−v n=0

𝛼 0 dt R3 Rotq,v (a, t) = a

Γ((n + 1)2q − v)

∞ 2n 2n 2nq+2q−v 𝛼 (n+1)2q−1−v ∑ i a i 0 dt t

Γ((n + 1)2q − v)

n=0

,

t > 0,

(8.122)

.

Application of equation (8.97) to this equation gives 𝛼 𝛼+1 0 dt R3 Rotq,v (a, t) = ai

∞ ∑ (−a2 )n i2nq+2q−1−v−𝛼 t (n+1)2q−1−v−𝛼 n=0

Γ((n + 1)2q − v − 𝛼)

,

2q > v, t > 0.

(8.123)

The summation is recognized as both an R-function and as an R3 Rot-function, yielding 𝛼 0 dt R3 Rotq,v (a, t)

= a i𝛼+1 R2q,v+𝛼 (−a2 , i t) = i𝛼 R3 Rotq,v+𝛼 (a, t),

2q > v, t > 0.

(8.124)

8.5.5 R3 Coflutq,v (a, k, t)

In this section, we determine the derivative of the R3 Coflutter-function. The 𝛼-order differintegral of R3 Coflq,v (a, k, t) is determined as 𝛼 0 dt R3 Coflq,v (a, k, t) ∞ ∑ a2n+1 t (n+1)2q−1−v = 0 dt𝛼 Γ((n + 1)2q − v) n=0

{ ( )} 𝜋 cos (2n(q + 1) + 2q − v) + 2𝜋 k , 2 t > 0, k = 0, 1, … , Dq Dv − 1.

(8.125)

Differintegrating term-by-term, 𝛼 0 dt R3 Coflq,v (a, k, t) ∞ ∑ a2n+1 0 dt𝛼 t (n+1)2q−1−v

=

n=0

{ ( )} 𝜋 cos (2n(q + 1) + 2q − v) + 2𝜋 k , Γ((n + 1)2q − v) 2 t > 0, k = 0, 1, … , Dq Dv − 1.

Application of equation (8.97) to this equation gives 𝛼 0 dt R3 Coflq,v (a, k, t) ∞ 2n+1 (n+1)2q−1−v−𝛼 ∑

=

n=0

{ ( )} a t 𝜋 cos (2n(q + 1) + 2q − v) + 2𝜋 k , Γ((n + 1)2q − v − 𝛼) 2 2q > v,

t > 0, k = 0, 1, … , Dq Dv − 1.

(8.126)

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

( ) Now, in equation (8.126), let x = (2n(q + 1) + 2q − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, also, ) ( let y = 𝛼 𝜋2 + 2𝜋k ; then applying equation (8.99), we have 𝛼 0 dt R3 Coflq,v (a, t) = cos(y)

∞ ∑ a2n+1 t (n+1)2q−1−v−𝛼 cos{x − y} Γ((n + 1)2q − v − 𝛼) n=0

− sin(y)

∞ ∑ n=0

a2n+1 t (n+1)q−1−v−𝛼 sin{x − y}, Γ((n + 1)2q − v − 𝛼) 2q > v,

t > 0, k = 0, 1, … , Dq Dv − 1.

The summations are recognized as R3 Coflq,v+𝛼 (a, k, t) and R3 Flutq,v+𝛼 (a, k, t) yielding the desired derivative form 𝛼 0 dt R3 Coflq,v (a, k, t)

where y = 𝛼

(

= cos(y)R3 Coflq,v+𝛼 (a, k, t) − sin(y)R3 Flutq,v+𝛼 (a, k, t), 2q > v, t > 0, k = 0, 1, … , Dq Dv − 1,

𝜋 2

(8.127)

) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R3 Coflq,v (a, k, t)

= −R3 Flutq,v+1 (a, k, t),

2q > v, t > 0, k = 0, 1, … , Dq Dv − 1. (8.128)

Taking q = 1 and v = 0, in equation (8.127), gives the derivative as 𝛼 0 dt R3 Cofl1,0 (a, k, t)

= cos(y)R3 Cofl1,𝛼 (a, k, t) − sin(y)R3 Flut1,𝛼 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1,

(8.129)

and with 𝛼 = 1 we have 0 dt R3 Cofl1,0 (a, k, t)

= −R3 Flut1,1 (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1.

However, R3 Cofl1,0 (a, k, t) = − sinh(at), thus t > 0, k = 0, 1, … , Dq Dv − 1. (8.130) The alternative R-function-based development uses equation (8.70) to obtain [ ] 2 2 R (−a , −it) − R (−a , it) 2q,v 2q,v 𝛼 𝛼 , t > 0, (8.131) 0 dt R3 Coflq,v (a, t) = 0 dt a 2i 0 dt R3 Cofl1,0 (a, k, t)

= −R3 Flut1,1 (a, k, t) = −a cosh(at),

by the differintegration equation (3.114) 𝛼 0 dt R3 Coflq,v (a, t)

=a

𝛼 0 dt R3 Coflq,v (a, t)

=a

(−i)𝛼 R2q,v+𝛼 (−a2 , −it) − i𝛼 R2q,v+𝛼 (−a2 , it) 2i i−𝛼 R2q,v+𝛼 (−a2 , −it) − i𝛼 R2q,v+𝛼 (−a2 , it) 2i

,

,

2q > v, t > 0, 2q > v, t > 0.

(8.132)

When 𝛼 = 1, we have for t > 0 0 dt R3 Coflq,v (a, t) = a

R2q,v+1 (−a2 , it) − R2q,v+1 (−a2 , −it)

as seen in equation (8.128).

−2

= −R3 Flutq,v+1 (a, t),

2q > v,

151

152

8 The R3 -Trigonometric Functions

8.5.6 R3 Flutq,v (a, k, t)

The 𝛼-order differintegral of R3 Flutq,v (a, k, t) is determined as 𝛼 0 dt R3 Flutq,v (a, k, t) ∞ ∑ a2n+1 t (n+1)2q−1−v = 0 dt𝛼 Γ((n + 1)2q − v) n=0

{ ( )} 𝜋 sin (2n(q + 1) + 2q − v) + 2𝜋 k , 2 t > 0, k = 0, 1, … , Dq Dv − 1,

(8.133)

𝛼 0 dt R3 Flutq,v (a, k, t) ∞ ∑ a2n+1 0 dt𝛼 t (n+1)2q−1−v

=

n=0

{ ( )} 𝜋 sin (2n(q + 1) + 2q − v) + 2𝜋 k , Γ((n + 1)2q − v) 2 t > 0, k = 0, 1, … , Dq Dv − 1.

Application of equation (8.97) to this equation gives 𝛼 0 dt R3 Flutq,v (a, k, t) ∞ 2n+1 (n+1)2q−1−v−𝛼 ∑

=

n=0

{ ( )} a t 𝜋 sin (2n(q + 1) + 2q − v) + 2𝜋 k , Γ((n + 1)2q − v − 𝛼) 2

2q > v, t > 0, k = 0, 1, … , Dq Dv − 1. ( ) x = (2n(q + 1) + 2q − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, also

Now, let ) ( 𝜋 y = 𝛼 2 + 2𝜋k ; then, applying equation (8.100), we have 𝛼 0 dt R2 Flutq,v (a, k, t) = cos(y)

let

∞ ∑ a2n+1 t (n+1)2q−1−v−𝛼 sin{x − y} Γ((n + 1)2q − v − 𝛼) n=0

∞ ∑ a2n+1 t (n+1)2q−1−v−𝛼 + sin(y) cos{x − y}, Γ((n + 1)2q − v − 𝛼) n=0

2q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

(8.134)

The summations are recognized as R3 Flutq,v+𝛼 (a, k, t) and R3 Cof lq,v+𝛼 (a, k, t), respectively, yielding the final result 𝛼 0 dt R3 Flutq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R3 Flutq,v+𝛼 (a, k, t) + sin(y)R3 Cof lq,v+𝛼 (a, k, t),

2q > v, t > 0, k = 0, 1, … , Dq Dv − 1, ) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R3 Flutq,v (a, k, t)

(8.135)

= R3 Cof lq,v+1 (a, k, t), 2q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

Taking q = 1 and v = 0, in equation (8.135), we obtain 𝛼 0 dt R3 Flut1,0 (a, k, t)

= cos(y)R3 Flut1,𝛼 (a, k, t) + sin(y)R3 Cof l1,𝛼 (a, k, t),

When we also have 𝛼 = 1, this becomes 0 dt R3 Flut1,0 (a, k, t)

= R3 Cof l1,1 (a, k, t),

t > 0.

However, R3 Flut1,0 (a, k, t) = 0, thus indicating that R3 Cof l1,1 (a, k, t) = 0,

t > 0, k = 0, 1, … , Dq Dv − 1.

t > 0.

(8.136)

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

The alternative R-function-based development uses equation (8.71) to obtain [ ] 2 2 R (−a , it) + R (−a , −it) 2q,v 2q,v 𝛼 𝛼 , t > 0, 0 dt R3 Flutq,v (a, t) = 0 dt a 2 by the differintegration equation (3.114) [ 𝛼 ] i R2q,v+𝛼 (−a2 , it) + i−𝛼 R2q,v+𝛼 (−a2 , −it) 𝛼 , 0 dt R3 Flutq,v (a, t) = a 2 When 𝛼 = 1, we have 0 dt R3 Flutq,v (a, t)

[ =a

R2q,v+1 (−a2 , −it) − R2q,v+1 (−a2 , it) 2i

(8.137)

2q > v, t > 0.

] ,

2q > v, t > 0,

which is recognized as 0 dt R3 Flutq,v (a, t)

= R3 Cof lq,v+1 (a, t),

2q > v, t > 0.

(8.138)

8.5.7 R3 Covibq,v (a, k, t)

In this section, we determine the derivative of the R3 Covibq,v (a, k, t)-function. The 𝛼-order differintegral of R3 Covibq,v (a, k, t) is determined as 𝛼 0 dt R3 Covibq,v (a, k, t) ∞ ∑ a2n t (n+1)2q−1−v−q 𝛼 = 0 dt Γ((n + 1)2q − v − q) n=0

{ ( )} 𝜋 cos (2n(q + 1) + q − 1 − v) + 2𝜋 k , 2 t > 0, k = 0, 1, … , Dq Dv − 1.

(8.139)

Differintegrating term-by-term gives 𝛼 0 dt R3 Covibq,v (a, k, t) ∞ ∑ a2n 0 dt𝛼 t (n+1)2q−1−v−q

=

n=0

{ ( )} 𝜋 cos (2n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)2q − v − q) 2 t > 0, k = 0, 1, … , Dq Dv − 1.

Application of equation (8.97) to this equation gives 𝛼 0 dt R3 Covibq,v (a, k, t) ∞ 2n (n+1)2q−1−v−q−𝛼 ∑

=

n=0

{ ( )} a t 𝜋 cos (2n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)2q − v − q − 𝛼) 2

(8.140) q > v, t > 0, k = 0, 1, … , Dq Dv − 1. ( ) Now, in equation (8.140), let x = (2n(q + 1) + q − 1 − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, ) ( also let y = 𝛼 𝜋2 + 2𝜋k ; then, applying equation (8.99), we have 𝛼 0 dt R3 Covibq,v (a, k, t) ∞ 2n (n+1)2q−1−v−q−𝛼 ∑

= cos(y)

a t cos{x − y} Γ((n + 1)q − v − q − 𝛼)

n=0 ∞

− sin(y)

∑ n=0

a2n t (n+1)2q−1−v−q−𝛼 sin{x − y}, Γ((n + 1)2q − v − q − 𝛼)

q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

153

154

8 The R3 -Trigonometric Functions

The summations are recognized as R3 Covibq,v+𝛼 (a, k, t) and R3 Vibq,v+𝛼 (a, k, t), respectively, yielding the final result: 𝛼 0 dt R3 Covibq,v (a, k, t)

where y = 𝛼

(

𝜋 2

= cos(y)R3 Covibq,v+𝛼 (a, k, t) − sin(y)R3 Vibq,v+𝛼 (a, k, t),

q > v, t > 0, k = 0, 1, … , Dq Dv − 1, ) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R3 Covibq,v (a, k, t)

(8.141)

q > v, t > 0, k = 0, 1, … , Dq Dv − 1. (8.142)

= −R3 Vibq,v+1 (a, k, t),

Setting q = 1 and v = 0, we obtain 0 dt R3 Covib1,0 (a, k, t)

= −R3 Vib1,1 (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1.

Because R3 Covib1,0 (a, k, t) = cosh(a t), we have R3 Vib1,1 (a, k, t) = −a sinh(a t),

t > 0, k = 0, 1, … , Dq Dv − 1.

(8.143)

The alternative R-function-based development of the results of equation (8.82) is obtained as [ ] R2q,v+q (−a2 , it) + R2q,v+q (−a2 , −it) 𝛼 𝛼 , t > 0. (8.144) 0 dt R3 Covibq,v (a, t) = 0 dt 2 By the differintegration equation (3.114) i𝛼 R2q,v+q+𝛼 (−a2 , it) + (−i)𝛼 R2q,v+q+𝛼 (−a2 , −it) 𝛼 d R Covib (a, t) = , q > v, t > 0, 0 t 3 q,v 2 i𝛼 R2q,v+q+𝛼 (−a2 , it) + i−𝛼 R2q,v+q+𝛼 (−a2 , −it) 𝛼 , q > v, t > 0. (8.145) 0 dt R3 Covibq,v (a, t) = 2 When 𝛼 = 1, we have R2q,v+q+1 (−a2 , it) − R2q,v+q+1 (−a2 , −it) d R Covib (a, t) = − 0 t 3 q,v 2i = −R3 Vibq,v+1 (a, t), q > v, t > 0, (8.146) and the equation is the same as equation (8.142). 8.5.8 R3 Vibq,v (a, k, t)

The 𝛼-order differintegral of R3 Vibq,v (a, k, t) is determined as 𝛼 0 dt R3 Vibq,v (a, k, t) ∞ ∑ a2n t (n+1)2q−1−v−q = 0 dt𝛼 Γ((n + 1)2q − v − q) n=0

{ ( )} 𝜋 sin (2n(q + 1) + q − 1 − v) + 2𝜋 k , 2 t > 0, k = 0, 1, … , Dq Dv − 1,

(8.147)

𝛼 0 dt R3 Vibq,v (a, k, t) ∞ ∑ a2n 0 dt𝛼 t (n+1)2q−1−v−q

=

n=0

{ ( )} 𝜋 sin (2n(q + 1) + q − 1 − v) + 2𝜋 k , Γ(2n(q + 1) + q − v) 2 t > 0, k = 0, 1, … , Dq Dv − 1.

Application of equation (8.97) to this equation gives 𝛼 0 dt R3 Vibq,v (a, k, t) ∞ 2n (n+1)2q−1−v−q−𝛼 ∑

=

n=0

{ ( )} a t 𝜋 sin (2n(q + 1) + q − 1 − v) + 2𝜋 k , Γ((n + 1)2q − v − q − 𝛼) 2 q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

(8.148)

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

( ) Now, in equation (8.148), let x = (2n(q + 1) + q − 1 − v) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv − 1, ) ( also let y = 𝛼 𝜋2 + 2𝜋k ; then, applying equation (8.100), we have 𝛼 0 dt R3 Vibq,v (a, k, t) = cos(y)

∞ ∑ n=0

+ sin(y)

a2n t (n+1)2q−1−v−q−𝛼 sin{x − y} Γ((n + 1)2q − v − q − 𝛼)

∞ ∑ n=0

a2n t (n+1)2q−1−v−q−𝛼 cos{x − y}, Γ((n + 1)2q − v − q − 𝛼)

q > v,

t > 0, k = 0, 1, … , Dq Dv − 1.

The summations are recognized as R3 Vibq,v+𝛼 (a, k, t) and R3 Covibq,v+𝛼 (a, k, t), respectively, yielding the final result: 𝛼 0 dt R3 Vibq,v (a, k, t)

where y = 𝛼

(

= cos(y)R3 Vibq,v+𝛼 (a, k, t) + sin(y)R3 Covibq,v+𝛼 (a, k, t), q > v, t > 0, k = 0, 1, … , Dq Dv − 1,

𝜋 2

(8.149)

) + 2𝜋k . Taking 𝛼 = 1, we have cos(y) = 0 and sin(y) = 1, giving

0 dt R3 Vibq,v (a, k, t)

= R3 Covibq,v+1 (a, k, t),

q > v, t > 0, k = 0, 1, … , Dq Dv − 1.

(8.150)

Now, taking q = 1 and v = 0 in equation (8.149), 𝛼 0 dt R3 Vib1,0 (a, k, t)

= cos(y)R3 Vib1,𝛼 (a, k, t) + sin(y)R3 Covib1,𝛼 (a, k, t), t > 0, k = 0, 1, … , Dq Dv − 1.

(8.151)

With 𝛼 = 1, q = 1, and v = 0, we have 0 dt R3 Vib1,0 (a, k, t)

= R3 Covib1,1 (a, k, t),

t > 0, k = 0, 1, … , Dq Dv − 1.

However, R3 Vib1,0 (a, k, t) = 0; therefore, we have R3 Covib1,1 (a, k, t) = 0. The alternative R-function-based development uses equation (8.82) giving ] [ R2q,v+q (−a2 , it) − R2q,v+q (−a2 , −it) 𝛼 𝛼 , t > 0, 0 dt R3 Vibq,v (a, t) = 0 dt 2i

(8.152)

by the differentiation equation (3.114) 𝛼 0 dt R3 Vibq,v (a, t) =

i𝛼 R2q,v+q+𝛼 (−a2 , it) − (−i)𝛼 R2q,v+q+𝛼 (−a2 , −it) 2i

,

q > v, t > 0.

When 𝛼 = 1, we have 0 dt R3 Vibq,v (a, t) =

R2q,v+q+1 (−a2 , it) + R2q,v+q+1 (−a2 , −it) 2

,

q > v, t > 0,

(8.153)

which is 0 dt R3 Vibq,v (a, t)

= R3 Covibq,v+1 (a, t),

q > v, t > 0.

(8.154)

Tables 8.1 and 8.2 summarize the various properties of the R3 -trigonometric functions.

155

For this table, 𝛼q = q

R3 Corq,v (a, t)

R3 Rotq,v (a, t)

R3 Cof lq,v (a, k, t)

R3 Flutq,v (a, k, t)

R3 Covibq,v (a, k, t)

R3 Vibq,v (a, k, t)

R3 Cosq,v (a, k, t)

(

sv (sin(𝛽q,v )sq − a sin(𝛽q,v − 𝛼q+1 ))

( ( )) ∑ an t (n+1)q−1−v 𝜋 sin (n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)q − v) 2 n=0

R3 Sinq,v (a, k, t)

( ( )) 𝜋 a2n t (n+1)2q−1−v−q cos (2n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)2q − v − q) 2

∑

,

s4q − 2a2 cos(𝛼2q+2 )s2q + a4

, ,

_

s4q − 2a2 cos(𝛼2q+2 )s2q + a4

a sv (cos(𝛽2q,v−1 )s2q − a2 cos(𝛽2q,v−1 − 𝛼2q+2 ))

s4q − 2a2 cos(𝛼2q+2 )s2q + a4

, ,

2q > v.

2q > v

q>v

a sv (sin(𝛽2q,v−1 )s2q − a2 sin(𝛽2q,v−1 − 𝛼2q+2 ))

s4q − 2a2 cos(𝛼2q+2 )s2q + a4

q>v

q>v

sv+q (cos(𝛽q,v )s2q − a2 cos(𝛽q,v − 𝛼2q+2 ))

∞ ∑ (i2(q+1) a2 )n t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0

( ) ) 𝜋 𝜋 + 2𝜋k , 𝛽q,v = (q − 1 − v) + 2𝜋 k , t > 0, and k = 0, 1, 2, … , Dq Dv − 1. 2 2

iq−1−v

s2q − 2a cos(𝛼q+1 )sq + a2

q>v

sv+q (sin(𝛽q,v )s2q − a2 sin(𝛽q,v − 𝛼2q+2 ))

_

i2q−v a

,

sv (cos(𝛽q,v )sq − a cos(𝛽q,v − 𝛼q+1 ))

∞ ∑ (i2(q+1) a2 )n t (n+1)2q−1−v Γ((n + 1)2q − v) n=0

∞ ( ( )) ∑ 𝜋 a2n+1 t (n+1)2q−1−v cos (2n(q + 1) + 2q − v) + 2𝜋 k Γ((n + 1)2q − v) 2 n=0

( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 sin (2n(q + 1) + 2q − v) + 2𝜋 k Γ((n + 1)2q − v) 2 n=0

n=0 ∞

n=0 ∞

( ( )) 𝜋 a2n t (n+1)2q−1−v−q sin (2n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)2q − v − q) 2

∞ ∑

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 cos (n(q + 1) + q − 1 − v) + 2𝜋k Γ((n + 1)q − v) 2 n=0

s2q − 2a cos(𝛼q+1 )sq + a2

Laplace transform

∞

Series definition

Function

Table 8.1 Summary of R3 -functions

8.5 Fractional Calculus Operations on the R3 -Trigonometric Functions

Table 8.2 Summary of R3 -functions Function

R3 Sinq,v (a, k, t) R3 Cosq,v (a, k, t) R3 Vibq,v (a, k, t) R3 Covibq,v (a, k, t)

q = 1, v = 0, k = 0

Im(Rq,v (ia, k, it))

0

Re(Rq,v (ia, k, it))

e−a t

Im(Ea (Rq,v (ia, k, it))) = Ea (Im(Rq,v (ia, k, it)))

0

Re(Ea (Rq,v (ia, k, it))) = Ea (Re(Rq,v (ia, k, it)))

cosh(at)

) )

Im(Oa (Rq,v (ia, k, it))) = Oa (Im(Rq,v (ia, k, it)))

Rq,v (ia, it) − Rq,v (−ia, −it) 2i Rq,v (ia, it) + Rq,v (−ia, −it) 2 R2q,v+q (−a2 , it) − R2q,v+q (−a2 , −it) 2i R2q,v+q (−a2 , it) + R2q,v+q (−a2 , −it) (

R3 Flutq,v (a, k, t)

Relation to Rq,v (ia, it)

R-Function equivalent

a (

2 R2q,v (−a2 , it) + R2q,v (−a2 , −it) 2 R2q,v (−a2 , −it) − R2q,v (−a2 , it)

Re(Oa (Rq,v (ia, k, it))) = Oa (Re(Rq,v (ia, k, it)))

sinh(at)

a iR2q,v (−a2 , i t)

Oa (Rq,v (ia, it))

− sinh(at)

R2q,v+q (−a , i t)

Ea (Rq,v (ia, it))

cosh(at)

R3 Cof lq,v (a, k, t)

a

R3 Rotq,v (a, t) R3 Corq,v (a, t)

2i

2

For this table, t > 0, k = 0, 1, 2, … , Dq Dv − 1.

8.5.9

Summary of Fractional Calculus Operations on the R3 -Trigonometric Functions

For ease of reference, the fractional calculus operations ( )are summarized here. The derivations are for t > 0, k = 0, 1, … , Dq Dv − 1, and y = 𝛼 𝜋2 + 2𝜋k : 𝛼 0 dt R3 Cosq,v (a, k, t)

= cos(y)R3 Cosq,v+𝛼 (a, k, t) − sin(y)R3 Sinq.v+𝛼 (a, k, t),

q > v,

(8.102)

𝛼 0 dt R3 Sinq,v (a, k, t)

= cos(y)R3 Sinq,v+𝛼 (a, k, t) + sin(y)R3 Cosq.v+𝛼 (a, k, t),

q > v,

(8.113)

i𝛼−1 (a, t), R Rot a 3 q,v+q+𝛼

𝛼 0 dt R3 Corq,v (a, t)

= i𝛼 R2q,v+q+𝛼 (−a2 , i t) =

𝛼 0 dt R3 Rotq,v (a, t)

= a i𝛼 R2q,v+𝛼 (−a2 , i t) = i𝛼−1 R3 Rotq,v+𝛼 (a, t),

𝛼 0 dt R3 Coflutq,v (a, k, t)

𝛼 0 dt R3 Flutq,v (a, k, t)

q > v,

2q > v,

= cos(y)R3 Coflutq,v+𝛼 (a, k, t) − sin(y)R3 Flutq,v+𝛼 (a, k, t),

(8.121) (8.124) 2q > v, (8.127)

= cos(y)R3 Flutq,v+𝛼 (a, k, t) + sin(y)R3 Cof lq,v+𝛼 (a, k, t),

2q > v, (8.135)

𝛼 0 dt R3 Covibq,v (a, k, t)

= cos(y)R3 Covibq,v+𝛼 (a, k, t) − sin(y)R3 Vibq,v+𝛼 (a, k, t),

q > v, (8.141)

𝛼 0 dt R3 Vibq,v (a, k, t)

= cos(y)R3 Vibq,v+𝛼 (a, k, t) + sin(y)R3 Covibq,v+𝛼 (a, k, t),

q > v. (8.149)

157

159

9 The Fractional Meta-Trigonometry In the previous four chapters, fractional generalizations of the integer-order trigonometric and hyperbolic functions have been developed. In Chapters 5 and 6, we have seen the relationship of the R1 -trigonometry to the R1 -hyperboletry. Such a direct relationship for the R2 -trigonometry, however, has not been found. After some study of this problem, it has become apparent that more complicated relationships exist between the bases for the various trigonometries. This chapter presents a generalization that helps explore these connections. Furthermore, the chapter contains the hyperboletry1 and all1 of the trigonometries that we have explored to this point. This chapter also presents new trigonometries (Lorenzo [82, 83]) not previously available and potentially allows the solution of linear fractional differential equations in terms of these special functions. The result then is the master fractional trigonometry based on the R-function. The bases of the trigonoboletries that we have considered to this point are as follows: Rq,v (a, t) ⇒ R1 -Hyperboletry

−

Chapter 5

Rq,v (ai, t) ⇒ R1 -Trigonometry

−

Chapter 6

Rq,v (a, it) ⇒ R2 -Trigonometry

−

Chapter 7

Rq,v (ai, it) ⇒ R3 -Trigonometry

−

Chapter 8.

Clearly, trigonometries can be created for, or associated with, all of the indexed forms of the R-function presented in Table 3.2, which is a total of 16 integer-valued bases. Many of these bases will likely have important application in the fractional calculus and in the solution of fractional differential equations. However, a more global approach is considered here. This chapter considers fractional meta-trigonometries based on Rq,v (ai𝛼 , i𝛽 t),

t > 0.

With 𝛼 and 𝛽 ranging between 0 and 4, all of the integer-valued bases are considered together with infinitely more noninteger possibilities. Multiplication of the a or t variables by i𝛼 rotates the variable in the complex plane by the amount 𝛼𝜋∕2 radians. This is shown graphically in Figure 9.1. These rotations can have profound effects on the real and imaginary projections, which relate to the generalized functions. This chapter is adapted from Lorenzo [82, 83], with permission of ASME.

1 Only for the principal functions for the R1 -trigonometry and hyperboletry. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

160

9 The Fractional Meta-Trigonometry

Figure 9.1 Graphical display of i𝛼 for 0 ≤ 𝛼 ≤ 4 in steps of 𝛼 = 1∕2. Source: Lorenzo 2009a [82]. Reproduced with permission of ASME.

i i 3/2

i1/2

i0 = i4

i2

i 7/2

i 5/2 i3

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity We start by separating Rq,v ( ai𝛼 , i𝛽 t) into real and imaginary parts. Thus, we consider Rq,v ( ai𝛼 , i𝛽 t) =

∞ ∑ (ai𝛼 )n (i𝛽 t)(n+1)q−1−v n=0 ∞

Rq,v ( ai𝛼 , i𝛽 t) =

Γ((n + 1)q − v)

,

t > 0,

∑ an t (n+1)q−1−v i𝛼n+𝛽[(n+1)q−1−v] n=0

Γ((n + 1)q − v)

,

(9.1) t > 0.

(9.2)

Now, for rational q, v, 𝛼, and 𝛽, we may write 𝛼n + 𝛽 [(n + 1)q − 1 − v] = 𝛼n + 𝛽 nq + 𝛽 q − 𝛽 − 𝛽 v nN𝛼 Dq Dv D𝛽 + n N𝛽 Nq Dv D𝛼 + N𝛽 Nq Dv D𝛼 − N𝛽 Dq Dv D𝛼 − N𝛽 Nv Dq D𝛼 = Dq Dv D𝛼 D𝛽 M = , D

(9.3)

where q = Nq ∕Dq , v = Nv ∕Dv , 𝛼 = N𝛼 ∕D𝛼 , and 𝛽 = N𝛽 ∕D𝛽 . Then, we may write i𝛼n+𝛽[(n+1)q−1−v] = cos(𝜉) + i sin(𝜉) = cis(𝜉), ( ) ( ) 𝜋 𝜋 + 2𝜋k or 𝜉 = (𝛼n + 𝛽[(n + 1)q − 1 − v]) + 2𝜋k and k = 0, 1, 2, … , D − where 𝜉 = M D 2 2 1 and k = 0, 1, 2, … , D − 1, and M/D is rational and in minimal form. Therefore, equation (9.2) may be written as ∞ ∑ an t (n+1)q−1−v Rq,v ( ai , i t) = cis(𝜉), Γ((n + 1)q − v) n=0 𝛼

𝛽

t > 0,

(9.4)

∞ ∞ ∑ ∑ an t (n+1)q−1−v an t (n+1)q−1−v Rq,v ( ai , k, i t) = cos(𝜉) + i sin(𝜉) Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0 𝛼

𝛽

t > 0, k = 0, 1, 2, … , D − 1,

(9.5)

where k is included in the R-function argument to reflect its presence in 𝜉 on the right-hand side. Then, similar to the earlier definitions, we define the generalized fractional

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity

functions as Cosq,v (a, 𝛼, 𝛽, k, t) ≡

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 cos (n (𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

= Re(Rq,v (ai𝛼 , i𝛽 t)),

t > 0, k = 0, 1, 2, … , D − 1,

(9.6)

Sinq,v (a, 𝛼, 𝛽, k, t) ≡

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 sin (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

= Im(Rq,v (ai𝛼 , i𝛽 t)),

t > 0, k = 0, 1, 2, … , D − 1.

(9.7)

These definitions generalize those for the trignoboletries that were developed in previous chapters. The notation is changed, and here drops the preceding R and its subscript. These fractional meta-trigonometric functions are discriminated from the traditional trigonometric functions by the capitalization and the subscripted order variables. Continuing from Ref. [82] with permission of ASME: As with the previous trigonometric functions, we define the principal functions for t > 0 as Cosq,v (a, 𝛼, 𝛽, t) ≡ Cosq,v (a, 𝛼, 𝛽, 0, t) ∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , cos (𝛼n + 𝛽 [(n + 1)q − 1 − v]) = Γ((n + 1)q − v) 2 n=0

t > 0,

(9.8)

t > 0.

(9.9)

t > 0, k = 0, 1, 2, … , D − 1,

(9.10)

Sinq,v (a, 𝛼, 𝛽, t) ≡ Sinq,v (a, 𝛼, 𝛽, 0, t) ∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , sin (𝛼n + 𝛽 [(n + 1)q − 1 − v]) = Γ((n + 1)q − v) 2 n=0 Combining equations (9.5)–(9.7) gives Rq,v ( ai𝛼 , k, i𝛽 t) = Cosq,v (a, 𝛼, 𝛽, k, t) + i Sinq,v (a, 𝛼, 𝛽, k, t),

where D is the product of the denominators of Dq , Dv , D𝛼 , and D𝛽 in minimal form. This is the generalized fractional Euler equation. A complimentary fractional meta-Euler equation is derived later as equation (9.173). 9.1.1

Alternate Forms

It should be noted that the form of equation (9.1) was chosen for the convenience that it allowed either the “a” and/or the “t” variables to be made complex. The cost of this convenience, however, was to introduce two new variables, 𝛼 and 𝛽, into the defining summation. Complexly proportional forms with one less variable in the summation are possible and are discussed in Appendix E. 9.1.2

Graphical Presentation – Complexity Functions

A complete graphical presentation for the Cosq,v (a, 𝛼, 𝛽, k, t)- and Sinq,v (a, 𝛼, 𝛽, k, t)-functions is, of course, impossible. There are seven parameters and variables, and the possible number of

161

9 The Fractional Meta-Trigonometry

1.5

Cos1.05,0(1,𝛼,0,t)

1 0.5 𝛼 = 2.0 0 1.4, 2.6 –0.5

1.2, 2.8 𝛼 = 1.0, 3.0

–1 0

2

4

6

8

10

t-Time

Figure 9.2 Effect of 𝛼 on Cos1.05,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, 𝛽 = 0, v = 0, k = 0.

1

Cos1.00,0(1,𝛼,0,t)

162

0.5 𝛼 = 2.0 0 1.4, 2.6 1.2, 2.8

–0.5

𝛼 = 1.0, 3.0 –1 0

2

4

6

8

10

t-Time

Figure 9.3 Effect of 𝛼 on Cos1.00,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, 𝛽 = 0, v = 0, k = 0. Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

charts is limitless. In previous chapters, the effects of variations of q and v have been presented with 𝛼 and 𝛽 values of zero and one. Therefore, the emphasis here is on the new variables introduced in this chapter, namely 𝛼 and 𝛽. Figures 9.2–9.5 show the effect of varying 𝛼 in the Cosq,v (a, 𝛼, 𝛽, t) for q = 1.05, 1.00, 0.75, and 0.50, with a = 1.0, v = 𝛽 = k = 0, and with 𝛼 variations from 1.0 to 3.0. For the Cosq,0 -function in Figure 9.3, we see that Cosq,0 (1, 2 − Δ𝛼, 0, 0, t) = Cosq,0 (1, 2 + Δ𝛼, 0, 0, t) over the range shown. The study is repeated for Sinq,v (a, 𝛼, 𝛽, t) in Figures 9.6–9.9. For the Sinq,0 -functions, in these figures, it can be seen that the behavior is symmetric around 𝛼 = 2.0; that is, Sinq,0 (1, 2 − Δ𝛼, 0, 0, t) = −Sinq,0 (1, 2 + Δ𝛼, 0, 0, t) for 0 ≤ Δ𝛼 ≤ 1. In Figure 9.7, the fractional trigonometric functions appear to be quite similar to exponentially damped sinusoids or cosinusoids. The common feature of these figures is the selection

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity

Figure 9.4 Effect of 𝛼 on Cos0.75,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, 𝛽 = 0, v = 0, k = 0. Cos0.75,0(1,𝛼,0,t)

0.5

𝛼 = 2.0 0 1.4, 2.6 1.2, 2.8 𝛼 = 1.0, 3.0 –0.5 0

2

4

6

8

10

6

8

10

8

10

t-Time

Figure 9.5 Effect of 𝛼 on Cos0.25,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 0.25, a = 1, 𝛽 = 0, v = 0, k = 0.

0.25 0.2

Cos0.50,0(1,𝛼,0,t)

0.15 0.1 0.05

𝛼 = 2.0

0

1.4, 2.6 1.2, 2.8

–0.05

𝛼 = 1.0, 3.0

–0.1 2

0

4 t-Time

Figure 9.6 Effect of 𝛼 on Sin1.05,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, 𝛽 = 0, v = 0, k = 0.

α = 1.0 1 Sin1.05,0 (1,α,0,t)

1.2 0.5 0

–0.5 2.8 –1

α = 3.0 0

2

4

6 t-Time

163

9 The Fractional Meta-Trigonometry

Figure 9.7 Effect of 𝛼 on Sin1.00,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, 𝛽 = 0, v = 0, k = 0. Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

α = 1.0

1

Sin1.05,0 (1,α,0,t)

1.2 0.5 0 –0.5 2.8 –1

α = 3.0

0

2

4

6

8

10

t-Time

Figure 9.8 Effect of 𝛼 on Sin0.75,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, 𝛽 = 0, v = 0, k = 0.

0.8 α = 1.0

0.6

Sin0.75,0 (1,α,0,t)

0.4 0.2 0 –0.2 –0.4 –0.6

α = 3.0

–0.8 0

2

4

6

8

10

t-Time

Figure 9.9 Effect of 𝛼 on Sin0.50,0 (1, 𝛼, 0, t) with 𝛼 = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, 𝛽 = 0, v = 0, k = 0.

0.8 0.6 α = 1.0

0.4 Sin0.50,0 (1,α,0,t)

164

0.2 0 –0.2 –0.4

α = 3.0

–0.6 –0.8

0

2

4

6 t-Time

8

10

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity

of variables, that is, q = 1 and v = 𝛽 = k = 0. Continuing from Ref. [84] with permission of ASME: Relative to Figure 9.7, we have from equation (9.7) ∞ ( ( )) ∑ an t n 𝜋 , sin (n(𝛼 + 𝛽)) Sin1,0 (a, 𝛼, 𝛽, 0, t) = Γ(n + 1) 2 n=0 ∞ ( ( )) ∑ (a t)n 𝜋 = , t > 0. sin (n(𝛼 + 𝛽)) n ! 2 n=0 Now, from Ref. [56], pp. 118–119, #632, b2 b3 b sin 𝜃 + sin 2𝜃 + sin 3𝜃 + · · · 2! 3! ∞ ∑ bn sin n𝜃 = exp(b cos 𝜃) sin(b sin 𝜃). = n! n=0

(9.11)

(9.12)

Thus, Sin1,0 (a, 𝛼, 𝛽, 0, t) = exp(a t cos((𝛼 + 𝛽)𝜋∕2) sin(a t sin((𝛼 + 𝛽)𝜋∕2),

t > 0,

(9.13)

a closed-form summation. Also, based on Ref. [56], pp. 116–117, #631, we can determine Cos1,0 (a, 𝛼, 𝛽, 0, t) = exp(a t cos((𝛼 + 𝛽)𝜋∕2) cos(a t sin((𝛼 + 𝛽)𝜋∕2),

t > 0.

(9.14)

Because ((𝛼 + 𝛽)𝜋∕2) is simply a constant for any individual curve, we see that for these special cases the functions are indeed exponentially damped sinusoids under the q = 1 and v = 𝛽 = k = 0 constraints. The effects of variations in 𝛽 on Cosq,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, is shown in Figures 9.10a–9.13. Symmetry around 𝛽 = 2, similar to that observed for the 𝛼 variations, is seen in Figure 9.11 for Cos1.00,0 (1, 0, 𝛽, t). The effect of 𝛽 variations on the fractional Sinq,0 -functions is presented in Figures 9.14–9.17. For values of q < 1, an increase in 𝛽 increases the apparent damping as seen in Figures 9.16 and 9.17. Phase plane plots of Figures 9.18–9.21 study the effects of variations in 𝛼 for values of q = 1.05, 1.00, 0.95, and 0.50. A significant change in the nature of these cross plots is seen with small variations of q around q = 1.0 in Figures 9.18–9.20. We notice in Figure 9.21 the loss of oscillation due to the small value of q. Figures 9.22 and 9.23 show the effect of 𝛼 on the phase plane behavior of Sin1.00,1.0 (1, 𝛼, 0, t) versus Cos1.00,1.0 (1, 𝛼, 0, t) with q = 1, 𝛽 = 0, a = 1, and k = 0. By comparing these two figures, the effect of v, v = ±1.0, is seen to have a strong effect on the function behavior. Figures 9.24 and 9.25 show the effect of 𝛽 on the fractional Sinq,0 for q = 1.15 and q = 0.95. Note, for q > 1 the functions depart from the origin and for q < 1 the functions arrive from infinity. Figure 9.26 shows the phase plane, Sin0.85,0 (1, 0, 𝛽, t) versus Cos0.85,0 (1, 0, 𝛽, t), for 𝛽 = 1 − 2. Here, all the functions arrive from infinity with increasing t. However, for 𝛽 < 1.5 the functions are captured by the origin, while for 𝛽 > 1.5 the functions are repelled by the unit circle. Because of symmetrical occurrences in some physical processes, Figures 9.27–9.31 present the Sinq,v (a, 𝛼, 𝛽, k, t)- versus Cosq,v (a, 𝛼, 𝛽, k, t)-functions together with their symmetric partners −Sinq,v (a, 𝛼, 𝛽, k, t) versus −Cosq,v (a, 𝛼, 𝛽, k, t). Figure 9.31 shows an instance of a barred spiral. Such spirals are of interest in astrophysics; see Chapter 18. The various other special cases are self-explanatory.

165

9 The Fractional Meta-Trigonometry

Cos1.05,0 (1,0,β,t)

1

0.5 β = 2.0 0 β = 1.2

–0.5

–1

β = 1.0 0

2

4

6

8

10

8

10

t-Time (a)

1

Cos1.05,0 (1,0,β,t)

166

0.5 β = 2.0 0 β = 2.6

–0.5

β = 2.8

–1

β = 3.0 0

2

4

6 t-Time (b)

Figure 9.10 Effect of 𝛽 on Cos1.05,0 (1, 0, 𝛽, t) with (a) 𝛽 = 1.0–2.0 and (b) 𝛽 = 2.0–3.0 in steps of 0.2, with q = 1.05, a = 1, 𝛼 = 0, v = 0, k = 0.

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity Continuing from Ref. [82] with permission of ASME: We now consider Rq,v ( ai𝛼 , i𝛽 t) based on parity of the exponent of a. Then, equation (9.1) is written as ∞ ∑ (ai𝛼 )n (i𝛽 t)(n+1)q−1−v , t > 0, (9.15) Rq,v ( ai𝛼 , i𝛽 t) = Γ((n + 1)q − v) n=0 Rq,v ( ai𝛼 , i𝛽 t) = t q−1−v i𝛽(q−1−v)

∞ ∑ (a)n (t q )n in(𝛼+𝛽q) n=0

Γ((n + 1)q − v)

,

t > 0,

(9.16)

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

Figure 9.11 Effect of 𝛽 on Cos1.00,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, 𝛼 = 0, v = 0, k = 0. Cos1.05,0 (1,0,β,t)

1

0.5 β = 2.0 0 1.4, 2.6 1.2, 2.8

–0.5

–1

β = 1.0, 3.0 0

2

4

6

8

10

6

8

10

6

8

10

t-Time

Figure 9.12 Effect of 𝛽 on Cos0.75,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, 𝛼 = 0, v = 0, k = 0.

1

Cos0.75,0 (1,0,β,t)

β = 1.0 0.5

0

β = 3.0

–0.5 –1

0

2

4 t-Time

Figure 9.13 Effect of 𝛽 on Cos0.50,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, 𝛼 = 0, v = 0, k = 0.

2 1.5

Cos0.50,0 (1,0,β,t)

1

β = 1.0

0.5 0 –0.5

β = 3.0

–1 –1.5 –2 0

2

4 t-Time

167

9 The Fractional Meta-Trigonometry

Figure 9.14 Effect of 𝛽 on Sin1.05,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, 𝛼 = 0, v = 0, k = 0.

2 1.5

Sin1.50,0 (1,0,β,t)

1

β = 1.0

0.5 0

–0.5 –1 –1.5 –2 0

β = 3.0 2

4

6

8

10

t-Time

Figure 9.15 Effect of 𝛽 on Sin1.00,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, 𝛼 = 0, v = 0, k = 0.

2 1.5

Sin1.00,0 (1,0,β,t)

1

β = 1.0

0.5 0 –0.5 –1

β = 3.0

–1.5 –2 0

2

4

6

8

10

t-Time

Figure 9.16 Effect of 𝛽 on Sin0.75,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, 𝛼 = 0, v = 0, k = 0.

2 1.5

β = 1.0

1 Sin0.75,0(1,0,β,t)

168

0.5 0

–0.5

β = 3.0

–1 –1.5 –2 0

2

4

6 t-Time

8

10

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

2 1.5

β = 1.0

Sin0.50,0(1,0,β,t)

1 0.5 0 β = 3.0 –0.5 –1 –1.5 –2

0

2

4

6

8

10

t-Time

Figure 9.17 Effect of 𝛽 on Sin0.50,0 (1, 0, 𝛽, t) with 𝛽 = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, 𝛼 = 0, v = 0, k = 0.

1.5

1

α = 1.0

1.2 Sin1.00,0(1,α,0,t)

0.5

1.4

0

–0.5

2.6 2.8 α = 3.0

–1

–1.5 –1.5

–1

0.5 0 –0.5 Cos1.00,0(1,α,0,t)

1

1.5

Figure 9.18 Phase plane Sin1.00,0 (1, 𝛼, 0, t) versus Cos1.00,0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with a = 1.0, q = 1.00, 𝛽 = 0, k = 0, v = 0. Arrows indicate increasing function. Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

169

9 The Fractional Meta-Trigonometry

2 α = 1.0

1.5 1.2

1 Sin1.05,0(1,α,0,t)

1.4 0.5 0 –0.5 2.6 –1

2.8

–1.5 –2

α = 3.0

–2

–1

0 Cos1.05,0(1,α,0,t)

2

1

Figure 9.19 Phase plane Sin1.05,0 (1, 𝛼, 0, t) versus Cos1.05,0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with a = 1.0, q = 1.05, k = 0, 𝛽 = 0, v = 0. Arrows indicate increasing function parameter. 1 0.8

α = 1.0

0.6 0.4 Sin0.95,0(1,α,0,t)

170

1.4

1.2

0.2 α = 2.0

0 –0.2

2.6

–0.4

2.8

–0.6 α = 3.0

–0.8 –1 –1

–0.5

0 0.5 Cos0.95,0(1,α,0,t)

1

Figure 9.20 Phase plane Sin0.95,0 (1, 𝛼, 0, t) versus Cos0.95,0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with a = 1.0, q = 0.95, k = 0, 𝛽 = 0, v = 0.

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

1 α = 1.0

1.2

Sin0.50,0(1,α,0,t)

0.5

1.4

α = 2.0

0

2.6

–0.5

2.8 α = 3.0

–1 –0.5

0

0.5 1 Cos0.50,0(1,α,0,t)

1.5

2

Figure 9.21 Phase plane Sin0.50,0 (1, 𝛼, 0, t) versus Cos0.50,0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with a = 1.0, q = 0.50, k = 0, 𝛽 = 0, v = 0. Arrows indicate increasing function parameter.

α = 1.0

1 1.2 1.6

Sin1.00,1.0(1,α,0,t)

0.5

α = 2.0

0

–0.5

2.4 2.8

–1

α = 3.0

–1

–0.5

0

0.5

1

Cos1.00,1.0(1,α,0,t)

Figure 9.22 Phase plane Sin1.00,1.0 (1, 𝛼, 0, t) versus Cos1.00,1.0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with v = 1.0, a = 1.0, q = 1.00, k = 0, 𝛽 = 0. Arrows indicate increasing function parameter. Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

171

9 The Fractional Meta-Trigonometry

Figure 9.23 Phase plane Sin1.00,−1.0 (1, 𝛼, 0, t) versus Cos1.00,−1.0 (1, 𝛼, 0, t) for 𝛼 = 1–3 in steps of 0.2, with v = −1.0, a = 1.0, q = 1.00, k = 0, 𝛽 = 0.

2.5 α = 1.0

2 1.5

1.25

Sin1.00,–1.0(1,α,0,t)

1 1.5 0.5 2.0

0 –0.5

2.5

–1 2.75 –1.5 –2.0 –2.5

α = 3.0

–1

0

1

Cos1.00,–1.0(1,α,0,t)

1 0.8 0.6

β = 1.0 β = 1.1 1.2

0.4 Sin1.15,0(1,0,β,t)

172

1.5

0.2 0 –0.2 –0.4 –0.6 –0.8 –1 –1

–0.5

0 Cos1.15,0(1,0,β,t)

0.5

1

Figure 9.24 Phase plane Sin1.15,0 (1, 𝛼, 0, t) versus Cos1.15,0 (1, 𝛼, 0, t) for 𝛽 = 1–1.5 in steps of 0.1, with q = 1.15, v = 0, a = 1.0, k = 0, 𝛼 = 0.

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

1.5

Sin0.95,0(1,0,β,t)

1

0.5

β = 1.0

0

β = 3.0

–0.5 –1 –1.5 –1.5

–1

–0.5

0 0.5 Cos0.95,0(1,0,β,t)

1

1.5

2

Figure 9.25 Phase plane Sin0.95,0 (1, 0, 𝛽, t) versus Cos0.95,0 (1, 0, 𝛽, t) for 𝛽 = 1–3 in steps of 0.25, with q = 0.95, v = 0, a = 1.0, k = 0, 𝛼 = 0.

5 4 3

1.6

Sin0.85,0(–1,0,β,t)

2

1.4 1.4

1

1.2

0

β = 1.0 –1 –2

β = 2.0

–3 1.8 –4 –5 –5

0

5

Cos0.85,0(–1,0,β,t)

Figure 9.26 Phase plane Sin0.85,0 (1, 0, 𝛽, t) versus Cos0.85,0 (1, 0, 𝛽, t) for 𝛽 = 1–2 in steps of 0.2, with a = −1.0, q = 0.85, v = 0, k = 0, 𝛼 = 0, t = 0–12.

173

9 The Fractional Meta-Trigonometry

Figure 9.27 Phase plane Sin1.10,−1 (1, 1, 0.2, t) versus Cos1.10,−1 (1, 1, 0.2, t) and −Sin1.10,−1 (1, 1, 0.2, t) versus −Cos1.10,−1 (1, 1, 0.2, t) for 𝛽 = 0.2, with a = 1.0, q = 1.10, v = −1, k = 0, 𝛼 = 1, t = 0–18.

2 +/+

1.5

Sin1.10,–1(1,1,0,2,t)

1

0.5

0

–0.5

–1

–1.5

–/–

–2 –1

0

1

Cos1.10,0(1,1,0,2,t)

+/+

1 0.8 0.6 Sin1.05,–0.6(1,1,0,2,t)

174

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1

–/– –0.5

0 0.5 Cos1.05,–0.6(1,1,0.2,t)

Figure 9.28 Phase plane Sin1.05,−0.6 (1, 0.71, 0, t) versus Cos1.05,−0.6 (1, 0.71, 0, t) and for −Sin1.05,−0.6 (1, 0.71, 0, t) versus −Cos1.05,−0.6 (1, 0.71, 0, t)𝛽 = 0, with a = 1.0, q = 1.05, v = 0.2, k = 0, 𝛼 = 1, t = 0–20.

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

1 +/+

0.8 0.6

Sin1.05,0(1,1,0,1,t)

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

–/–

–1 –1

–0.5

0

0.5

1

Cos1.05,0(1,1,0.1,t)

Figure 9.29 Phase plane Sin1.05,0.1 (1, 1, 0.1, t) versus Cos1.05,0 (1, 1, 0.1, t) and for −Sin1.05,0.1 (1, 1, 0.1, t) versus −Cos1.05,0 (1, 1, 0.1, t), 𝛽 = 0.1, with a = 1.0, q = 1.05, v = 0, k = 0, 𝛼 = 1, t = 0–20.

Sin1.05,0.2(1,1,0,1,t)

1

+/+

0.5

0 –0.5 –1

–/–

–1.5

–1

0 0.5 –0.5 Cos1.05,0.2(1,1,0.1,t)

1

1.5

Figure 9.30 Phase plane Sin1.05,0.2 (1, 1, 0.1, t) versus Cos1.05,0.2 (1, 1, 0.1, t) and for −Sin1.05,0.2 (1, 1, 0.1, t) versus −Cos1.05,0.2 (1, 1, 0.1, t) 𝛽 = 0.1, with a = 1.0, q = 1.05, v = 0.2, k = 0, 𝛼 = 1, t = 0–20.

The summation becomes 1 a t q i(𝛼+𝛽q) a2 t 2q i2(𝛼+𝛽q) a3 t 3q i3(𝛼+𝛽q) a4 t 4q i4(𝛼+𝛽q) + + + + + · · ·. Γ(q − v) Γ(2q − v) Γ(3q − v) Γ(4q − v) Γ(5q − v)

(9.17)

Forming two summations by separating the even and odd powers of a, we have ⎧∑ ⎫ ∞ a2n (t q )2n i2n(𝛼+𝛽q) ⎪ ⎪ ⎪ 𝛼 𝛽 𝛽 q−1−v ⎪n=0 Γ((2n + 1)q − v) Rq,v ( ai , i t) = (i t) ⎨ ⎬, ∞ 2n+1 q 2n+1 (2n+1)(𝛼+𝛽q) (t ) i ⎪ +∑ a ⎪ ⎪ Γ((2n + 2)q − v) ⎪ n=0 ⎩ ⎭

(9.18)

175

9 The Fractional Meta-Trigonometry

20 –/–

15 10

Sin1.05,0.05(1,0.7,t)

176

5 0 –5 –10 –15

+/+

–20 –20

–10

0

10

20

Cos1.05,0.2(1,0.7,t)

Figure 9.31 Phase plane Sin1.05,0.05 (1, 0.7, 0, t) versus Cos1.05,0.05 (1, 0.7, 0, t) and for −Sin1.05,0.05 (1, 0.7, 0, t) versus −Cos1.05,0.05 (1, 0.7, 0, t), 𝛽 = 0, with a = 1.0, q = 1.05, v = 0.05, k = 0, 𝛼 = 0.7, t = 0–10.

⎧∑ ⎫ ∞ 2n (n+1)2q−1−v−q 2n(𝛼+𝛽q)+𝛽(q−1−v) i ⎪ a t ⎪ Γ((n + 1)2q − v − q) ⎪n=0 ⎪ 𝛼 𝛽 Rq,v ( ai , i t) = ⎨ , ∞ 2n+1 (n+1)2q−1−v (2n+1)(𝛼+𝛽q)+𝛽(q−1−v) ⎬ ∑ a t i ⎪ + ⎪ ⎪ ⎪ Γ((n + 1)2q − v) n=0 ⎩ ⎭

t > 0.

(9.19)

This also may be expressed as Rq,v ( ai𝛼 , i𝛽 t) = R2q,v+q (a2 i2𝛼 , i𝛽 t) + ai𝛼 R2q,v (a2 i2𝛼 , i𝛽 t),

t > 0.

(9.20)

The summations of equation (9.19) contain the even and odd powers of a, respectively. In parallel with the previous development of the R-trigonometric functions, we define the generalized or meta-Corotation and Rotation functions as Corq,v (a, 𝛼, 𝛽, t) ≡

∞ ∑ a2n t (n+1)2q−1−v−q i2n(𝛼+𝛽q)+𝛽(q−1−v) , Γ((n + 1)2q − v − q) n=0

= Ea (Rq,v (ai𝛼 , i𝛽 t)),

t > 0,

∑ (i2(𝛼+𝛽 q) a2 )n t (n+1)2q−1−v−q

(9.21)

∞

= i𝛽(q−1−v)

n=0

Γ((n + 1)2q − v − q)

= R2q,v+q (i2𝛼 a2 , i𝛽 t),

= i𝛽(q−1−v) R2q,v+q (i2(𝛼+𝛽 q) a2 , t)

t > 0,

where the nomenclature Ea (x) means the terms with even powers of a in (x).

(9.22)

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

Similarly, Rotq,v (a, 𝛼, 𝛽, t) = = i𝛼+𝛽(2q−1−v)

∞ ∑ a2n+1 t (n+1)2q−1−v i(2n+1)(𝛼+𝛽q)+𝛽(q−1−v) , Γ((n + 1)2q − v) n=0

∞ ∑ a(i2(𝛼+𝛽 q) a2 )n t (n+1)2q−1−v n=0

Γ((n + 1)2q − v)

= ai𝛼 R2q,v (i2𝛼 a2 , i𝛽 t) = Oa (Rq,v (ai𝛼 , i𝛽 t)),

t > 0,

(9.23)

= a i𝛼+𝛽(2q−1−v) R2q,v (i2(𝛼+𝛽 q) a2 , t) t > 0,

(9.24)

and where the nomenclature Oa (x) means the terms with odd powers of a in (x). As in the previous trigonometries, the generalized Corotation and Rotation functions are also, in general, complex. Clearly, we may also write Rq,v ( ai𝛼 , i𝛽 t) = Corq,v (a, 𝛼, 𝛽, t) + Rotq,v (a, 𝛼, 𝛽, t), t > 0 ( ( ) ) = R2q,v+q a2 i2𝛼 , i𝛽 t + ai𝛼 R2q,v a2 i2𝛼 , i𝛽 t , ( ) = i𝛽(q−1−v) R2q,v+q a2 i2(𝛼+𝛽q) , t ( ) + ai𝛼+𝛽(2q−1−v) R2q,v a2 i2 (𝛼+𝛽q) , t , t > 0.

(9.25)

The real and imaginary parts of these functions are now used to define the four new real fractional meta-trigonometric functions. The Corq,v (a, 𝛼, 𝛽, t)-function is given in equation (9.21) as ∞ ∑ a2n t (n+1)2q−1−v−q i2n(𝛼+𝛽q)+𝛽(q−1−v) , t > 0. (9.26) Corq,v (a, 𝛼, 𝛽, t) = Γ((n + 1)2q − v − q) n=0 Now, applying equation (3.123) to i2n(𝛼+𝛽q)+𝛽(q−1−v) with q, v, 𝛼, and 𝛽 rational, we have 2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v) 2nN𝛼 Dq Dv D𝛽 + 2nN𝛽 Nq Dv D𝛼 + N𝛽 Nq Dv D𝛼 − N𝛽 Dq Dv D𝛼 − N𝛽 Nv Dq D𝛼 = Dq Dv D𝛼 D𝛽 M = , (9.27) D with M/D a rational number in minimal form. Thus, the Corq,v (a, 𝛼, 𝛽, t) may be written as Corq,v (a, 𝛼, 𝛽, k, t) =

∞ ∑ n=0

+i

( ( )) a2n t (n+1)2q−1−v−q 𝜋 cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v − q) 2

∞ ∑ n=0

a2n t (n+1)2q−1−v−q Γ((n + 1)2q − v − q)

)) ( ( 𝜋 + 2𝜋k , t > 0, (9.28) × sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 where k = 0, 1, 2, … , D − 1. The real part of the Corq,v (a, 𝛼, 𝛽, t) is now defined as the generalized or meta-Covibration function Covibq,v (a, 𝛼, 𝛽, k, t) ≡

∞ ∑ n=0

( ( )) a2n t (n+1)2q−1−v−q 𝜋 cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k , Γ((n + 1)2q − v − q) 2

= Re(Ea (Rq,v (ai𝛼 , i𝛽 t))), with t > 0, and k = 0, 1, 2, … , D − 1.

(9.29)

177

178

9 The Fractional Meta-Trigonometry

The meta-Vibration function, Vibq,v (a, 𝛼, 𝛽, t), is defined as the imaginary part of the Corq,v (a, 𝛼, 𝛽, t)-function; thus, Vibq,v (a, 𝛼, 𝛽, k, t) ≡

∞ ∑ n=0

( ( )) a2n t (n+1)2q−1−v−q 𝜋 sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k , Γ((n + 1)2q − v − q) 2

= Im((Ea (Rq,v (ai𝛼 , i𝛽 t))),

(9.30)

with t > 0, and k = 0, 1, 2, … , D − 1. Then, we have Corq,v (a, 𝛼, 𝛽, k, t) = Covibq,v (a, 𝛼, 𝛽, t) + i Vibq,v (a, 𝛼, 𝛽, t),

t > 0, k = 0, 1, 2, … , D − 1.

(9.31)

The generalized or meta-Rotation function similarly defines two new functions based on its real and imaginary parts; these are the meta-Flutter and meta-Coflutter functions, that is, ∞ ∑ a2n+1 t (n+1)2q−1−v Cof lq,v (a, 𝛼, 𝛽, k, t) ≡ Γ((n + 1)2q − v) n=0 )) ( ( 𝜋 + 2𝜋k , × cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 = Re(Oa (Rq,v (ai𝛼 , i𝛽 t))), t > 0, k = 0, 1, 2, … , D − 1, (9.32) and ∞ ∑ a2n+1 t (n+1)2q−1−v Flutq,v (a, 𝛼, 𝛽, k, t) ≡ Γ((n + 1)2q − v) n=0 )) ( ( 𝜋 + 2𝜋k , × sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 = Im(Oa (Rq,v (ai𝛼 , i𝛽 ))), t > 0, k = 0, 1, 2, … , D − 1,

(9.33)

and where D is the product of the denominators Dq , Dv , D𝛼 , D𝛽 . In parallel with equation (9.31), we also have Rotq,v (a, 𝛼, 𝛽, k, t) = Cof lq,v (a, 𝛼, 𝛽, t) + i Flutq,v (a, 𝛼, 𝛽, t), t > 0, k = 0, 1, 2, … , D − 1. (9.34) Table 9.1 presents special values for the fractional meta-trigonometric functions when q = 1 and v = 0. The row 𝛼 + 𝛽 = 0 can apply to the R1 -hyperboletry, while the row 𝛼 + 𝛽 = 1 can Table 9.1 Special values of the generalized trignobolic functions 𝜶+𝜷

Cos1,0 (· · ·)

Sin1,0 (· · ·)

Cor1,0 (· · ·)

Rot1,0 (· · ·)

Cov1,0 (· · ·)

Vib1,0 (· · ·)

Cof l1,0 (· · ·)

Flut1,0 (· · ·)

0

eat

0

cosh(at)

sinh(at)

cosh(at)

0

sinh(at)

0

1

cos(at)

sin(at)

cos(at)

i sin(at)

cos(at)

0

0

sin(at)

2

e−at

0

cosh(at)

− sinh(at)

cosh(at)

0

− sinh(at)

0

3

cos(at)

sin(at)

cosh(−iat)

sinh(−iat)

cos(at)

0

0

− sin(at)

4

eat

0

cosh(at)

sinh(at)

cosh(at)

0

sinh(at)

0

1/2

–

–

cosh(i1∕2 at)

sinh(i1∕2 at)

–

–

–

–

For this table (a, 𝛼, 𝛽, k, t) ⇒ (· · ·), also q = 1, v = 0, t > 0, – indicates only series description found. Source: Lorenzo 2009a [82]. Reproduced with permission of ASME.

9.3 Commutative Properties of the Complexity and Parity Operations

apply to the R1 - and R2 -trigonometries. The R3 -trigonometry is a special case of the 𝛼 + 𝛽 = 2 row. Of course, any of these rows may also be applied to noninteger values of 𝛼 and 𝛽. Continuing from Ref. [82] with permission of ASME: Some observations may be made from the table. For example, for the Cor1,0 (a, 𝛼, 𝛽, t)function column all terms are of the form cosh(ai(𝛼+𝛽) t). This may be shown as follows: Cor1,0 (a, 𝛼, 𝛽, t) = =

∞ ∑ a2n t 2n i2n(𝛼+𝛽) Γ(2n + 1) n=0

t > 0,

(9.35)

∞ ∑ (ai(𝛼+𝛽) t)2n = cosh(ai(𝛼+𝛽) t), (2n)! n=0

t > 0.

(9.36)

Similarly, for the Rot1,0 (a, 𝛼, 𝛽, t) column, all terms are of the form sinh(ai(𝛼+𝛽) t), since ∞ ∑ a2n+1 t 2n+1 i(2n+1)(𝛼+𝛽) , Rot1,0 (a, 𝛼, 𝛽, t) = Γ(2n + 2) n=0

t > 0,

∞ ∑ (ai(𝛼+𝛽) t)2n+1 = = sinh(ai𝛼+𝛽 t), (2n + 1)! n=0

(9.37) t > 0.

(9.38)

Thus, we see that these functions are backward compatible with the hyperbolic functions with imaginary arguments. Other relationships are observed for the remaining columns.

9.3 Commutative Properties of the Complexity and Parity Operations In this section, we demonstrate that the operations of determining the real/imaginary parts and even/odd parts of the parity functions may be interchanged. We start with the real part of Rq,v (ai𝛼 , i𝛽 t) from equation (9.6); then, for t > 0, we have ∞ ∞ ∑ ∑ an t (n+1)q−1−v an t nq q−1−v Re Rq,v ai , i t = cos(𝜉n ) = t cos(𝜉n ), (9.39) Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0 ( ) where 𝜉n = (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) 𝜋2 + 2𝜋k . Expanding the summation yields } { cos(𝜉0 ) (at q ) cos(𝜉1 ) (at q )2 cos(𝜉2 ) (at q )3 cos(𝜉3 ) + + + +··· . (9.40) = t q−1−v Γ(q − v) Γ(2q − v) Γ(3q − v) Γ(4q − v)

(

{

𝛼

𝛽

)}

Collecting even and odd powers of a gives ∞ ∑ a2n t (n+1)2q−1−v−q a2n+1 t (n+1)2q−1−v cos(𝜉2n ) + cos(𝜉2n+1 ), Γ((n + 1)2q − v − q) Γ((n + 1)2q − v) n=0 n=0 ∞ ( ( )) ∑ a2n t (n+1)2q−1−v−q 𝜋 = cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k , + Γ((n + 1)2q − v) 2 n=0 ( ( ( 𝛼 𝛽 ))) ( ( ( 𝛼 𝛽 ))) + Oa Re Rq,v ai , i t . (9.41) = Ea Re Rq,v ai , i t

=

∞ ∑

179

180

9 The Fractional Meta-Trigonometry

However, we also observe that ( ( ( ))) Ea Re Rq,v ai𝛼 , i𝛽 t = Covibq,v (a, 𝛼, 𝛽, k, t), and that

( ( ( ))) = Cof lq,v (a, 𝛼, 𝛽, k, t), Oa Re Rq,v ai𝛼 , i𝛽 t

t > 0,

(9.42)

t > 0.

(9.43)

Now, from equations (9.29) and (9.42), we have ( ( ( ( ( ( ))) ))) Covibq,v (a, 𝛼, 𝛽, k, t) = Re Ea Rq,v ai𝛼 , i𝛽 t = Ea Re Rq,v ai𝛼 , i𝛽 t ,

t > 0,

(9.44)

and from equations (9.32) and (9.43) ( ( ( ( ( ( ))) ))) Cof lq,v (a, 𝛼, 𝛽, k, t) = Re Oa Rq,v ai𝛼 , i𝛽 t = Oa Re Rq,v ai𝛼 , i𝛽 t ,

t > 0,

(9.45)

proving the assertion for these functions. For the remaining functions, we begin with the imaginary part of Rq,v (ai𝛼 , i𝛽 t), from equation (9.7) ∞ ∞ ∑ ∑ an t (n+1)q−1−v an t nq q−1−v Im Rq,v ai , i t = sin(𝜉n ) = t sin(𝜉n ), (9.46) Γ((n + 1)q − v) Γ((n + 1)q − v) n=0 n=0 ( ) where 𝜉n = (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) 𝜋2 + 2𝜋k and t > 0. Expanding this summation yields

(

{

𝛼

𝛽

{

=t

q−1−v

)}

sin(𝜉0 ) (at q ) sin(𝜉1 ) (at q )2 sin(𝜉2 ) (at q )3 sin(𝜉3 ) + + + +··· Γ(q − v) Γ(2q − v) Γ(3q − v) Γ(4q − v)

}

.

(9.47)

Again, collecting even and odd powers of a gives ∑ a2n+1 t (n+1)2q−1−v a2n t (n+1)2q−1−v−q sin(𝜉2n ) + sin(𝜉2n+1 ) Γ((n + 1)2q − v − q) Γ((n + 1)2q − v) n=0 n=0 ∞ ( ( )) ∑ a2n t (n+1)2q−1−v−q 𝜋 = sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k , + Γ((n + 1)2q − v) 2 n=0

=

∞ ∑

∞

= Ea (Im(Rq,v (ai𝛼 , i𝛽 t))) + Oa (Im(Rq,v (ai𝛼 , i𝛽 t))).

(9.48)

Here, we observe Ea (Im(Rq,v (ai𝛼 , i𝛽 t))) = Vibq,v (a, 𝛼, 𝛽, k, t),

t > 0,

(9.49)

Oa (Im(Rq,v (ai𝛼 , i𝛽 t))) = Flutq,v (a, 𝛼, 𝛽, k, t),

t > 0.

(9.50)

and Now, using equations (9.30) with (9.49) Vibq,v (a, 𝛼, 𝛽, k, t) = Im(Ea (Rq,v (ai𝛼 , i𝛽 t))) = Ea (Im(Rq,v (ai𝛼 , i𝛽 t))),

t > 0,

(9.51)

t > 0,

(9.52)

and from equations (9.33) and (9.50), Flutq,v (a, 𝛼, 𝛽, k, t) = Im(Oa (Rq,v (ai𝛼 , i𝛽 t))) = Oa (Im(Rq,v (ai𝛼 , i𝛽 t))),

completing the demonstration of the complexity–parity commutivity properties of Rq,v ( ai𝛼 , i𝛽 t). These properties are summarized for the meta-trigonometric functions

9.3 Commutative Properties of the Complexity and Parity Operations

with t > 0 [83]: Cosq,v (a, 𝛼, 𝛽, k, t) = Re(Rq,v (ai𝛼 , i𝛽 t)), 𝛼

(9.53)

𝛽

Sinq,v (a, 𝛼, 𝛽, k, t) = Im(Rq,v (ai , i t)),

(9.54)

𝛼

𝛽

𝛼

𝛽

𝛼

𝛽

𝛼

𝛽

𝛼

𝛽

𝛼

𝛽

𝛼

𝛽

Covibq,v (a, 𝛼, 𝛽, k, t) = Re(Ea (Rq,v (ai , i t))) = Ea (Re(Rq,v (ai , i t))),

(9.55)

Vibq,v (a, 𝛼, 𝛽, k, t) = Im(Ea (Rq,v (ai , i t))) = Ea (Im(Rq,v (ai , i t))),

(9.56)

Cof lq,v (a, 𝛼, 𝛽, k, t) = Re(Oa (Rq,v (ai , i t))) = Oa (Re(Rq,v (ai , i t))), 𝛼

(9.57)

𝛽

Flutq,v (a, 𝛼, 𝛽, k, t) = Im(Oa (Rq,v (ai , i t))) = Oa (Im(Rq,v (ai , i t))),

(9.58)

𝛼

𝛽

(9.59)

𝛼

𝛽

(9.60)

Rotq,v (a, 𝛼, 𝛽, t) = Oa (Rq,v (ai , i t)), Corq,v (a, 𝛼, 𝛽, k, t) = Ea (Rq,v (ai , i t)), where Oa and Ea refer to terms containing the odd and even powers of a, respectively. 9.3.1

Graphical Presentation – Parity Functions

Once again, it is not possible to show a representative display of the parity functions. The problem is exacerbated because the number of functions is doubled. Thus, the focus is on the effect of the generalizing variables 𝛼 and 𝛽. Figures 9.32–9.35 show the effect of the primary order variable, q, on the parity functions, Covibq,0 (a, 𝛼, 𝛽, k, t), Vibq,0 (a, 𝛼, 𝛽, k, t), Cof lq,0 (a, 𝛼, 𝛽, k, t), and Flutq,0 (a, 𝛼, 𝛽, k, t) as a function of t time, with 𝛼 = 0.5 and 𝛽 = 0.5. Because 𝛼 = 1, 𝛽 = 0 corresponds to the R1 -trigonometry and 𝛼 = 0, 𝛽 = 1 corresponds to the R2 -trigonometry, the choice of 𝛼 = 0.5 and 𝛽 = 0.5 examines a new trigonometry sharing aspects of both R1 - and R2 -trigonometries. An observed feature of these figures is that increasing the order q increases the oscillatory behavior of the function in most cases. Note that the reversal of the first peak amplitude for q ≥≈ 0.8 for the Vibq,0 - and Cof lq,0 -functions. The effect of 𝛼 and 𝛽, where 𝛼 + 𝛽 = 1, for Covibq,0 (a, 𝛼, 𝛽, k, t), Vibq,0 (a, 𝛼, 𝛽, k, t), Cof lq,0 (a, 𝛼, 𝛽, k, t), and Flutq,0 (a, 𝛼, 𝛽, k, t) functions is shown in Figures 9.36–9.39. For Figure 9.32 The effect of q for Covibq,0 (a, 𝛼, 𝛽, k, t), q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, 𝛼 = 0.5, 𝛽 = 0.5.

1 0.8

Covibq,0(a,0.50,0.50,t)

0.6 0.4 0.2 q = 0.1

0 –0.2 –0.4 –0.6 –0.8 –1

q = 1.0 0

1

2

3 4 t-Time

5

6

7

181

9 The Fractional Meta-Trigonometry

0.6

Figure 9.33 The effect of q for Vibq,0 (a, 𝛼, 𝛽, k, t), q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, 𝛼 = 0.5, 𝛽 = 0.5.

q = 0.1

Vibq,0(a,0.5,0.5,t)

0.4

0.2 q = 0.1, 0.2

0

q = 1.0

–0.2

q = 0.9

–0.4 0

1

2

3 4 t-Time

5

6

7

Figure 9.34 The effect of q for Cof lq,0 (a, 𝛼, 𝛽, k, t), q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, 𝛼 = 0.5, 𝛽 = 0.5.

0.8

Coflq,0(1,0.5,0.5,t)

0.6

0.4

0.2 q = 1.0

0 q = 0.1

q = 0.9

–0.2

q = 0.8

0

1

2

3 4 t-Time

5

6

7

Figure 9.35 The effect of q for Flutq,0 (a, 𝛼, 𝛽, k, t), q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, 𝛼 = 0.5, 𝛽 = 0.5.

1.5 q = 1.0

1

Flutq,0(1,0.5,0.5,t)

182

0.5

q = 0.1

0 –0.5 –1 0

1

2

3 4 t-Time

5

6

7

9.3 Commutative Properties of the Complexity and Parity Operations

Figure 9.36 The effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Covibq,0 (a, 𝛼, 𝛽, k, t), with 𝛼 = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

1 α = 1.0

Covib0.5,0(1,α,1–α,t)

0.5

0

α = 0.0, 2.0

–0.5

α = 0.8, 1.2

0.6, 1.4

–1 0

Figure 9.37 The effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Vibq,0 (a, 𝛼, 𝛽, k, t), with 𝛼 = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

1

2

3 4 t-Time

5

6

7

5

6

7

5

6

7

1

Vib0.5,0(1,α,1–α,t)

α = 0.0 α = 0.2

0.5

0

–0.5

α = 1.8 α = 2.0

–1 0

Figure 9.38 The effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Cof lq,0 (a, 𝛼, 𝛽, k, t), with 𝛼 = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

1

4 t-Time

α = 1.8

0.5

0

3

α = 2.0

1

Cofl0.5,0(1,α,1–α,t)

2

1.0

α = 0.2

–0.5

–1

α = 0.0 0

1

2

3 4 t-Time

183

9 The Fractional Meta-Trigonometry

Figure 9.39 The effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Flutq,0 (a, 𝛼, 𝛽, k, t), with 𝛼 = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

α = 0.0, 2.0

1

Flut0.5,0(1,α,1–α,t)

0.2, 1.8 0.5 α = 1.0 0

–0.5

–1 0

0.6

1

2

3

4 t-Time

5

6

7

Figure 9.40 The effect of a for Vibq,0 (a, 𝛼, 𝛽, k, t), with a = 0.25–2.0 in steps of 0.25, and with q = 0.5, 𝛼 = 0.5, 𝛽 = 0.5, v = 0, k = 0.

a = 2.0

0.4 0.2

Vib0.5,0(1,0.5,0.5,t)

184

0 –0.2

a = 0.25

–0.4 –0.6 –0.8 –1

0

1

2

3 4 t-Time

5

6

7

these figures, 𝛼 = 0.0–2.0 in steps of 0.2, and q = 0.5, a = 1, v = 0, k = 0. Note that the responses for the Vibq,0 (a, 𝛼, 𝛽, k, t),- and Cof lq,0 (a, 𝛼, 𝛽, k, t)-functions are symmetric around 𝛼 = 1, and the Covibq,0 (a, 𝛼, 𝛽, k, t)- and Flutq,0 (a, 𝛼, 𝛽, k, t)-functions overlay themselves for 𝛼 = 1 ± Δ𝛼, where Δ𝛼 = 0.2. Figures 9.40 and 9.41 study the effect of a for Vibq,0 (a, 𝛼, 𝛽, k, t)- and Flutq,0 (a, 𝛼, 𝛽, k, t)functions with a = 0.25– 2.0 in steps of 0.25, and with q = 0.5, 𝛼 = 0.5, 𝛽 = 0.5, v = 0, k = 0. Figures 9.42 and 9.43 study the effect of the differintegration variable, v, for the Vibq,v (a, 𝛼, 𝛽, k, t)- and Flutq,v (a, 𝛼, 𝛽, k, t)-functions with v = −1.0 to 1.0 in steps of 0.25, and with q = 0.5, a = 1, 𝛼 = 0.5, 𝛽 = 0.5, k = 0. Figure 9.44 considers the effect of the index variable k, for k = 0–4, with q = 3/5, a = 1, 𝛼 = 0.5, 𝛽 = 0.5, v = 0. Note again the untypical symmetric responses resulting from the particular choice of q. The remaining figures (Figures 9.45–9.52) are phase plane plots chosen to expose a few of the many varied possibilities. Figures 9.45 and 9.46 are phase planes showing the effect of 𝛼 and 𝛽, where 𝛼 + 𝛽 = 1, for Vibq,0 (a, 𝛼, 𝛽, k, t) versus Covibq,0 (a, 𝛼, 𝛽, k, t). For Figure 9.45, 𝛼 = 0.0–1.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0. For Figure 9.46, 𝛼 = 1.0 to 2.0 in steps of

9.3 Commutative Properties of the Complexity and Parity Operations

Figure 9.41 The effect of a for Flutq,0 (a, 𝛼, 𝛽, k, t), with a = 0.25–2.0 in steps of 0.25, and with q = 0.5, 𝛼 = 0.5, 𝛽 = 0.5, v = 0, k = 0.

a = 2.0 1.2

Flut0.5,0(1,0.5,0.5,t)

1 0.8 0.6 0.4

a = 0.50

0.6 0 –0.2

Figure 9.42 The effect of v for Vibq,v (a, 𝛼, 𝛽, k, t), with v = −1.0–1.0 in steps of 0.25, and with q = 0.5, a = 1, 𝛼 = 0.5, 𝛽 = 0.5, k = 0.

a = 0.25 0

1

2

3

4 t-Time

5

6

7

5

6

7

5

6

7

Vib0.5,𝜈(1,0.5,0.5,t)

1

𝜈 = –1.00

0.5

𝜈 = –0.75

0 𝜈 = 1.00

–0.5 0

Figure 9.43 The effect of v for Flutq,v (a, 𝛼, 𝛽, k, t), with v = −1.0–1.0 in steps of 0.25, and with q = 0.5, a = 1, 𝛼 = 0.5, 𝛽 = 0.5, k = 0.

1

2

4 t-Time

𝜈 = –1.00

0.8 Flut0.5,𝜈(1,0.5,0.5,t)

3

𝜈 = –0.75 0.6 𝜈 = –0.50 0.4 0.2 0 –0.2 0

𝜈 = 1.00 1

2

3 4 t-Time

185

9 The Fractional Meta-Trigonometry

Figure 9.44 Effect of k, for k = 0–4, with q = 3/5, a = 1, 𝛼 = 0.5, 𝛽 = 0.5, v = 0.

10 k=3 Cov0.6,0(1,0.5,0.5,t)

5 k=1 k=0 0 k=2 –5 k=4 –10

0

1

2

3 4 t-Time

5

6

7

1 0.8 α = 1.0

0.6

Figure 9.45 Phase plane showing the effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Vibq,0 (a, 𝛼, 𝛽, k, t) versus Covibq,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.0–1.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0, t = 0–9.

0.4

Vib0.5,0(1,α,β,t)

186

0.2 0 –0.2 –0.4 –0.6

α = 0.0

–0.8 –1 –1

–0.5

0 0.5 Covib0.5,0(1,α,β,t)

1

0.1, also with q = 0.5, a = 1, v = 0, k = 0. All cases arrive from infinity and spiral into the origin except 𝛼 = 2.0, which is attracted to the unit circle. From Lorenzo [83]: A particularly interesting, and important, pair of plots is found in Figures 9.47 and 9.48 for the Flut0.5,0 (a, 𝛼, 𝛽, k, t) versus Cof l0.5,0 (a, 𝛼, 𝛽, k, t) phase plane. For both cases, 𝛼 = 1.0–2.0 and q = 0.5, a = 1, v = 0, k = 0. While it is not obvious because of the change of scale between the pair, in Figure 9.47 as 𝛼 is varied over the range 0 < 𝛼 < 1, with 𝛽 = 3, the responses start from the unit circle and diverge to infinity. In Figure 9.48 with 𝛽 = 1 and all other variables the same, the responses start from the same identical points as in Figure 9.47 but are attracted to the origin. Furthermore, the slopes across the unit circle are preserved for each response. This behavior and its interpretation are discussed in more detail in Section 9.12.

9.3 Commutative Properties of the Complexity and Parity Operations

Figure 9.46 Phase plane showing the effect of 𝛼 and 𝛽, 𝛼 + 𝛽 = 1, for Vibq,0 (a, 𝛼, 𝛽, k, t) versus Covibq,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0, t = 0–9.

1 0.8 α = 2.0

0.6 Vib0.5,0(1,α,β,t)

0.4 0.2 0 –0.2 –0.4 α = 1.0

–0.6 –0.8 –1 –0.5

–1

Figure 9.47 Phase plane showing the effect of 𝛼 for Flut0.5,0 (a, 𝛼, 𝛽, k, t) versus Cof l0.5,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, 𝛽 = 1, k = 0. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

0 0.5 Covib0.5,0(1,α,β,t)

1

3 2.5 α = 1.5

2

1.4

1.3

Flut0.5,0(1,α,β,t)

1.5

1.2

1 0.5 0

1.0

1.6

α = 1.1 2.0

1.7

–0.5 –1 –1.5 –3

1.8 –2

1.9 0 –1 Cofl0.5,0(1,α,β,t) 0

1

Figure 9.49 shows an interesting spiral converging to the origin that is similar to some of the classical spirals [64], pp. 188, 189. Figures 9.50–9.53 are phase plane plots of several barred spirals based on the parity functions. Figure 9.51 presents a phase plane for Flut1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus Cof l1.04,−0.3 (a, 𝛼, 𝛽, k, t) and −Flut1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus −Cof l1.04,−0.3 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.5–0.8. The spirals start at the origin and spiral out to infinity with increasing rate as 𝛼 increases. Figure 9.52 is a very interesting phase plane plot for Vib1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus Covib1.04,−0.3 (a, 𝛼, 𝛽, k, t) and −Vib1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus −Covib1.04,−03 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.5–0.8. The unusual feature here is that the departure is from the end of the spiral bar

187

9 The Fractional Meta-Trigonometry

Figure 9.48 . Phase plane showing the effect of 𝛼 for Flut0.5,0 (a, 𝛼, 𝛽, k, t) versus Cof l0.5,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, 𝛽 = 3, k = 0. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

α = 1.1

1

1.0

0.8

1.5

Flut0.5,0(1,α,β,t)

0.6 0.4 0.2

1.9

0

2.0

–0.2 –0.4 –0.6 –0.8 –1 –1

–0.5

0 0.5 Cofl0.5,0(1,α,β,t)

1

Figure 9.49 Phase plane for Flut0.8,0 (a, 𝛼, 𝛽, k, t) versus Covib0.8,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.16, 𝛽 = 1.25, and with q = 0.80, a = 0.8, v = 0, k = 0, t = 0–40.

1 Flut0.8,0(0.8,0.16,1.25,t)

188

0.5

0

–0.5

–1 –1

–0.5

0 0.5 1 Cofl0.8,0(0.8,0.16,1.25,t)

1.5

2

as 𝛼 increases. Such barred spiral behavior is of particular interest in the astrophysics of the galaxies. More such behaviors are studied in Chapter 18.

9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions The development of the Laplace transform for the generalized functions parallels that of the previous trigonometries. {∞ } ∑ an t (n+1)q−1−v { } L Cosq,v (a, 𝛼, 𝛽, k, t) = L cos(𝜉) , (9.61) Γ((n + 1)q − v) n=0 ( ) ( ) where 𝜉 = n(𝛼 + 𝛽q) 𝜋2 + 2𝜋k + 𝛽[q − 1 − v] 𝜋2 + 2𝜋k = n𝜎 + 𝜆, k = 0, 1, 2, … , D − 1, and where D is the product of the denominators Dq , Dv , D𝛼 , D𝛽 , in equation (9.3), in minimal

9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions

Figure 9.50 Phase plane for Flut1.09,0 (a, 𝛼, 𝛽, k, t) versus Covib1.09,0 (a, 𝛼, 𝛽, k, t) with 𝛼 = 𝛽 = 0.5, and with q = 1.09, a = 1.0, v = 0, k = 0, t = 0–19.

10

Flut1.09,0(1,0.5,0.5,t)

5

0

–5

–10

–15 –15

–5 0 Covib1.09,0(1,0.5,0.5,t)

5

5

10

+/+ α = 0.8

Flut1.04,–0.3(1,α,0.6,t)

Figure 9.51 Phase plane for Flut1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus Cof l1.04,−0.3 (a, 𝛼, 𝛽, k, t) and −Flut1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus −Cof l1.04,−0.3 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.5–0.8, 𝛽 = 0.6, and with q = 1.04, v = −0.3, a = 1.0, k = 0, t = 0–4.8.

–10

0.7

0.6

0.5

0

0.5

0.6 0.7

–5 –6

α = 0.8 –/–

–4

0 2 –2 Cofl1.04,–0.3(1,α,0.6,t)

4

6

form. Because the series converges uniformly (see Sections 3.16 and 14.3), we may transform term-by-term. Thus, ∞ { } ∑ L Cosq,v (a, 𝛼, 𝛽, k, t) = an L n=0

{

t (n+1)q−1−v Γ((n + 1)q − v)

} cos(n𝜎 + 𝜆),

(9.62)

Continuing from Ref. [83]: ∞ { } ∑ L Cosq,v (a, 𝛼, 𝛽, k, t) = n=0

an s(n+1)q−v

ei(n𝜎+𝜆) + e−i(n𝜎+𝜆) , q > v, 2

(9.63)

189

9 The Fractional Meta-Trigonometry

5

Figure 9.52 Phase plane for Vib1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus Covib1.04,−0.3 (a, 𝛼, 𝛽, k, t) and −Vib1.04,−0.3 (a, 𝛼, 𝛽, k, t) versus −Covib1.04,−0.3 (a, 𝛼, 𝛽, k, t) with 𝛼 = 0.5–0.8, 𝛽 = 0.6, and with q = 1.04, v = −0.3, a = 1.0, k = 0, t = 0–4.8. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

–/– α = 0.8

Vib1.04,–0.3(1,α,0.6,t)

190

0.7 0.6

0.5

0 0.6

0.5

0.7

–5 –6

α = 0.8 +/+

–4

0 2 –2 Covib1.04,–0.3(1,α,0.6,t)

4

Rq,v (a,t)

6

1

Rq,v (aiα,iβt)

Reals

Complexity functions

Cosq,v (a,α,β,t)

Imgs

Sinq,v (a,α,β,t)

Evens

2

Parity functions

Corq,v (a,α,β,t)

Odds

Rotq,v (a,α,β,t)

3

4

Covq,v

Coflq,v

Vibq,v

Flutq,v

Covq,v

Vibq,v

Coflq,v

Flutq,v

Real & Even

Real & Odd

Img & Even

Img & Odd

Even & Real

Even & Img

Odd & Real

Odd & Img

Figure 9.53 Taxonomy of the fractional meta-trigonometric functions. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME. ∞ ∞ { } ei𝜆 ∑ an ein𝜎 e−i𝜆 ∑ an e−in𝜎 L Cosq,v (a, 𝛼, 𝛽, k, t) = q−v + , 2s snq 2sq−v n=0 snq n=0 )n )n ∞ ( ∞ ( { } ei𝜆 ∑ a ei𝜎 e−i𝜆 ∑ a e−i𝜎 L Cosq,v (a, 𝛼, 𝛽, k, t) = q−v + q−v , q > v. 2s sq 2s sq n=0 n=0

Recognizing the summations using equation (7.52) gives { } ei𝜆 e−i𝜆 sq sq L Cosq,v (a, 𝛼, 𝛽, k, t) = q−v q + q−v q , i𝜎 2s [ s − a e 2s s −] a e−i𝜎 { } sv e−i𝜆 ei𝜆 L Cosq,v (a, 𝛼, 𝛽, k, t) = + , 2 sq − a ei𝜎 sq − a e−i𝜎 [ ] } sv ei𝜆 (sq − ae−i𝜎 ) + e−i𝜆 (sq − aei𝜎 ) { , L Cosq,v (a, 𝛼, 𝛽, k, t) = 2 s2q − a ei𝜎 sq − a e−i𝜎 sq + a2

(9.64) (9.65)

9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions

[ ] { } sv sq (ei𝜆 + e−i𝜆 ) − a(ei(𝜆−𝜎) + e−i(𝜆−𝜎) ) L Cosq,v (a, 𝛼, 𝛽, k, t) = , 2 s2q − 2a cos(𝜎)sq + a2 [ q ] } { s cos(𝜆) − a cos(𝜆 − 𝜎) , q > v, (9.66) L Cosq,v (a, 𝛼, 𝛽, k, t) = sv s2q − 2a cos(𝜎)sq + a2 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k , and k = 0, 1, 2, · · · D − 1. The two forms, those of equations (9.65) and (9.66), are particularly useful. The derivation for the Laplace transform of Sinq,v (a, 𝛼, 𝛽, k, t) proceeds in a similar manner: {∞ } ∑ an t (n+1)q−1−v { } L Sinq,v (a, 𝛼, 𝛽, k, t) = L sin(𝜉) , (9.67) Γ((n + 1)q − v) n=0 ) ( ) ( with 𝜉 = n(𝛼 + 𝛽q) 𝜋2 + 2𝜋k + 𝛽[q − 1 − v] 𝜋2 + 2𝜋k = n𝜎 + 𝜆, k = 0, 1, 2, … , D − 1 and giving the results [ ] { } sv e−i𝜆 ei𝜆 L Sinq,v (a, 𝛼, 𝛽, k, t) = − (9.68) 2i sq − a ei𝜎 sq − a e−i𝜎 and

] sq sin(𝜆) − a sin(𝜆 − 𝜎) L Sinq,v (a, 𝛼, 𝛽, k, t) = s , q − v > 0, (9.69) s2q − 2 a cos(𝜎)sq + a2 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k and k = 0, 1, 2, … , D − 1. The Laplace transform for the Covibq,v (a, 𝛼, 𝛽, k, t) follows. Continuing from Ref. [83]: {

}

[

v

∞ { } ∑ L Covibq,v (a, 𝛼, 𝛽, k, t) = a2n L n=0 ∞

{ } ∑ L Covibq,v (a, 𝛼, 𝛽, k, t) = n=0

{

}

L Covibq,v (a, 𝛼, 𝛽, k, t) =

{

t (n+1)2q−1−v−q Γ((n + 1)2q − v − q)

a2n s(n+1)2q−v−q

} cos(2n𝜎 + 𝜆),

(9.70)

ei(2n𝜎+𝜆) + e−i(2n𝜎+𝜆) , q > v, 2

∞ ∞ ei𝜆 ∑ a2n ei2n𝜎 e−i𝜆 ∑ a2n e−i2n𝜎 + , 2sq−v n=0 s2nq 2sq−v n=0 s2nq

{ } ei𝜆 e−i𝜆 s2q s2q L Covibq,v (a, 𝛼, 𝛽, k, t) = q−v 2q + q−v 2q , 2 i2𝜎 2s 2s s −a e s − a2 e−i2𝜎 1 ei𝜆 sq+v 1 e−i𝜆 sq+v L{Covibq,v (a, 𝛼, 𝛽, k, t)} = + , q > v, 2q 2 i2𝜎 2 s [− a e 2 s2q − a2 e−i2𝜎 ] sq+v ei𝜆 (s2q − a2 e−i2𝜎 ) + e−i𝜆 (s2q − a2 ei2𝜎 ) L{Covibq,v (a, 𝛼, 𝛽, k, t)} = , 2 s4q − a2 (ei2𝜎 + e−i2𝜎 )s2q + a4 [ ] sq+v s2q (ei𝜆 + e−i𝜆 ) − a2 (ei(𝜆−2𝜎) + e−i(𝜆−2𝜎) ) , L{Covibq,v (a, 𝛼, 𝛽, k, t)} = 2 s4q − a2 (ei2𝜎 + e−i2𝜎 )s2q + a4 [ 2q ] s cos(𝜆) − a2 cos(𝜆 − 2𝜎) L{Covibq,v (a, 𝛼, 𝛽, k, t)} = sq+v , q > v, s4q − 2a2 cos(2𝜎)s2q + a4

(9.71)

(9.72)

where k = 0, 1, 2, … , D − 1, and D is the product of the denominators Dq , Dv , D𝛼 , D𝛽 , in ) ( ) ( minimal form and where 𝜎 = (𝛼 + 𝛽 q) 𝜋2 + 2𝜋k and 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k .

191

192

9 The Fractional Meta-Trigonometry

Similarly, for the Vibq,v (a, 𝛼, 𝛽, k, t), we have { } L Vibq,v (a, 𝛼, 𝛽, k, t) =

e−i𝜆 sq+v ei𝜆 sq+v − , q > v, (9.73) 2 i2𝜎 2q − a e ) 2i (s − a2 e−i2𝜎 ) [ 2q ] { } s sin(𝜆) − a2 sin(𝜆 − 2𝜎) L Vibq,v (a, 𝛼, 𝛽, k, t) = sq+v , q > v, (9.74) s4 − 2a2 cos(2𝜎)s2q + a4 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k , and k = 0, 1, 2, … , D − 1. The Laplace transform for the Cof lq,v (a, 𝛼, 𝛽, k, t) follows: 2i (s2q

{ } a ei(𝜆+𝜎) sv a e−i(𝜆+𝜎) sv L Cof lq,v (a, 𝛼, 𝛽, k, t) = + , 2q > v, (9.75) 2 i2𝜎 2 s2q 2 s2q − a2 e−i2𝜎 [ −2q a e ] { } s cos(𝜆 + 𝜎) − a2 cos(𝜆 − 𝜎) L Cof lq,v (a, 𝛼, 𝛽, k, t) = a sv , 2q > v, (9.76) s4 − 2a2 cos(2𝜎)s2q + a4 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k , and k = 0, 1, 2, … , D − 1. The derivation for the Flutq,v (a, 𝛼, 𝛽, k, t) is similar to that for Cof lq,v (a, 𝛼, 𝛽, k, t) and yields { } a ei(𝜆+𝜎) sv a e−i(𝜆+𝜎) sv L Flutq,v (a, 𝛼, 𝛽, k, t) = − , (9.77) 2q 2 i2𝜎 2i s[ − a e 2i s2q − a2 e−i2𝜎 ] { } s2q sin(𝜆 + 𝜎) − a2 sin(𝜆 − 𝜎) L Flutq,v (a, 𝛼, 𝛽, k, t) = a sv , 2q > v, (9.78) s4 − 2a2 cos(2𝜎)s2q + a4 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[2q − 1 − v] 𝜋2 + 2𝜋k , and k = 0, 1, 2, … , D − 1. This collection of Laplace transforms generalizes those derived for the previous trigonometries. Comparison of these transforms with the parallel results for the R3 -trigonometric functions shows that the transforms are structurally the same and the R3 results may be had by the substitutions 𝛼 = 𝛽 = 1. Table 9.2 summarizes the Laplace transforms of the meta-trigonometric functions.

9.5 R-Function Representation of the Fractional Meta-Trigonometric Functions The fractional exponential or R-function representation of the trigonometric functions is useful for analysis and numerical computation. Continuing from Ref. [83]: Now, from the definition of Cosq,v , we have for t > 0 Cosq,v (a, 𝛼, 𝛽, k, t) =

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 cos (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, 2, … , D − 1 . Now,

)) ( ( 𝜋 + 2𝜋k cos (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) 2 ei((n(𝛼+𝛽q)+𝛽(q−1−v))(𝜋∕2+2𝜋k)) + e−i((n(𝛼+𝛽q)+𝛽(q−1−v))(𝜋∕2+2𝜋k)) . = 2

(9.79)

(

𝜋 2

+ 2𝜋k

)

Γ((n + 1)2q − v)

, 𝜆 = 𝛽[q − 1 − v]

(

𝜋 2

] sq sin(𝜆) − a sin(𝜆 − 𝜎) , q>v s2q − 2 a cos(𝜎)sq + a2 [ q ] s cos(𝜆) − a cos(𝜆 − 𝜎) sv , q>v s2q − 2a cos(𝜎)sq + a2 [ 2q ] s sin(𝜆) − a2 sin(𝜆 − 2𝜎) sq+v 4q , q>v s − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s cos(𝜆) − a2 cos(𝜆 − 2𝜎) , q>v sq+v s4q − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s sin(𝜆 + 𝜎) − a2 sin(𝜆 − 𝜎) a sv , 2q > v s4q − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s cos(𝜆 + 𝜎) − a2 cos(𝜆 − 𝜎) , 2q > v a sv s4q − 2a2 cos(2𝜎)s2q + a4

–

–

sv

[

Laplace transform

) + 2𝜋k , k = 0, 1, 2, … , D − 1, t > 0; D is the product of the denominators of Dq , Dv , D𝛼 , D𝛽 with repeated

∑ a2n t (n+1)2q−1−v−q i2n(𝛼+𝛽q)+𝛽(q−1−v) Γ((n + 1)2q − v − q) n=0

n=0 ∞

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 sin (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k Γ((n + 1)q − v) 2 n=0 ∞ ( ( )) ∑ 𝜋 an t (n+1)q−1−v cos (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k Γ((n + 1)q − v) 2 n=0 ∞ ( ( )) ∑ 𝜋 a2n t (n+1)2q−1−v−q sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 ∞ ( ( )) ∑ a2n t (n+1)2q−1−v−q 𝜋 cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v − q) 2 n=0 ∞ ( ( )) ∑ 𝜋 a2n+1 t (n+1)2q−1−v sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v) 2 n=0 ∞ ( ( )) ∑ a2n+1 t (n+1)2q−1−v 𝜋 cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋k Γ((n + 1)2q − v) 2 n=0 ∞ ∑ a2n+1 t (n+1)2q−1−v i(2n+1)(𝛼+𝛽q)+𝛽(q−1−v)

Series definition

multipliers removed. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

𝜎 = (𝛼 + 𝛽q)

Corq,v (a, 𝛼, 𝛽, t)

Rotq,v (a, 𝛼, 𝛽, t)

Cof lq,v (a, 𝛼, 𝛽, k, t)

Flutq,v (a, 𝛼, 𝛽, k, t)

Covibq,v (a, 𝛼, 𝛽, k, t)

Vibq,v (a, 𝛼, 𝛽, k, t)

Cosq,v (a, 𝛼, 𝛽, k, t)

Sinq,v (a, 𝛼, 𝛽, k, t)

Function

Table 9.2 Summary of the meta-trigonometric functions.

194

9 The Fractional Meta-Trigonometry

Applying equation (3.124), namely im∕n = ei(m∕n(𝜋∕2+2𝜋k)) , yields )) ( ( 𝜋 + 2𝜋k cos (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) 2 i(n(𝛼+𝛽q)+𝛽(q−1−v)) + i−(n(𝛼+𝛽q)+𝛽(q−1−v)) . = 2 Then, ∞ i𝛽(q−1−v) ∑ (ai(𝛼+𝛽q) )n t (n+1)q−1−v Cosq,v (a, 𝛼, 𝛽, t) = 2 Γ((n + 1)q − v) n=0 +

(9.80)

∞ i−𝛽(q−1−v) ∑ (ai−(𝛼+𝛽q) )n t (n+1)q−1−v , 2 Γ((n + 1)q − v) n=0

t > 0.

The summations are recognized as R-functions giving the result i𝛽(q−1−v) i−𝛽(q−1−v) Rq,v (ai(𝛼+𝛽q) , t) + Rq,v (ai−(𝛼+𝛽q) , t). 2 2 Applying equation (3.121) simplifies the result to Cosq,v (a, 𝛼, 𝛽, t) =

Cosq,v (a, 𝛼, 𝛽, t) =

(9.81)

Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t)

, t > 0. (9.82) 2 The remaining functions are determined in a similar manner and are listed in two forms, t real and complex; thus, we have Sinq,v (a, 𝛼, 𝛽, t) =

i𝛽(q−1−v) Rq,v (ai𝛼+𝛽q , t) − i−𝛽(q−1−v) Rq,v (ai−(𝛼+𝛽q) , t) 2i

,

(9.83)

or Sinq,v (a, 𝛼, 𝛽, t) = Covibq,v (a, 𝛼, 𝛽, t) =

Rq,v (ai𝛼 , i𝛽 t) − Rq,v (ai−𝛼 , i−𝛽 t) 2i

,

t > 0.

1 𝛽(q−1−v) R2q,v+q (a2 i2(𝛼+𝛽q) , t) i 2 1 + i−𝛽(q−1−v) R2q,v+q (a2 i−2(𝛼+𝛽q) , t), 2

(9.84)

t > 0,

(9.85)

or Covibq,v (a, 𝛼, 𝛽, t) = Vibq,v (a, 𝛼, 𝛽, t) =

R2q,v+q ((ai𝛼 )2 , i𝛽 t) + R2q,v+q ((ai−𝛼 )2 , i−𝛽 t) 2

,

t > 0.

(9.86)

i𝛽(q−1−v) R2q,v+q (a2 i2(𝛼+𝛽q) , t) − i−𝛽(q−1−v) R2q,v+q (a2 i−2(𝛼+𝛽q) , t) 2i

,

(9.87)

or Vibq,v (a, 𝛼, 𝛽, t) =

R2q,v+q ((ai𝛼 )2 , i𝛽 t) − R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

2i a 𝛼+𝛽(2q−1−v) 2 2(𝛼+𝛽q) Cof lq,v (a, 𝛼, 𝛽, t) = i R2q,v (a i , t) 2 a −𝛼−𝛽(2q−1−v) + i R2q,v (a2 i−2(𝛼+𝛽q) , t), 2

,

t > 0,

t > 0,

(9.88)

(9.89)

or Cof lq,v (a, 𝛼, 𝛽, t) = a

i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) + i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

2 a 𝛼+𝛽(2q−1−v) 2 2(𝛼+𝛽q) Flutq,v (a, 𝛼, 𝛽, t) = i R2q,v (a i , t) 2i a − i−𝛼−𝛽(2q−1−v) R2q,v (a2 i−2(𝛼+𝛽q) , t), 2i

.

(9.90)

t > 0,

(9.91)

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

or Flutq,v (a, 𝛼, 𝛽, t) = a

i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) − i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

2i Finally, summarizing the results of equations (9.22) and (9.24)

.

(9.92)

Corq,v (a, 𝛼, 𝛽, t) = R2q,v+q (a2 i2𝛼 , i𝛽 t) = i𝛽(q−1−v) R2q,v+q (i2(𝛼+𝛽 q) a2 , t), 𝛼

t > 0,

(9.93)

𝛽

Rotq,v (a, 𝛼, 𝛽, t) = ai R2q,v (a i , i t) 2 2𝛼

= a i𝛼+𝛽(2q−1−v) R2q,v (i2(𝛼+𝛽 q) a2 , t),

t > 0.

(9.94)

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions For the fractional differintegrations that follow, it is assumed that the integrated function, and all of its derivatives, are identically zero for all t < 0. Continuing from Ref. [83]: 9.6.1 Cosq,v (a, 𝜶, 𝜷, k, t)

The u-order differintegral of Cosq,v (a, 𝛼, 𝛽, k, t) is determined for t > 0 as u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t) ∞ ∑ an t (n+1)q−1−v u = 0 dt Γ((n + 1)q − v) n=0

t > 0,

{ ( )} 𝜋 cos (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋 k , 2

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.95)

We may differintegrate term-by-term (see Section 3.16); thus, u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t) ∞ ∑ an 0 dtu t (n+1)q−1−v

=

n=0

{ ( )} 𝜋 cos (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋 k , Γ((n + 1)q − v) 2

t > 0,

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.96)

Applying equation (5.37), that is, 𝜆 p 0 dt x

=

Γ(p + 1)xp−𝜆 , p > −1 . Γ(p − 𝜆 + 1)

(9.97)

Thus equation (9.96) becomes u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t) ∞ n (n+1)q−1−v−u ∑

=

n=0

{ ( )} a t 𝜋 cos (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) + 2𝜋 k , Γ((n + 1)q − v − u) 2

q > v,

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.98)

From the sum and difference formulas for the integer-order trigonometry, the following identities may be derived: cos(x) = cos(y) cos(x − y) − sin(y) sin(x − y),

(9.99)

195

196

9 The Fractional Meta-Trigonometry

and sin(x) = cos(y) sin(x − y) + sin(y) cos(x − y). (9.100) ( ) ( ) Now, in equation (9.98), let x = (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k , and y = 𝛽 u 𝜋2 + 2𝜋k , with k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1; then applying equation (9.99) gives u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t) = cos(y)

∞ ∑

an t (n+1)q−1−v−u cos{x − y} Γ((n + 1)q − v − u)

n=0 ∞

− sin(y)

∑ n=0

an t (n+1)q−1−v−u sin{x − y} , Γ((n + 1)q − v − u)

q > v, t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.101)

The summations are recognized as Cosq,v+u (a, 𝛼, 𝛽, k, t) and Sinq,v+u (a, 𝛼, 𝛽, k, t), respectively, yielding the final result u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Cosq,v+u (a, 𝛼, 𝛽, k, t) − sin(y)Sinq.v+u (a, 𝛼, 𝛽, k, t), q > v, t > 0,

where y = 𝛽 u giving

(

𝜋 2

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1,

)

(9.102)

+ 2𝜋k . Taking u = (1∕𝛽) ≥ 0, we have cos(y) = 0 and sin(y) = 1

1∕𝛽 0 dt Cosq,v (a, 𝛼, 𝛽, k, t)

= −Sinq.v+1∕𝛽 (a, 𝛼, 𝛽, k, t), q > v, t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.103)

Now, taking q = 1 and v = 0 1∕𝛽 0 dt Cos1,0 (a, 𝛼, 𝛽, k, t)

= −Sin1,1∕𝛽 (a, 𝛼, 𝛽, k, t), t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

An alternative development of the result of equation (9.103) is obtained as follows: [ ] 𝛼 𝛽 −𝛼 −𝛽 R (ai , i t) + R (ai , i t) q,v q,v u u , t > 0, 0 dt Cosq,v (a, 𝛼, 𝛽, t) = 0 dt 2

(9.104)

(9.105)

by the differentiation equation (3.114) u 0 dt Cosq,v (a, 𝛼, 𝛽, t) =

i𝛽 u Rq,v+u (ai𝛼 , i𝛽 t) + i−𝛽 u Rq,v+u (ai−𝛼 , i−𝛽 t) 2

,

q > v.

(9.106)

q > v, t > 0,

(9.107)

When u = 1∕𝛽, we have 1∕𝛽

0 dt Cosq,v (a, 𝛼, 𝛽, t) =

iRq,v+1∕𝛽 (ai𝛼 , i𝛽 t) − iRq,v+1∕𝛽 (ai−𝛼 , i−𝛽 t)

=−

2 Rq,v+1∕𝛽 (ai𝛼 , i𝛽 t) − Rq,v+1∕𝛽 (ai−𝛼 , i−𝛽 t) 2i

, ,

which is recognized as 1∕𝛽 0 dt Cosq,v (a, 𝛼, 𝛽, t)

= −Sinq,v+1∕𝛽 (a, 𝛼, 𝛽, t),

q > v, t > 0,

which is identically equation (9.104) when k = 0, q = 1, and v = 0.

(9.108)

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

9.6.2 Sinq,v (a, 𝜶, 𝜷, k, t)

Determination of the differintegral for the Sinq,v (a, 𝛼, 𝛽, k, t)-function proceeds in a manner similar to that of the Cosq,v (a, 𝛼, 𝛽, k, t). Then, the u-order differintegral of Sinq,v (a, 𝛼, 𝛽, k, t) is determined as ∞ ∑ an t (n+1)q−1−v u u d Sin (a, 𝛼, 𝛽, k, t) = d 0 t q,v 0 t Γ((n + 1)q − v) n=0 )} { ( 𝜋 + 2𝜋 k , × sin (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.109) u 0 dt Sinq,v (a, 𝛼, 𝛽, k, t)

=

∞ ∑ an 0 dtu t (n+1)q−1−v

Γ((n + 1)q − v) )} { ( 𝜋 + 2𝜋 k , × sin (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. n=0

Application of equation (9.97) to this equation gives ∞ ∑

an t (n+1)q−1−v−u Γ((n + 1)q − v − u) n=0 )} { ( 𝜋 + 2𝜋 k , × sin (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 q > v, t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. (9.110) ) ( ) ( In equation (9.110), let x = (n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k and y = 𝛽 u 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1; then applying equation (9.100), we have u 0 dt Sinq,v (a, 𝛼, 𝛽, k, t) =

u 0 dt Sinq,v (a, 𝛼, 𝛽, k, t) = cos(y)

∞ ∑

an t (n+1)q−1−v−u sin{x − y} Γ((n + 1)q − v − u)

n=0 ∞

+ sin(y)

∑ n=0

q > v,

an t (n+1)q−1−v−u cos{x − y} , Γ((n + 1)q − v − 𝜆)

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.111)

The summations are seen to be Sinq,v+u (a, 𝛼, 𝛽, k, t) and Cosq,v+u (a, 𝛼, 𝛽, k, t), respectively, yielding the final result u 0 dt Sinq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Sinq,v+u (a, 𝛼, 𝛽, k, t) + sin(y)Cosq.v+u (a, 𝛼, 𝛽, k, t),

where y = 𝛽 u

(

𝜋 2

)

q > v,

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1,

(9.112)

+ 2𝜋k . Taking u = 1∕𝛽, we have cos( y) = 0 and sin( y) = 1, giving

1∕𝛽 0 dt Sinq,v (a, 𝛼, 𝛽, k, t)

= Cosq.v+1∕𝛽 (a, 𝛼, 𝛽, k, t), q > v,

t > 0 , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.113)

Furthermore, taking q = 1 and v = 0 gives 1∕𝛽 0 dt Sin1,0 (a, 𝛼, 𝛽, k, t)

= Cos1,1∕𝛽 (a, 𝛼, 𝛽, k, t), t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1 .

(9.114)

197

198

9 The Fractional Meta-Trigonometry

The R-function-based development of the result of equation (9.113) is obtained based on equation (9.84) as [ ] Rq,v (ai𝛼 , i𝛽 t) − Rq,v (ai−𝛼 , i−𝛽 t) u u , q > v, t > 0, (9.115) 0 dt Sinq,v (a, 𝛼, 𝛽, t) = 0 dt 2i by the differentiation equation (3.114) [ 𝛽u ] i Rq,v+u (ai𝛼 , i𝛽 t) − i−𝛽 u Rq,v+u (ai−𝛼 , i−𝛽 t) u , 0 dt Sinq,v (a, 𝛼, 𝛽, t) = 2i

q > v, t > 0.

(9.116)

When u = 1∕𝛽 and k = 0, this yields the same result as equation (9.113). 9.6.3 Corq,v (a, 𝜶, 𝜷, t)

The u-order differintegral for the Corq,v (a, 𝛼, 𝛽, t)-function is determined, using definition (9.2.8), as u u 0 dt Corq,v (a, 𝛼, 𝛽, t) = 0 dt

u 0 dt Corq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n (t)(n+1)2q−1−v−q i2n(𝛼+𝛽q)+𝛽(q−1−v)

Γ((n + 1)2q − v − q)

n=0 ∞ 2n 2n(𝛼+𝛽q)+𝛽(q−1−v) ∑

a i

u (n+1)2q−1−v−q 0 dt (t)

Γ((n + 1)2q − v − q)

n=0

,

t > 0,

,

t > 0.

(9.117)

Application of equation (9.97) to this equation gives u 𝛽u 0 dt Corq,v (a, 𝛼, 𝛽, t) = i

∞ ∑ a2n i2n(𝛼+𝛽q)+𝛽(q−1−v−u) (t)(n+1)2q−1−v−q−u

Γ((n + 1)2q − v − q − u)

n=0 u 0 dt Corq,v (a, 𝛼, 𝛽, t)

= i𝛽u Corq,v+u (a, 𝛼, 𝛽, t),

,

t > 0,

t > 0,

(9.118) (9.119)

with q > v, and where the summation has been recognized as an Corq,v+u (a, 𝛼, 𝛽, t)-function, to yield the final result. In terms of the R-function, we have u 0 dt Corq,v (a, 𝛼, 𝛽, t)

or u 0 dt Corq,v (a, 𝛼, 𝛽, t)

= 0 dtu R2q,v+q ((ai𝛼 )2 , i𝛽 t),

= i𝛽 u R2q,v+q+u ((ai𝛼 )2 , i𝛽 t),

t > 0,

(9.120)

t > 0, q > v.

(9.121)

9.6.4 Rotq,v (a, 𝜶, 𝜷, t)

Determination of the differintegral for the Rotq,v (a, 𝛼, 𝛽, t)-function proceeds in a manner similar to that of Corq,v (a, 𝛼, 𝛽, t). Then, the u-order differintegral of Rotq,v (a, 𝛼, 𝛽, t) is determined as ∞ ∑ a2n+1 t (n+1)2q−1−v i(2n+1)(𝛼+𝛽q)+𝛽(q−1−v) u u d Rot (a, 𝛼, 𝛽, t) = d , t > 0, (9.122) 0 t q,v 0 t Γ((n + 1)2q − v) n=0 u 0 dt Rotq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n+1 0 dtu t (n+1)2q−1−v i(2n+1)(𝛼+𝛽q)+𝛽(q−1−v) , Γ((n + 1)2q − v) n=0

t > 0.

(9.123)

t > 0, 2q > v.

(9.124)

Application of equation (9.97) to this equation gives u 0 dt Rotq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n+1 t (n+1)2q−1−v−u i(2n+1)(𝛼+𝛽q)+𝛽(q−1−v) , Γ((n + 1)2q − v − u) n=0

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

The summation is seen to be i𝛽u Rotq,v+u (a, 𝛼, 𝛽, t), yielding u 0 dt Rotq,v (a, 𝛼, 𝛽, t)

= i𝛽 u Rotq,v+u (a, 𝛼, 𝛽, t),

t > 0, u ≥ 0, 2q > v.

(9.125)

In terms of the R-function representation, we have u u 𝛼 2 2𝛼 𝛽 0 dt Rotq,v (a, 𝛼, 𝛽, t) = 0 dt ai R2q,v (a i , i t) = i𝛽 u (ai𝛼 R2q,v+u (a2 i2𝛼 , i𝛽 t)) = ai𝛼+𝛽u R2q,v+u (a2 i2𝛼 , i𝛽 t),

t > 0, 2q > v.

(9.126)

9.6.5 Coflutq,v (a, 𝜶, 𝜷, k, t)

In this section, we determine the differintegral of the Coflutter function. The u-order differintegral of Coflutq,v (a, 𝛼, 𝛽, k, t) is determined as ∞ ∑ a2n+1 t (n+1)2q−1−v Γ((n + 1)2q − v) n=0 )} { ( 𝜋 + 2𝜋 k , × cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.127)

u u 0 dt Coflutq,v (a, 𝛼, 𝛽, k, t) = 0 dt

u 0 dt Coflutq,v (a, 𝛼, 𝛽, k, t)

=

∞ ∑ a2n+1 0 dtu t (n+1)2q−1−v

Γ((n + 1)2q − v) )} { ( 𝜋 + 2𝜋 k , × cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. n=0

Application of equation (9.97) to this equation gives ∞ ∑ a2n+1 t (n+1)2q−1−v−u Γ((n + 1)2q − v − u) n=0 )} { ( 𝜋 + 2𝜋 k , × cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.128) ( ) with 2q − v > 0. Now, in equation (9.128), let y = 𝛽 u 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, ) ( also, let x = ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k ; then, applying equation (9.99), we have ∞ ∑ a2n+1 t (n+1)2q−1−v−u u d Coflut (a, 𝛼, 𝛽, k, t) = cos(y) cos(x − y) 0 t q,v Γ((n + 1)2q − v − u) n=0 u 0 dt Coflutq,v (a, 𝛼, 𝛽, k, t) =

− sin(y)

∞ ∑ a2n+1 t (n+1)2q−1−v−u sin(x − y), Γ((n + 1)2q − v − u) n=0

t > 0, 2q > v,

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.129)

The summations are recognized as Cof lq,v+u (a, 𝛼, 𝛽, k, t) and Flutq,v+u (a, 𝛼, 𝛽, k, t), yielding the desired differintegral form u 0 dt Coflutq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Coflutq,v+u (a, 𝛼, 𝛽, k, t) − sin(y)Flutq,v+u (a, 𝛼, 𝛽, k, t), k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1,

t > 0, 2q > v, (9.130)

199

200

9 The Fractional Meta-Trigonometry

where y = 𝛽u giving

(

𝜋 2

) + 2𝜋k . For the special case with 𝛽 = 1∕u, we have cos(y) = 0 and sin(y) = 1,

u 0 dt Coflutq,v (a, 𝛼, 1∕u, k, t)

= −Flutq,v+1 (a, 𝛼, 1∕u, k, t), q > v,

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.131)

Taking q = 1 and v = 0 in equation (9.130) gives the general form for the differintegral of the Coflutter function as u 0 dt Coflut1,0 (a, 𝛼, 𝛽, k, t)

= cos(y)Coflut1,u (a, 𝛼, 𝛽, k, t) − sin(y)Flut1,u (a, 𝛼, 𝛽, k, t),

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. (9.132)

Furthermore, with 𝛽 = u = 1, we have 0 dt Coflut1,0 (a, 𝛼, 1, k, t)

= −Flut1,1 (a, 𝛼, 1, k, t), t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.133)

The alternative R-function-based development of the results of equation (9.90) is obtained as [ 𝛼 ] i R2q,v ((ai𝛼 )2 , i𝛽 t) + i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t) u u , (9.134) 0 dt Coflutq,v (a, 𝛼, 𝛽, t) = 0 dt a 2 by the differentiation equation (3.114) [ 𝛼+𝛽 u ] i R2q,v+u ((ai𝛼 )2 , i𝛽 t) + i−𝛼−𝛽 u R2q,v+u ((ai−𝛼 )2 , i−𝛽 t) u , 0 dt Coflutq,v (a, 𝛼, 𝛽, t) = a 2

2q > v. (9.135)

For the case 𝛽 = u = 1,

[

i i𝛼 R2q,v+𝜆 ((ai𝛼 )2 , i𝛽 t) − i i−𝛼 R2q,v+𝜆 ((ai−𝛼 )2 , i−𝛽 t)

0 dt Coflutq,v (a, 𝛼, 1, t)

=a

0 dt Coflutq,v (a, 𝛼, 1, t)

= −Flutq,v+1 (a, 𝛼, 1, t),

2 2q > v,

t > 0.

] , 2q > v, (9.136) (9.137)

9.6.6 Flutq,v (a, 𝜶, 𝜷, k, t)

The u-order differintegral of Flutq,v (a, 𝛼, 𝛽, k, t) is determined as ∞ ∑ a2n+1 t (n+1)2q−1−v Γ((n + 1)2q − v) n=0 )) ( ( 𝜋 + 2𝜋k , × sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2

u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t)

= 0 dtu

t > 0, k = 0, 1, 2, … , D − 1, (9.138)

where D = Dq Dv D𝛼 D𝛽 is in minimal form. ∞ ∑ a2n+1 0 dtu t (n+1)2q−1−v Γ((n + 1)2q − v) n=0 )) ( ( 𝜋 + 2𝜋k , × sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2

u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t) =

k = 0, 1, 2, … , D − 1.

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

Applying equation (9.97) to this equation gives ∞ ∑ a2n+1 t (n+1)2q−1−v−u = Γ((n + 1)2q − v − u) n=0 )) ( ( 𝜋 + 2𝜋k , 2q > v, × sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 k = 0, 1, 2, … , D − 1. (9.139) ( ) Now, in equation (9.139), let y = 𝛽 u 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1 and let ) ( x = ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k ; then, applying equation (9.100), we have u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t)

u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t) = cos(y)

∞ ∑ a2n+1 t (n+1)2q−1−v−u sin(x − y) Γ((n + 1)2q − v − u) n=0

+ sin(y) t > 0,

∞ ∑ a2n+1 t (n+1)2q−1−v−u cos(x − y), Γ((n + 1)2q − v − u) n=0

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1,

(9.140)

with 2q − v > 0. The summations are recognized as Flutq,v+u (a, 𝛼, 𝛽, k, t) and Cof lq,v+u (a, 𝛼, 𝛽, k, t), respectively, yielding the final result u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Flutq,v+u (a, 𝛼, 𝛽, k, t) + sin(y) Cof lq,v+u (a, 𝛼, 𝛽, k, t), 2q > v,

(

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1,

)

(9.141)

where y = 𝛽 u 𝜋2 + 2𝜋k . For the special case where 𝛽 = 1∕u, we have cos(y) = 0 and sin(y) = 1, giving u 0 dt Flutq,v (a, 𝛼, 1∕u, k, t)

= Cof lq,v+u (a, 𝛼, 1∕u, k, t), 2q > v,

t > 0 , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.142)

The alternative R-function-based development of the results of equation (9.92) is obtained as follows: [ 𝛼 ] 𝛼 2 𝛽 −𝛼 −𝛼 2 −𝛽 i R ((ai ) , i t) − i R ((ai ) , i t) 2q,v 2q,v u u , t > 0. (9.143) 0 dt Flutq,v (a, 𝛼, 𝛽, t) = 0 dt a 2i Application of the differentiation equation (3.114) gives the result u 0 dt Flutq,v (a, 𝛼, 𝛽, t)

[

=a

i𝛼+𝛽 u R2q,v+u ((ai𝛼 )2 , i𝛽 t) − i−𝛼−𝛽 u R2q,v+u ((ai−𝛼 )2 , i−𝛽 t) 2i

] , 2q > v,

t > 0.

With 𝛽 = 1∕u, we have u 0 dt Flutq,v (a, 𝛼, 1∕u, t)

[

=a

ii𝛼 R2q,v+u ((ai𝛼 )2 , i1∕u t) − i−1 i−𝛼 R2q,v+u ((ai−𝛼 )2 , i−1∕u t) 2i

] , 2q > v,

t > 0,

(9.144)

or u 0 dt Flutq,v (a, 𝛼, 1∕u, t)

= Cof lq,v+u (a, 𝛼, 1∕u, t), 2q > v,

t > 0.

(9.145)

201

202

9 The Fractional Meta-Trigonometry

9.6.7 Covibq,v (a, 𝜶, 𝜷, k, t)

The u-order differintegral of the Covibq,v (a, 𝛼, 𝛽, k, t) is determined as follows: u u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t) = 0 dt

∞ ∑ n=0

a2n t (n+1)2q−1−v−q Γ((n + 1)2q − v − q)

)} { ( 𝜋 + 2𝜋 k , × cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.146) ∞ ∑ a2n 0 dtu t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0 )} { ( 𝜋 + 2𝜋 k , × cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t) =

Application of equation (9.97) to this equation gives ∞ ∑

a2n t (n+1)2q−1−v−q−u Γ((n + 1)2q − v − q − u) n=0 )} { ( 𝜋 + 2𝜋 k , × cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 q > v, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. (9.147)

u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t) =

) + 2𝜋k , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1 in equation (9.147) and let ) ( x = (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k , k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1; then, applying equation (9.99), we have Now let y = 𝛽 u

(

𝜋 2

u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t)

= cos(y)

∞ ∑

a2n t (n+1)2q−1−v−q−u cos(x − y) Γ((n + 1)2q − v − q − u)

n=0 ∞

− sin(y)

∑ n=0

a2n t (n+1)2q−1−v−q−u sin(x − y) , q > v, t > 0. Γ((n + 1)2q − v − q − u) (9.148)

The summations are seen to be Covibq,v+u (a, 𝛼, 𝛽, k, t) and Vibq,v+u (a, 𝛼, 𝛽, k, t), respectively, yielding the key result u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Covibq,v+u (a, 𝛼, 𝛽, k, t) − sin(y) Vibq,v+u (a, 𝛼, 𝛽, k, t), q > v,

where y = 𝛽 u

(

𝜋 2

t > 0, k = 0, 1, … Dq Dv D𝛼 D𝛽 − 1, ) + 2𝜋k . When 𝛽 = 1∕u we have cos(y) = 0 and sin(y) = 1; thus,

u 0 dt Covibq,v (a, 𝛼, 1∕u, k, t)

(9.149)

= −Vibq,v+u (a, 𝛼, 1∕u, k, t), q > v,

t > 0,

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

The R-function-based differintegral is obtained as [ ] R2q,v+q ((ai𝛼 )2 , i𝛽 t) + R2q,v+q ((ai−𝛼 )2 , i−𝛽 t) u u , 0 dt Covibq,v (a, 𝛼, 𝛽, t) = 0 dt 2

t > 0.

(9.150)

(9.151)

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

By the differentiation equation (3.114) u 0 dt Covibq,v (a, 𝛼, 𝛽, t)

=

i𝛽u R2q,v+q+u ((ai𝛼 )2 , i𝛽 t) + i−𝛽u R2q,v+q+u ((ai−𝛼 )2 , i−𝛽 t) 2

, q > v,

t > 0.

When 𝛽 = 1∕u, we have u 0 dt Covibq,v (a, 𝛼, 1∕u, t) =

−R2q,v+q+u ((ai𝛼 )2 , i1∕u t) + R2q,v+q+u ((ai−𝛼 )2 , i−1∕u t)

, (9.152) 2i The right-hand side is seen to be −Vibq,v+u (a, 𝛼, 1∕u, t), thus validating equation (9.150). 9.6.8 Vibq,v (a, 𝜶, 𝜷, k, t)

The u-order differintegral of the generalized vibration function, Vibq,v (a, 𝛼, 𝛽, k, t), is derived as follows: ∞ ∑ a2n t (n+1)2q−1−v−q u u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t) = 0 dt Γ((n + 1)2q − v − q) n=0 )} { ( 𝜋 + 2𝜋 k , × sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.153) u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t) =

∞ ∑ a2n 0 dtu t (n+1)2q−1−v−q

Γ((n + 1)2q − v − q) )} { ( 𝜋 + 2𝜋 k , × sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. n=0

Application of equation (9.97) to this equation gives ∞ ∑

a2n t (n+1)2q−1−v−q−u Γ((n + 1)2q − v − q − u) n=0 )} { ( 𝜋 + 2𝜋 k , × sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 2 t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.154) ( ) ( ) with q − v > 0. Now, let y = 𝛽 u 𝜋2 + 2𝜋k , x = (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) 𝜋2 + 2𝜋k , k = 0, 1, … Dq Dv D𝛼 D𝛽 − 1, in equation (9.154). Then, applying equation (9.100), we have u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t)

u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t)

=

= cos(y)

∞ ∑

a2n t (n+1)2q−1−v−q−u sin{x − y} Γ((n + 1)2q − v − q − u)

n=0 ∞

+ sin(y)

∑ n=0

q > v,

a2n t (n+1)2q−1−v−q−u cos{x − y} , Γ((n + 1)2q − v − q − u)

t > 0,

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1 .

(9.155)

The summations are recognized as Vibq,v+u (a, 𝛼, 𝛽, k, t) and Covibq,v+u (a, 𝛼, 𝛽, k, t), respectively, yielding the key result u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Vibq,v+u (a, 𝛼, 𝛽, k, t) + sin(y)Covibq,v+u (a, 𝛼, 𝛽, k, t),

q > v, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1, (9.156)

203

204

9 The Fractional Meta-Trigonometry

where y = 𝛽 u

(

𝜋 2

) + 2𝜋k . When 𝛽 = 1∕u, we have cos(y) = 0 and sin(y) = 1, giving

u 0 dt Vibq,v (a, 𝛼, 1∕u, k, t)

= Covibq,v+u (a, 𝛼, 1∕u, k, t), q > v,

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.157)

Now, taking q = 1 and v = 0 in equation (9.156), u 0 dt Vib1,0 (a, 𝛼, 𝛽, k, t)

= cos(y)Vib1,u (a, 𝛼, 𝛽, k, t) + sin(y)Covib1,u (a, 𝛼, 𝛽, k, t),

t > 0, k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1. (9.158)

With both u = 1 and with q = 1 and v = 0, we have 0 dt Vib1,0 (a, 𝛼, 𝛽, k, t)

= cos(y)Vib1,1 (a, 𝛼, 𝛽, k, t) + sin(y)Covib1,1 (a, 𝛼, 𝛽, k, t), t > 0,

k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1.

(9.159)

However, if additionally 𝛼 + 𝛽 = 1, Vib1,0 (a, 𝛼, 1 − 𝛼, t) = 0; therefore, we have Vib1,1 (a, 𝛼, 1 − 𝛼, t) = − tan(y)Covib1,1 (a, 𝛼, 1 − 𝛼, t),

t > 0,

(9.160)

where y = (1 − 𝛼) 𝜋∕2. The alternative R-function-based development of the results of equation (9.88) is obtained as [ ] R2q,v+q ((ai𝛼 )2 , i𝛽 t) − R2q,v+q ((ai−𝛼 )2 , i−𝛽 t) u u , 0 dt Vibq,v (a, 𝛼, 𝛽, t) = 0 dt 2i t > 0,

q > v,

(9.161)

by the differintegration equation (3.114) [ 𝛽u ] i R2q,v+q+u ((ai𝛼 )2 , i𝛽 t) − i−𝛽 u R2q,v+q+u ((ai−𝛼 )2 , i−𝛽 t) u , 0 dt Vibq,v (a, 𝛼, 𝛽, t) = 2i q > v, When u = 1, we have 0 dt Vibq,v (a, 𝛼, 𝛽, t)

[ =

t > 0.

(9.162)

i𝛽 R2q,v+q+1 ((ai𝛼 )2 , i𝛽 t) − i−𝛽 R2q,v+q+1 ((ai−𝛼 )2 , i−𝛽 t) 2i

q > v,

] ,

t > 0.

(9.163)

Tables 9.2 and 9.3 summarize the various properties of the meta-trigonometric functions. 9.6.9

Summary of Fractional Calculus Operations on the Meta-Trigonometric Functions

For ease of reference the(fractional)calculus operations are summarized here. The derivations are for t > 0, and y = 𝛽 u 𝜋2 + 2𝜋k and k = 0, 1, … , Dq Dv D𝛼 D𝛽 − 1: u 0 dt Cosq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Cosq,v+u (a, 𝛼, 𝛽, k, t) − sin(y)Sinq.v+u (a, 𝛼, 𝛽, k, t), q > v,

u 0 dt Sinq,v (a, 𝛼, 𝛽, k, t)

(9.164)

= cos(y)Sinq,v+u (a, 𝛼, 𝛽, k, t) + sin(y)Cosq.v+u (a, 𝛼, 𝛽, k, t), q > v,

(9.165)

Oa (Rq,v (ai𝛼 , i𝛽 t)) Ea (Rq,v (ai𝛼 , i𝛽 t))

ai𝛼 R2q,v (a2 i2𝛼 , i𝛽 t)

R2q,v+q (a2 i2𝛼 , i𝛽 t)

Rotq,v (a, 𝛼, 𝛽, t)

Corq,v (a, 𝛼, 𝛽, t)

k = 0, 1, 2, … , D − 1, t > 0; D is the product of the denominators of Dq , Dv , D𝛼 , D𝛽 in minimum form. Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

2

Re(Oa (Rq,v (ai𝛼 , i𝛽 t))) = Oa (Re(Rq,v (ai𝛼 , i𝛽 t)))

a

2i i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) + i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

Cof lq,v (a, 𝛼, 𝛽, t)

Im(Oa (Rq,v (ai𝛼 , i𝛽 t))) = Oa (Im(Rq,v (ai𝛼 , i𝛽 t)))

Re(Ea (Rq,v (ai𝛼 , i𝛽 t))) = Ea (Re(Rq,v (ai𝛼 , i𝛽 t)))

Im(Ea (Rq,v (ai𝛼 , i𝛽 t))) = Ea (Im(Rq,v (ai𝛼 , i𝛽 t)))

Re(Rq,v (ai𝛼 , i𝛽 t))

Im(Rq,v (ai𝛼 , i𝛽 t))

Relation to Rq,v (a, it)

a

2 i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) − i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

2i R2q,v+q ((ai𝛼 )2 , i𝛽 t) + R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

2 R2q,v+q ((ai𝛼 )2 , i𝛽 t) − R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

2i Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t)

Rq,v (ai𝛼 , i𝛽 t) − Rq,v (ai−𝛼 , i−𝛽 t)

R-Function equivalent

Flutq,v (a, 𝛼, 𝛽, t)

Covibq,v (a, 𝛼, 𝛽, t)

Vibq,v (a, 𝛼, 𝛽, t)

Cosq,v (a, 𝛼, 𝛽, t)

Sinq,v (a, 𝛼, 𝛽, k, t)

Function

Table 9.3 Summary of the meta-trigonometric functions.

–

–

2,

− sinh(a t)

0

1,

sin (a t) 0

2,

cosh (a t)

2, 1,

0 cos (a t)

2, 1,

exp(−a t) 0

2, 1,

0 cos (a t)

1,

sin (a t) 2,

1,

q = 1, v = 0, k = 0 𝜶 + 𝜷 = 1 or 2

206

9 The Fractional Meta-Trigonometry u 0 dt Corq,v (a, 𝛼, 𝛽, t) u 0 dt Rotq,v (a, 𝛼, 𝛽, t)

= i𝛽 u Corq,v+u (a, 𝛼, 𝛽, t), q > v,

u 0 dt Coflutq,v (a, 𝛼, 𝛽, k, t)

= cos(y)Coflutq,v+u (a, 𝛼, 𝛽, k, t)

𝛽u

= i Rotq,v+u (a, 𝛼, 𝛽, t) , 2q > v, − sin(y)Flutq,v+u (a, 𝛼, 𝛽, k, t), 2q > v,

u 0 dt Flutq,v (a, 𝛼, 𝛽, k, t)

(9.166) (9.167) (9.168)

= cos(y)Flutq,v+u (a, 𝛼, 𝛽, k, t) + sin(y) Cof lq,v+u (a, 𝛼, 𝛽, k, t), 2q > v,

(9.169)

u 0 dt Covibq,v (a, 𝛼, 𝛽, k, t) = cos(y)Covibq,v+u (a, 𝛼, 𝛽, k, t)

− sin(y) Vibq,v+u (a, 𝛼, 𝛽, k, t), q > v, u 0 dt Vibq,v (a, 𝛼, 𝛽, k, t)

(9.170)

= cos(y)Vibq,v+u (a, 𝛼, 𝛽, k, t) + sin(y)Covibq,v+u (a, 𝛼, 𝛽, k, t), q > v .

(9.171)

9.7 Special Topics in Fractional Differintegration In this, and previous chapters, fractional differintegrals of the fractional trigonometric functions have been derived. In all cases, the fractional differintegration has started from t = 0. Because the functions are zero for t < 0, initialization or history effects have not been an issue. However, when a function does not start from zero or is segmented, such as { fi (t) b = 0 < t < c f (t) = Cosq,v (a, 𝛼, 𝛽, k, t) t > c and preceded by another different function, fractional differintegrations must consider the initialization effects caused by the preceding function. Clearly, there will be times when it is desired to start a fractional differintegration at times other than t = 0. The reader is cautioned that fractional differintegration differs from integer-order integration in this important regard. Detailed analysis of these important differences is addressed in Appendix D.

9.8 Meta-Trigonometric Function Relationships This section determines generalized identities for the fractional meta-trigonometric functions in terms of the fractional exponential functions. These are parallel to the very basic identities presented in equations (1.18) and (1.19). The results will, of course, apply to all particular fractional trigonometries that may be derived by specializing the results of this chapter. 9.8.1 Cosq,v (a, 𝜶, 𝜷, t) and Sinq,v (a, 𝜶, 𝜷, t) Relationships

The generalized fractional Euler equation, with k = 0, for the Cosq,v (a, 𝛼, 𝛽, t)- and Sinq,v (a, 𝛼, 𝛽, t)-functions was determined in Section 9.1 as Vibq,v (a, 𝛼, 𝛽, k, t) =

∞ ∑ n=0

) ( a2n t (n+1)2q−1−v−q 𝜋 . sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) Γ((n + 1)2q − v − q) 2

(9.172)

The complementary generalized fractional Euler equation, with k = 0, can be determined by expanding Rq,v ( ai−𝛼 , i−𝛽 t), in a manner similar to the expansion from equation (9.1) to (9.5), or by subtracting equation (9.84) from (9.82), which gives

9.8 Meta-Trigonometric Function Relationships

( ) Rq,v ai−𝛼 , i−𝛽 t = Cosq,v (a, 𝛼, 𝛽, t) − i Sinq,v (a, 𝛼, 𝛽, t) ,

t > 0.

(9.173)

Continuing from Ref. [83]:

Squaring equations (9.82) and (9.84) yields Cos2q,v (a, 𝛼, 𝛽, t) R2q,v (ai𝛼 , k, i𝛽 t) + R2q,v (ai−𝛼 , i−𝛽 t) + 2 Rq,v (ai𝛼 , i𝛽 t) Rq,v (ai−𝛼 , i−𝛽 t)

=

Sin2q,v (a, 𝛼, 𝛽, t) −R2q,v (ai𝛼 , i𝛽 t) =

,

t > 0.

(9.174)

,

t > 0.

(9.175)

4 − R2q,v (ai−𝛼 , i−𝛽 t) + 2 Rq,v (ai𝛼 , i𝛽 t) Rq,v (ai−𝛼 , i−𝛽 t) 4

Adding and subtracting the resultant equations provides the following two identities: Cos2q,v (a, 𝛼, 𝛽, t) + Sin2q,v (a, 𝛼, 𝛽, t) = Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t), Cos2q,v (a, 𝛼, 𝛽, t) − Sin2q,v (a, 𝛼, 𝛽, t) =

R2q,v ( ai𝛼 , i𝛽 t) + R2q,v ( ai−𝛼 , i−𝛽 t) 2

t > 0,

(9.176)

,

(9.177)

t > 0.

From the result of equation (9.176), it is clear that because Cos2q,v (a, 𝛼, 𝛽, t) and Sin2q,v (a, 𝛼, 𝛽, t) are real, the product Rq,v ( ai𝛼 , i𝛽 , t)Rq,v ( ai−𝛼 , i−𝛽 t) must also be real. Thus, we note that Rq,v ( ai−𝛼 , i−𝛽 t) is the complex conjugate of Rq,v ( ai𝛼 , i𝛽 t). These identities (9.176) and (9.177) are broad generalizations of cos2 (x) + sin2 (x) = 1 and cos2 (x) − sin2 (x) = cos(2x) from the ordinary trigonometry. This may be shown by taking q = a = 𝛼 = 1, and 𝛽 = v = 0 in equations (9.176) and (9.177). 9.8.2 Corq,v (a, 𝜶, 𝜷, t) and Rotq,v (a, 𝜶, 𝜷, t) Relationships

Consider, from equation (9.124) with u = q 𝛼+𝛽q 0 dt Rotq,v (a, 𝛼, 𝛽, t) = ai q

∞ ∑ a2n t (n+1)2q−1−v−q i2n(𝛼+𝛽q)+𝛽(q−1−v) , Γ((n + 1)2q − v − q) n=0

t > 0.

(9.178)

The summation is recognized as Corq,v (a, 𝛼, 𝛽, t) yielding q 0 dt Rotq,v (a, 𝛼, 𝛽, t)

= ai𝛼+𝛽q Corq,v (a, 𝛼, 𝛽, t),

t > 0.

(9.179)

From equation (9.125), we have q 0 dt Rotq,v (a, 𝛼, 𝛽, t)

= i𝛽 q Rotq,v+q (a, 𝛼, 𝛽, t),

t > 0, 2q > v.

(9.180)

Therefore, Rotq,v+q (a, 𝛼, 𝛽, t) = ai𝛼 Corq,v (a, 𝛼, 𝛽, t), Continuing from Ref. [83]:

t > 0, 2q > v.

(9.181)

207

208

9 The Fractional Meta-Trigonometry

9.8.3 Covibq,v (a, 𝜶, 𝜷, t) and Vibq,v (a, 𝜶, 𝜷, t) Relationships

Equations (9.86) and (9.88) are rewritten in the following form: R2q,v+q ((ai𝛼 )2 , i𝛽 t) + R2q,v+q ((ai−𝛼 )2 , i−𝛽 t) = 2Covibq,v (a, 𝛼, 𝛽, t), 𝛼 2

𝛽

R2q,v+q ((ai ) , i t) − R2q,v+q ((ai ) , i t) = 2iVibq,v (a, 𝛼, 𝛽, t), −𝛼 2

t > 0, t>0

−𝛽

(9.182) (9.183)

Adding these equations gives the generalized fractional semi-Euler equation for the Vibration functions as R2q,v+q ((ai𝛼 )2 , i𝛽 t) = Covibq,v (a, 𝛼, 𝛽, t) + iVibq,v (a, 𝛼, 𝛽, t),

t>0

(9.184)

or Rq,v+q∕2 ((ai𝛼 )2 , i𝛽 t) = Covibq∕2,v (a, 𝛼, 𝛽, t) + iVibq∕2,v (a, 𝛼, 𝛽, t),

t > 0.

(9.185)

While subtracting yields the generalized complementary semi-Euler equation R2q,v+q ((ai−𝛼 )2 , i−𝛽 t) = Covibq,v (a, 𝛼, 𝛽, t) − iVibq,v (a, 𝛼, 𝛽, t),

t > 0,

(9.186)

or Rq,v+q∕2 ((ai−𝛼 )2 , i−𝛽 t) = Covibq∕2,v (a, 𝛼, 𝛽, t) − iVibq∕2,v (a, 𝛼, 𝛽, t),

t > 0.

(9.187)

The terminology semi-Euler is used because of the q to q/2 relationship between the right- and left-hand sides of the equations. Squaring equations (9.86) and (9.88) yields Covib2q,v (a, 𝛼, 𝛽, t) =

R22q,v+q ((ai𝛼 )2 , i𝛽 t) + R22q,v+q ((ai−𝛼 )2 , i−𝛽 t)

+

4 2R2q,v+q ((ai𝛼 )2 , i𝛽 t) R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

Vib2q,v (a, 𝛼, 𝛽, t) =

4

,

t > 0,

(9.188)

t > 0.

(9.189)

−R22q,v+q ((ai𝛼 )2 , i𝛽 t) − R22q,v+q ((ai−𝛼 )2 , i−𝛽 t)

+

4 2R2q,v+q ((ai𝛼 )2 , i𝛽 t) R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

4 Adding and subtracting these equations give

,

Covib2q,v (a, 𝛼, 𝛽, t) + Vib2q,v (a, 𝛼, 𝛽, t) = R2q,v+q ((ai𝛼 )2 , i𝛽 t) R2q,v+q ((ai−𝛼 )2 , i−𝛽 t),

t > 0,

(9.190)

Covib2q,v (a, 𝛼, 𝛽, t) − Vib2q,v (a, 𝛼, 𝛽, t) R22q,v+q ((ai𝛼 )2 , i𝛽 t) + R22q,v+q ((ai−𝛼 )2 , i−𝛽 t)

, t > 0, 2 meta-identities relating the Covibq,v (a, 𝛼, 𝛽, t) and Vibq,v (a, 𝛼, 𝛽, t)-functions. =

(9.191)

9.8.4 Coflq,v (a, 𝜶, 𝜷, t) and Flutq,v (a, 𝜶, 𝜷, t) Relationships

The Cof lq,v - and Flutq,v -functions can also be related. We rewrite equations (9.90) and (9.92) as follows: 2Cof lq,v (a, 𝛼, 𝛽, t) = ai𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) + ai−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t),

t>0

(9.192)

9.8 Meta-Trigonometric Function Relationships

and 2iFlutq,v (a, 𝛼, 𝛽, t) = ai𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) − a i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t),

t > 0.

(9.193)

Adding equations (9.192) and (9.193) gives R2q,v ((ai𝛼 )2 , i𝛽 t) =

1 (Cof lq,v (a, 𝛼, 𝛽, t) + iFlutq,v (a, 𝛼, 𝛽, t)), ai𝛼

t > 0.

(9.194)

Alternatively, i−𝛼 (9.195) (Cof lq∕2,v (a, 𝛼, 𝛽, t) + iFlutq∕2,v (a, 𝛼, 𝛽, t)), t > 0. a the generalized fractional semi-Euler equation for the Flutter functions. Subtracting equation (9.193) from (9.192) yields Rq,v ((ai𝛼 )2 , i𝛽 t) =

R2q,v ((ai−𝛼 )2 , i−𝛽 t) =

1 (Cof lq,v (a, 𝛼, 𝛽, t) − iFlutq,v (a, 𝛼, 𝛽, t)), ai−𝛼

t > 0,

(9.196)

or

i𝛼 (9.197) (Cof lq∕2,v (a, 𝛼, 𝛽, t) − iFlutq∕2,v (a, 𝛼, 𝛽, t)), t > 0, a the generalized complementary fractional semi-Euler equation for the Flutter functions. The functions as represented in equations (9.90) and (9.92) are squared to give Rq,v ((ai−𝛼 )2 , i−𝛽 t) =

2 Coflq,v (a, 𝛼, 𝛽, t); = a2

× +

i2𝛼 R22q,v ((ai𝛼 )2 , i𝛽 t) + i−2𝛼 R22q,v ((ai−𝛼 )2 , i−𝛽 t) 4 2R2q,v ((ai𝛼 )2 , i𝛽 t) R2q,v ((ai−𝛼 )2 , i−𝛽 t)

2 Flutq,v (a, 𝛼, 𝛽, t) = a2

× +

4

,

t > 0,

(9.198)

−i2𝛼 R22q,v ((ai𝛼 )2 , i𝛽 t) − i−2𝛼 R22q,v ((ai−𝛼 )2 , i−𝛽 t) 4 2R2q,v ((ai𝛼 )2 , i𝛽 t) R2q,v ((ai−𝛼 )2 , i−𝛽 t)

4 The sum and difference of these equations yield

,

t > 0.

(9.199)

2 2 (a, 𝛼, 𝛽, t) + Flutq,v (a, 𝛼, 𝛽, t) Coflq,v

= a2 R2q,v ((ai𝛼 )2 , i𝛽 t) R2q,v ((ai−𝛼 )2 , i−𝛽 t),

t > 0,

2 2 Coflq,v (a, 𝛼, 𝛽, t) − Flutq,v (a, 𝛼, 𝛽, t) 2𝛼 𝛼 2 𝛽 i R2q,v ((ai ) , i t) + i−2𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t) 2

=a

2 generalized identities relating the Cof lq,v and Flutq,v -functions.

(9.200)

,

t > 0,

(9.201)

9.8.5 Coflq,v (a, 𝜶, 𝜷, t) and Vibq,v (a, 𝜶, 𝜷, t) Relationships

Taking u = q and k = 0 in equation (9.139) gives q

0 dt Flutq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n+1 t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0

× sin(((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2),

t > 0.

(9.202)

209

210

9 The Fractional Meta-Trigonometry

Now, ∞ ∑ a2n+1 t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0

aVibq,v (a, 𝛼, 𝛽, t) =

× sin((2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2),

t > 0.

(9.203)

Let x = (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2 and y1 = −(𝛼 + 𝛽q)𝜋∕2 with sin(x) = cos(y1 ) sin(x − y1 ) + sin(y1 ) cos(x − y1 ).

(9.204)

Then, we may write equation (9.203) as aVibq,v (a, 𝛼, 𝛽, t) = cos(y1 )

∞ ∑ a2n+1 t (n+1)2q−1−v−q sin(x − y1 ) Γ((n + 1)2q − v − q) n=0

+ sin(y1 )

∞ ∑ a2n+1 t (n+1)2q−1−v−q cos(x − y1 ), Γ((n + 1)2q − v − q) n=0

t > 0.

(9.205)

More succinctly, q

aVibq,v (a, 𝛼, 𝛽, t) = cos(y1 )0 dt Flutq,v (a, 𝛼, 𝛽, t) q

+ sin(y1 )0 dt Cof lq,v (a, 𝛼, 𝛽, t),

t > 0.

(9.206)

Taking 𝛿 = 1.0, and k = 0 in equation (9.204) gives q

0 dt Coflutq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n+1 t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0

× cos{((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2},

t > 0.

(9.207)

Now, aCovibq,v (a, 𝛼, 𝛽, t) =

∞ ∑ a2n+1 t (n+1)2q−1−v−q Γ((n + 1)2q − v − q) n=0

× cos((2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2),

t > 0.

(9.208)

Using the same substitutions x = ((2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v))𝜋∕2) and y1 = −(𝛼 + 𝛽q)𝜋∕2 with cos(x) = cos(y1 ) cos(x − y1 ) − sin(y1 ) sin(x − y1 ) into equation (9.208) gives q

aCovibq,v (a, 𝛼, 𝛽, t) = cos(y1 )0 dt Coflutq,v (a, 𝛼, 𝛽, t) q

+ sin(y1 )0 dt Flutq,v (a, 𝛼, 𝛽, t).

(9.209)

Adding equations (9.206) and (9.209) yields the interesting result q 0 dt (Coflutq,v (a, 𝛼, 𝛽, t)

+ Flutq,v (a, 𝛼, 𝛽, t)) a = (Covibq,v (a, 𝛼, 𝛽, t) + Vibq,v (a, 𝛼, 𝛽, t)), cos(y1 ) + sin(y1 )

t > 0,

(9.210)

where y1 = −(𝛼 + 𝛽q)𝜋∕2. Subtracting equation (9.206) from (9.209) gives the complementary result, which may be combined into the single identity q 0 dt (Coflutq,v (a, 𝛼, 𝛽, t)

± Flutq,v (a, 𝛼, 𝛽, t)) a = (Covibq,v (a, 𝛼, 𝛽, t) ± Vibq,v (a, 𝛼, 𝛽, t)), cos(y1 ) ± sin(y1 )

where again y1 = −(𝛼 + 𝛽q)𝜋∕2.

t > 0,

(9.211)

9.8 Meta-Trigonometric Function Relationships

9.8.6 Cosq,v (a, 𝜶, 𝜷, t) and Sinq,v (a, 𝜶, 𝜷, t) Relationships to Other Functions

The following relationships are presented without proof. They may be easily shown by substitution into the defining series and with the use of equations (9.99) and (9.100): Sin2q,v+q (a2 , 2𝛼, 𝛽, k, t) = Vibq,v (a, 𝛼, 𝛽, k, t),

t > 0,

Cos2q,v+q (a , 2𝛼, 𝛽, k, t) = Covibq,v (a, 𝛼, 𝛽, k, t), 2

(9.212)

t > 0.

(9.213)

For the principal functions, we have aSin2q,v (a2 , 2𝛼, 𝛽, t) = cos(y)Flutq,v (a, 𝛼, 𝛽, t) + sin(y)Cof lq,v (a, 𝛼, 𝛽, t),

t > 0,

(9.214)

t > 0,

(9.215)

where y = −𝛼𝜋∕2. aCos2q,v (a2 , 2𝛼, 𝛽, t) = cos(y)Cof lq,v (a, 𝛼, 𝛽, t) − sin(y)Flutq,v (a, 𝛼, 𝛽, t),

where y = −𝛼𝜋∕2. These are but a few of the many relationships possible for the meta-trigonometric functions. Many other relations paralleling the multiple- and fractional-angle formulas from the integer-order trigonometry and more are yet to be derived. 9.8.7

Meta-Identities Based on the Integer-order Trigonometric Identities

Because the series definitions of the meta-trigonometric functions contain the periodic cosine and sine functions, inter-relationships may be determined for these functions. This section determines meta-identities based on these relationships for the Cosq,v (a, 𝛼, 𝛽, t)and Sinq,v (a, 𝛼, 𝛽, t)-functions. Identities such as cos(−x) = cos(x) and sin(−x) = − sin(x) are possible bases, as are the following identities provided by Spanier and Oldham [116], pp. 298–299 from the integer-order trigonometry: (

m𝜋 cos x ± 2 (

m𝜋 sin x ± 2

)

)

⎧cos(x) ⎪∓ sin(x) =⎨ − cos(x) ⎪ ⎩± sin(x)

m = 0, 4, 8, … m = 1, 5, 9, … m = 2, 6, 10, … m = 3, 7, 11, …

(9.216)

⎧sin(x) ⎪± cos(x) =⎨ − sin(x) ⎪ ⎩∓ cos(x)

m = 0, 4, 8, … m = 1, 5, 9, … m = 2, 6, 10, … m = 3, 7, 11, …

(9.217)

Identities for the Cosq,v (a, 𝛼, 𝛽, t)-functions are considered first. 9.8.7.1 The cos(−x) = cos(x)-Based Identity for Cosq,v (a, 𝜶, 𝜷, t)

The question to be answered is: For given values of q and v what values for 𝛼2 and 𝛽2 will provide an identical function with parameters 𝛼1 and 𝛽1 ? In other words, under what conditions will Cosq,v (a, 𝛼1 , 𝛽1 , t) = Cosq,v (a, 𝛼2 , 𝛽2 , t),

t > 0.

(9.218)

From the definition of Cosq,v (a, 𝛼, 𝛽, t), Cosq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , cos (𝛼n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

t > 0. (9.219)

This will occur when for all n ( ) ( ) 𝜋 𝜋 cos (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) = cos (𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]) . 2 2

(9.220)

211

212

9 The Fractional Meta-Trigonometry

Now, from the classical trigonometry using cos(−x) = cos(x), this requires (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) = −(𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]).

(9.221)

This equality must also hold for n = 0; thus, 𝛽1 [q − 1 − v] = −𝛽2 [q − 1 − v] ⇒ 𝛽1 = −𝛽2 . This result substituted into equation (9.221) yields 𝛼1 n + 𝛽1 [(n + 1)q − 1 − v] = −𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]

(9.222)

from which 𝛼2 = −𝛼1 . Thus, the final result is Cosq,v (a, 𝛼1 , 𝛽1 , t) = Cosq,v (a, −𝛼1 , −𝛽1 , t). 9.8.7.2

(9.223)

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

Here, we wish to find conditions under which Sinq,v (a, 𝛼1 , 𝛽1 , t) = Sinq,v (a, 𝛼2 , 𝛽2 , t),

t > 0.

(9.224)

From the definition of Sinq,v (a, 𝛼, 𝛽, t), Sinq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , sin (𝛼n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

t > 0.

(9.225)

Considering the sine arguments (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) = −(𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]).

(9.226)

This equality must also hold for n = 0; thus, 𝛽1 [q − 1 − v] = −𝛽2 [q − 1 − v] ⇒ 𝛽2 = −𝛽1 . This result is now substituted into equation (9.226) giving 𝛼1 n + 𝛽1 [(n + 1)q − 1 − v] = −𝛼2 n + 𝛽1 [(n + 1)q − 1 − v] ⇒ 𝛼2 = −𝛼1 . Combing the argument result with the sign change for sin(−x) = − sin(x) yields the final result Sinq,v (a, 𝛼1 , 𝛽1 , t) = −Sinq,v (a, −𝛼1 , −𝛽1 , t) . 9.8.7.3

(9.227)

The Cosq,v (a, 𝜶, 𝜷, t) ⇔ Sinq,v (a, 𝜶, 𝜷, t) Identity

Selected arguments of the classical cosine and sine functions in the Cosq,v (a, 𝛼, 𝛽, t)- and Sinq,v (a, 𝛼, 𝛽, t)-functions also lead to identities. For this study, we consider Cosq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , cos (𝛼n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

t > 0, (9.228)

and ∞ ) ( ∑ an t (n+1)q−1−v 𝜋 Sinq,v (a, 𝛼, 𝛽, t) = , sin (𝛼n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

t > 0.

Then Sinq,v (a, 𝛼1 , 𝛽1 , t) = Cosq,v (a, 𝛼2 , 𝛽2 , t) when ) ( ) ( 𝜋 𝜋 = cos (𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]) . sin (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) 2 2

(9.229)

(9.230)

9.8 Meta-Trigonometric Function Relationships

From the classical trigonometry (equation (9.217)), we have ± cos(x) = sin(x ± m 𝜋∕2), where m = 1, 5, 9, …. Applying the identity to equation (9.230) gives ) ( ) ( 𝜋 𝜋 𝜋 = sin (𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]) + m . (9.231) sin (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) 2 2 2 Comparing arguments 𝛼1 n + 𝛽1 [(n + 1)q − 1 − v] = 𝛼2 n + 𝛽2 [(n + 1)q − 1 − v] + m, (𝛼1 − 𝛼2 )n + (𝛽1 − 𝛽2 ) [(n + 1)q − 1 − v] = m.

(9.232)

Again, the relationship must hold for n = 0; thus, m m ⇒ 𝛽 2 = 𝛽1 − . (𝛽1 − 𝛽2 ) = q−1−v q−1−v This result substituted into equation (9.232) yields m (𝛼1 − 𝛼2 )n + [(n + 1)q − 1 − v] = m q−1−v mnq (𝛼1 − 𝛼2 )n + +m=m q−1−v mq 𝛼2 = 𝛼1 + . q−1−v Thus, the resulting meta-trigonometric identity is given by ) ( mq m , 𝛽1 − , t Sinq,v (a, 𝛼1 , 𝛽1 , t) = Cosq,v a, 𝛼1 + q−1−v q−1−v 9.8.7.4

(9.233)

m = 1, 5, 9, … (9.234)

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

For Sinq,v (a, 𝛼, 𝛽, t), identities will occur when

Sinq,v (a, 𝛼1 , 𝛽1 , t) = Sinq,v (a, 𝛼2 , 𝛽2 , t).

(9.235)

Based on the definition of Sinq,v (a, 𝛼, 𝛽, t), ∞ ) ( ∑ an t (n+1)q−1−v 𝜋 ,t > 0 Sinq,v (a, 𝛼, 𝛽, t) = sin (𝛼n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

.

(9.236)

We require ( ) ( ) 𝜋 𝜋 sin (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) = sin (𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]) . (9.237) 2 2 Here, from equation (9.217), we use the identity sin(x) = sin(x ± m𝜋∕2), where m = 0, 4, 8, …; thus, ) {( )} ( 𝜋 𝜋 𝜋 = sin (𝛼2 n + 𝛽2 [(n + 1)q − 1 − v]) ± m . sin (𝛼1 n + 𝛽1 [(n + 1)q − 1 − v]) 2 2 2 (9.238) Comparing arguments 𝛼1 n + 𝛽1 [(n + 1)q − 1 − v] = 𝛼2 n + 𝛽2 [(n + 1)q − 1 − v] ± m.

(9.239)

Since the relationship must also hold when n = 0, (𝛽1 − 𝛽2 ) =

±m , m2 = 0, 4, 8, … . q−1−v

(9.240)

213

214

9 The Fractional Meta-Trigonometry

Substituting equation (9.240) into (9.239) gives ±m n(𝛼1 − 𝛼2 ) + [nq + q − 1 − v] = ±m, m = 0, 4, 8, … q−1−v mnq n(𝛼1 − 𝛼2 ) ± ± m = ±m, m = 0, 4, 8, … q−1−v mq 𝛼2 − 𝛼1 = ± , m2 = 0, 4, 8, … q−1−v Thus, we have the identity Sinq,v (a, 𝛼1 , 𝛽1 , t) = Sinq,v

) mq m ,𝛽 ∓ ,t , a, 𝛼1 ± q−1−v 1 q−1−v

(9.241)

(

t > 0, m = 0, 4, 8, … .

(9.242) Clearly other, possibly stronger, identities of this type are possible. Importantly, because the series defining the parity functions also contain integer-order sine and cosine terms, similar identities may be created for them. Continuing from Ref. [83]:

9.9 Fractional Poles: Structure of the Laplace Transforms Consider the Laplace transform of the Covibration function. From equations (9.72) and (9.71), we have [ 2q ] 2 q+v s cos(𝜆) − a cos(𝜆 − 2𝜎) L{Covibq,v (a, 𝛼, 𝛽, k, t)} = s , q > v, s4q − 2a2 cos(2𝜎)s2q + a4 1 ei𝜆 sq+v 1 e−i𝜆 sq+v L{Covibq,v (a, 𝛼, 𝛽, k, t)} = + . (9.243) 2 s2q − a2 ei2𝜎 2 s2q − a2 e−i2𝜎 This can be further resolved to L{Covibq,v (a, 𝛼, 𝛽, k, t)} [ ] 1 ei(𝜆−𝜎) sq+v ei(𝜆−𝜎) sq+v e−i(𝜆−𝜎) sq+v e−i(𝜆−𝜎) sq+v = − + − , 4a (sq − aei𝜎 ) (sq + aei𝜎 ) (sq − ae−i𝜎 ) (sq + ae−i𝜎 )

q > v.

(9.244)

From the transform pair (equation (9.244)), it may be seen that the Covibq,v (a, 𝛼, 𝛽, k, t) may be composed by a weighted sum of four R-functions or four complex fractional poles. The transforms of all of the fractional meta-trigonometric functions may be similarly decomposed.

9.10 Comments and Issues Relative to the Meta-Trigonometric Functions The previous sections of this chapter have presented the definitions, Laplace transforms, special properties, and identities for the meta-trigonometry based on the fractional exponential function, Rq,v (a, t). From these meta-definitions and properties, the analyst may specialize the variables q, v, 𝛼, and 𝛽 to obtain particular fractional trigonometries or hyperboletries. The same variables substituted into the meta-transforms, meta-identities, and so on will yield those transforms and identities specialized to the particular fractional trigonometry. The mathematical process that was used to generate the meta-functions is shown graphically in Figure 9.53. The a and t variables of fractional exponential function, Rq,v (a, t)-function

9.12 Discussion

expressed in series form, are generalized to the complex forms shown in the level 2 box. Two paths are shown from level 2 to level 3. The left path takes the real and imaginary parts of the series while the right path takes the even and odd powered terms of the series. From level 3 to 4, following the left path, four new functions are generated by taking the even and odd powered terms of the cosine and sine series. Following the right path, the same four functions may be generated by taking the real and imaginary parts of the complex corotation and rotation functions. Thus, the process described produces eight functions, six of which are real, and the complex corotation and rotation functions. It is noted that at level three, the leading term of the Laplace transform denominators is 2q while at level four it doubles to 4q. It is noted that the mathematical process observed here is fractal and may be repeated indefinitely to finer and finer resolution of Rq,v (a, t).

9.11 Backward Compatibility to Earlier Fractional Trigonometries The generalized fractional or meta-trigonometry as defined here contains both the R2 and R3 -fractional trigonometries discussed previously. For example, to obtain the R2 -functions, we need only set 𝛼 = 0 and 𝛽 = 1 in the meta-functions; that is, the basis for the trigonometry is Rq,v (a, i t). Similarly, for the R3 -trigonobolic functions, we set 𝛼 = 𝛽 = 1, giving the basis Rq,v (ai, it). The generalized R1 -trigonometry is not, however, directly compatible with definitions forwarded for the R1 -trigonometry, which was conceived prior to the generalized theory. However, the principal R1 -trigonometric functions are compatible with the meta-trigonometry. It is noted that both versions have the basis Rq,v (ai, t). The choice q = 1, v = 0, 𝛼 = 1, and 𝛽 = 0 will yield the functions and identities associated with the classical trigonometry (albeit with new names). The Covibration function is the real and even part with respect to the powers of a of Rq,v (ai𝛼 , i𝛽 t) and, therefore, is a proper successor to the cosine function of the classical trigonometry and the Flut function is imaginary and odd part with respect to the powers of a and, therefore, is an appropriate successor for the sine function. Thus, the Cosq,v -function in the meta-trigonometry is only the real part of the Rq,v (ai𝛼 , i𝛽 t)-function and Sinq,v is only the imaginary part of the Rq,v (ai𝛼 , i𝛽 t)-function; that is, the parity operations have not been applied. The taxonomy chart of Figure 9.53 illustrates the relationships.

9.12 Discussion The meta-trigonometry expands the fractional trigonometries and fractional hyperboletry from the four bases of Chapters 5–8 to the complete infinite set. The defined functions are useable for the study of many dynamic processes and for the solution of many classes of fractional differential equations via the Laplace transforms of the meta-functions. Considerable study will be required to determine which of the bases are important and where they may be applied. While the fractional trignoboletries release us from the constraints associated with the unit circle, many challenges must be met for the area to develop to mathematical maturity, some follow. The foregoing material has only dealt with the primary functions: Sinq,v , Cosq,v , and so on. The ratio and reciprocal functions (i.e., tan, cotan, etc.) associated with the six primary

215

216

9 The Fractional Meta-Trigonometry

functions still need to be defined and to have their properties developed. The task is daunting as there are 36 such functions. The next chapter provides some thoughts on this topic. Development of inverse function definitions (in series form) is problematic since in most cases no repetitive principal cycle exists, and the inverse functions are one-to-many mappings. Software methods, however, may allow us to temporarily bypass this step for practical application. A more detailed discussion of this topic is found in Chapter 20. The existence of orthogonal function sets outside of the classical trigonometry will need to be explored. Furthermore, will the classical definitions of orthogonality be adequate or will new fractional-based definitions be needed?

217

10 The Ratio and Reciprocal Functions Our attention to this point has largely been focused on the primary fractional trigonometric functions, that is, the generalizations of the sine and cosine functions. In this chapter, we expand our considerations to include their reciprocals and ratios. When both the complexity and the parity functions are considered, there are a total of 36 new functions defined. We limit the consideration to the principal, k = 0, meta-trigonometric functions. From the previous chapter, they are, for t > 0: Cosq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , cos (𝛼 n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

Sinq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ an t (n+1)q−1−v 𝜋 , sin (𝛼 n + 𝛽 [(n + 1)q − 1 − v]) Γ((n + 1)q − v) 2 n=0

Covibq,v (a, 𝛼, 𝛽, t) =

∞ ∑ n=0

Vibq,v (a, 𝛼, 𝛽, k, t) =

∞ ∑ n=0

) ( a2n t (n+1)2q−1−v−q 𝜋 , cos (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) Γ((n + 1)2q − v − q) 2 ) ( a2n t (n+1)2q−1−v−q 𝜋 , sin (2n(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) Γ((n + 1)2q − v − q) 2

Cof lq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ a2n+1 t (n+1)2q−1−v 𝜋 , cos ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) Γ((n + 1)2q − v) 2 n=0

Flutq,v (a, 𝛼, 𝛽, t) =

∞ ) ( ∑ a2n+1 t (n+1)2q−1−v 𝜋 . sin ((2n + 1)(𝛼 + 𝛽q) + 𝛽(q − 1 − v)) Γ((n + 1)2q − v) 2 n=0

10.1 Fractional Complexity Functions The naming convention for the complexity fractional meta-trigonometric functions will follow that for the integer-order trigonometry; thus, we have Tanq,v (a, 𝛼, 𝛽, t) ≡ Cotq,v (a, 𝛼, 𝛽, t) ≡

Sinq,v (a, 𝛼, 𝛽, t) Cosq,v (a, 𝛼, 𝛽, t) Cosq,v (a, 𝛼, 𝛽, t) Sinq,v (a, 𝛼, 𝛽, t)

,

t > 0,

(10.1)

,

t > 0,

(10.2)

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

10 The Ratio and Reciprocal Functions

Cosecq,v (a, 𝛼, 𝛽, t) ≡

1 , Sinq,v (a, 𝛼, 𝛽, t)

t > 0,

(10.3)

Secq,v (a, 𝛼, 𝛽, t) ≡

1 , Cosq,v (a, 𝛼, 𝛽, t)

t > 0.

(10.4)

To minimize the number of plots, the graphic displays for the complexity functions are based on the natural quency simplification forms of equations (E.21) and (E.22). Figures 10.1–10.6 4 q = 1.5

Sinq,ν (Ωn,δ,t)

2

0

q = 0.25

–2

–4

–6

0

2

4

6

8

10

t-Time

Figure 10.1 Effect of q with; Sinq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10.

2

0

Cosq,ν (Ωn,δ,t)

218

q = 0.25

–2 –4 –6 –8

–10

q = 1.5

0

2

4

6

8

10

t-Time

Figure 10.2 Effect of q with; Cosq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10.

10.2 The Parity Reciprocal Functions

20

q = 1.5

0.25 15

Tanq,ν (Ωn,δ,t)

10 5 0 –5 –10 –15 –20

0

2

4

6

8

10

t-Time

Figure 10.3 Effect of q with; Tanq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10. 20

q = 1.5

15

CoTanq,ν (Ωn,δ,t)

10 5 0 q = 0.25 –5 –10 –15 –20

0

2

4

6

8

10

t-Time

Figure 10.4 Effect of q with; Co ta nq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10.

show the effect of variations in q for the fractional complexity meta-trigonometric functions for 𝛿 = 1, Ωn = 0.5, v = 0, q = 0.25–1.5 in steps of 0.25, t = 0–10.

10.2 The Parity Reciprocal Functions The parity reciprocal functions are listed as 1 , Covibq,v (a, 𝛼, 𝛽, t)

t > 0,

219

10 The Ratio and Reciprocal Functions

20 15

q = 0.25

q = 1.5

Cosecq,ν (Ωn,δ,t)

10 5 0 –5 –10 –15 –20

0

2

4

6

8

10

t-Time

Figure 10.5 Effect of q with; Cosecq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10. 20

.25

q = 1.5

15 10 Secq,ν (Ωn,δ,t)

220

5 0 –5 –10

–15 –20

0

2

4

6

8

10

t-Time

Figure 10.6 Effect of q with; Secq,0 (Ωn , 𝛿, t) versus t-Time for Ωn = 0.5, v = 0, 𝛿 = 1.0, and q = 0.25–1.5 in steps of 0.25, t = 0–10.

1 , Vibq,v (a, 𝛼, 𝛽, k, t)

and

t > 0,

1 , Cof lq,v (a, 𝛼, 𝛽, t)

t > 0,

1 , Flutq,v (a, 𝛼, 𝛽, t)

t > 0,

10.3 The Parity Ratio Functions

Figure 10.7 Effect of q with Covibq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

10 8

Covibq,0 (1,0.75,0,0,t)

6 4 2 0

q = 1.25 q = 0.25

–2 –4 –6 –8

0

2

4

6

8

10

t-Time 10

1/Covibq,0 (1,0.75,0,0,t)

8 q = 0.25 6

q = 1.25

4 2 0 –2 –4 –6 –8

0

2

4

6

8

10

t-Time

Figure 10.8 Effect of q with 1∕Covibq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

when they exist. Figures 10.7–10.14 show the effect of variations in q for each of the parity meta-trigonometric functions and their reciprocals with 𝛼 = 0.75 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

10.3 The Parity Ratio Functions We consider only 2 of the 36 possible ratio functions for the parity meta-trigonometric functions. They are as follows: Vibq,v (a, 𝛼, 𝛽, t) Covibq,v (a, 𝛼, 𝛽, t)

,

t > 0,

and

Flutq,v (a, 𝛼, 𝛽, t) Cof lq,v (a, 𝛼, 𝛽, t)

,

t > 0.

(10.5)

221

10 The Ratio and Reciprocal Functions

8 6

Vibq,0 (1,0.75,0,0,t)

4 q = 1.25

2 0

q = 0.25

–2 –4 –6

0

2

4

6

8

10

t-Time

Figure 10.9 Effect of q with Vibq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

8 q = 1.25 6

1/Vibq,0 (1,0.75,0,0,t)

222

4 2

q = 0.25

0 –2 –4 –6

0

2

4

6

8

10

t-Time

Figure 10.10 Effect of q with 1∕Vibq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

10.3 The Parity Ratio Functions

4 3

Coflq,0 (1,0.75,0,0,t)

2 1

q = 1.25

0

q = 0.25

–1 –2 –3 –4

0

2

4

6

8

10

t-Time

Figure 10.11 Effect of q with Cof lq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

10 q = 1.25

1/Coflq,0 (1,0.75,0,0,t)

5 q = 0.25 0

–5

–10

0

2

6

4

8

10

t-Time

Figure 10.12 Effect of q with 1/Cof lq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

223

10 The Ratio and Reciprocal Functions

10

Flutq,0 (1,0.75,0,0,t)

5 q = 1.25 0 q = 0.25

–5

–10

0

2

6

4

8

10

t-Time

Figure 10.13 Effect of q with Flutq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

10

q = 1.25 5 1/Flutq,0 (1,0.75,0,0,t)

224

0

q = 0.25

–5

–10

0

2

4

6

8

10

t-Time

Figure 10.14 Effect of q with 1/Flutq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75,𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.25–1.25 in steps of 0.25, t = 0–10.

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

Vibq,0 (1,0.75,0,0,t) / Covibq,0 (1,0.75,0,0,t)

20

15

10 q = 1.25

0.5 5

0

–5 –10

0

2

4

6

8

10

t-Time

Figure 10.15 Effect of q with Vibq,0 (a, 𝛼, 𝛽, k, t)∕Covibq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.5–1.25 in steps of 0.25, t = 0–10.

Figures 10.15 and 10.16 show the effect of q for these ratio functions versus t-Time for 𝛼 = 0.75 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.5–1.25 in steps of 0.25, t = 0–10. The general appearance of these function is similar to that of the Tanq,v (a, 𝛼, 𝛽, t), as seen in Figure 10.3.

10.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions The R-function representations for the ratio and reciprocal functions are easily determined using the results from Section 9.5. Thus, we have for t > 0 Tanq,v (a, 𝛼, 𝛽, t) ≡ Cotq,v (a, 𝛼, 𝛽, t) ≡

Sinq,v (a, 𝛼, 𝛽, t) Cosq,v (a, 𝛼, 𝛽, t) Cosq,v (a, 𝛼, 𝛽, t) Sinq,v (a, 𝛼, 𝛽, t)

=

𝛼 𝛽 −𝛼 −𝛽 1 Rq,v (ai , i t) − Rq,v (ai , i t) , i Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t)

=i

Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t) Rq,v (ai𝛼 , i𝛽 t) − Rq,v (ai−𝛼 , i−𝛽 t)

,

(10.6)

(10.7)

Cosecq,v (a, 𝛼, 𝛽, t) =

1 2i = , Sinq,v (a, 𝛼, 𝛽, t) Rq,v (ai𝛼 , i𝛽 t) − Rq,v (ai−𝛼 , i−𝛽 t)

(10.8)

Secq,v (a, 𝛼, 𝛽, t) =

1 2 = , Cosq,v (a, 𝛼, 𝛽, t) Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t)

(10.9)

2 1 = , Covibq,v (a, 𝛼, 𝛽, t) R2q,v+q ((ai𝛼 )2 , i𝛽 t) + R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

(10.10)

2i 1 = , Vibq,v (a, 𝛼, 𝛽, t) R2q,v+q ((ai𝛼 )2 , i𝛽 t) − R2q,v+q ((ai−𝛼 )2 , i−𝛽 t)

(10.11)

225

10 The Ratio and Reciprocal Functions

10 0.5 Flutq,0 (1,0.75,0,0,t) / Coflq,0 (1,0.75,0,0,t)

226

q = 1.25

5

0

–5

–10

0

2

4

6

8

10

t-Time

Figure 10.16 Effect of q with Flutq,0 (a, 𝛼, 𝛽, k, t)∕Cof lq,0 (a, 𝛼, 𝛽, k, t) versus t-Time for 𝛼 = 0.75, 𝛽 = 0.0, a = 1.0, v = k = 0, q = 0.5–1.25 in steps of 0.25, t = 0–10.

1 1 2 = , Cof lq,v (a, 𝛼, 𝛽, t) a i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) + i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

(10.12)

1 2i 1 = . Flutq,v (a, 𝛼, 𝛽, t) a i𝛼 R2q,v ((ai𝛼 )2 , i𝛽 t) − i−𝛼 R2q,v ((ai−𝛼 )2 , i−𝛽 t)

(10.13)

10.5 Relationships From equation (9.176), we have Cos2q,v (a, 𝛼, 𝛽, t) + Sin2q,v (a, 𝛼, 𝛽, t) = Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t) ,

t > 0.

(10.14)

Dividing by Cos2q,v (a, 𝛼, 𝛽, t) gives 1+

Sin2q,v (a, 𝛼, 𝛽, t) Cos2q,v (a, 𝛼, 𝛽, t)

=

1 + Tan2q,v (a, 𝛼, 𝛽, t) =

Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t) Cos2q,v (a, 𝛼, 𝛽, t) Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t) Cos2q,v (a, 𝛼, 𝛽, t)

,

t > 0,

(10.15)

,

t > 0,

(10.16)

or 1 + Tan2q,v (a, 𝛼, 𝛽, t) = Sec2q,v (a, 𝛼, 𝛽, t){Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t)}, 2

t > 0.

(10.17)

2

This is a broad generalization of the trigonometric identity 1 + tan x = sec x. Dividing equation (10.14) by Sin2q,v (a, 𝛼, 𝛽, t) yields Cos2q,v (a, 𝛼, 𝛽, t) Sin2q,v (a, 𝛼, 𝛽, t)

+1=

Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t) Sin2q,v (a, 𝛼, 𝛽, t)

,

t > 0,

(10.18)

10.6 Discussion

or

2 (a, 𝛼, 𝛽, t) = Cosec2q,v (a, 𝛼, 𝛽, t){Rq,v ( ai𝛼 , i𝛽 t)Rq,v ( ai−𝛼 , i−𝛽 t)}. 1 + Cotq,v 2

(10.19)

2

This is a generalization of the identity 1 + cot x = cosec x. Many similar identities may also be written for the parity functions.

10.6 Discussion There is much to be done in this area and this presentation only begins to scratch the surface. Because of the many possible ratio and reciprocal functions, it is important that a simple naming convention be established that is based on the numerator and denominator functions. The following possibility is offered. Consider the associations Sinq,v (•) ⇒ S Cosq,v (•) ⇒ C Vibq,v (•) ⇒ V CoVibq,v (•) ⇒ CV Flutq,v (•) ⇒ F CoFlutq,v (•) ⇒ CF. Then, using lower case “r” to represent “the reciprocal of,” we have rS ⇒ 1∕Sinq,v (•) , rC ⇒ 1∕Cosq,v (•), and so on. Furthermore, SoC ⇒ Sinq,v (•) ∕Cosq,v (•), VoCV ⇒ Vibq,v (•)∕CoVib (•), and so on for the names of the ratio functions with “o” representing “over.” Importantly, we need to know what special properties may be associated with these functions, whether series expansions can be developed, and whether there are geometric interpretations beyond triangles that may be useful.

227

229

11 Further Generalized Fractional Trigonometries The G- and H-functions (introduced in Chapter 3) are needed to address the condition of repeated poles in the Laplace transforms representing system dynamics in Chapter 12. These functions are generalizations of the F- and R-functions and as such allow generalizations of the fractional meta-trigonometries. Although it is beyond the scope of this book to provide an in-depth coverage of such trigonometries, we briefly consider the subject here.

11.1 The G-Function-Based Trigonometry In Chapter 3, we introduced the G-function; it is defined as { } ∞ ∑ (−r)(−1 − r) · · · (1 − j − r) (−a)j t (r+j)q−𝜈−1 Gq,𝜈,r [a, t] = , j! Γ((r + j)q − 𝜈) j=0

t > 0,

(11.1)

or in terms of the Pochhammer polynomial, (r)j , Gq,𝜈,r [a, t] =

∞ ∑ (r)j (a)j t (r+j)q−𝜈−1 j=0

where (r)j =

j! Γ((r + j)q − 𝜈)

,

t > 0,

(−1)j Γ(1 − r) = (−r)(−1 − r) · · · (1 − r − j)(−1)j , Γ(1 − j − r)

Its Laplace transform (equation (3.95)) is given as { } s𝜈 L Gq,v,r [a, t] = q , Re(qr − v) > 0, Re(s) > 0, (s − a)r

(11.2)

j = 1, 2, … .

| a| | | < 1. | sq | | |

(11.3)

The G-function-based trigonometry is based on Gq,𝜈,r [ai𝛼 , i𝛽 t], that is, [

𝛼

𝛽

]

Gq,𝜈,r ai , i t =

∞ ∑ (r)j (ai𝛼 )j (i𝛽 t)(r+j)q−𝜈−1 j=0

j! Γ((r + j)q − 𝜈)

,

t > 0,

∞ (r)j (a)j (t)(r+j)q−𝜈−1 𝛼j+𝛽((r+j)q−𝜈−1) [ ] ∑ Gq,𝜈,r ai𝛼 , i𝛽 t = , t > 0. i j! Γ((r + j)q − 𝜈) j=0

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(11.4)

230

11 Further Generalized Fractional Trigonometries

Now, consider the exponent of i, assuming rational parameters (including r) 𝛼 j + 𝛽((r + j)q − 𝜈 − 1) N𝛽 Nr Nq N𝛽 Nq N𝛽 Nv N𝛽 N +j − − , = 𝛼 j+ D𝛼 D𝛽 Dr Dq D𝛽 Dq D𝛽 Dv D𝛽 M = 𝛼 j + 𝛽((r + j)q − 𝜈 − 1) = 𝛼 j + 𝛽rq + 𝛽jq − 𝛽𝜈 − 𝛽 D N𝛼 D𝛽 Dq Dv Dr j + N𝛽 Nq Nr D𝛼 Dv + jN𝛽 Nq D𝛼 Dv Dr − N𝛽 Nv D𝛼 Dq Dr − N𝛽 D𝛼 Dq Dv Dr , = D𝛼 D𝛽 Dq Dv Dr Gq,𝜈,r [ai𝛼 , i𝛽 t] =

∞ ∑ (r)j (a)j (t)(r+j)q−𝜈−1 ( j=0

j!Γ((r + j)q − 𝜈)

( cis

M D

(

𝜋 + 2𝜋k 2

)))

, k = 0, 1, 2, … , D − 1, t > 0,

(11.5)

where M∕D = M∕(D𝛼 D𝛽 Dq Dv Dr ) is rational with common factors removed. From this form, the G-trigonometric functions are defined as ∞ ( ( )) ∑ (r)j (a)j (t)(r+j)q−𝜈−1 𝜋 cos (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] ≡ j!Γ((r + j)q − 𝜈) 2 j=0 t > 0, k = 0, 1, 2, … , D − 1, (11.6) and GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] ≡

( ( )) 𝜋 sin (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , j!Γ((r + j)q − 𝜈) 2

∞ ∑ (r)j (a)j (t)(r+j)q−𝜈−1 j=0

t > 0,

k = 0, 1, 2, … , D − 1. (11.7)

The principal G-trigonometric functions, k = 0, are GCosq,𝜈,r [a, 𝛼, 𝛽, t] =

( ( )) 𝜋 , cos (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) j! Γ((r + j)q − 𝜈) 2

∞ ∑ (r)j (a)j (t)(r+j)q−𝜈−1 j=0

t > 0,

(11.8)

( ( )) 𝜋 , t > 0. sin (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) j!Γ((r + j)q − 𝜈) 2

(11.9)

GSinq,𝜈,r [a, 𝛼, 𝛽, t] =

∞ ∑ (r)j (a)j (t)(r+j)q−𝜈−1 j=0

11.2 Laplace Transforms for the G-Trigonometric Functions Consider the Laplace transform of the GCosq,𝜈,r [a, 𝛼, 𝛽, k, t]-function, then transforming term-by-term { } ∞ (r)j (a)j } ∑ { (t)(r+j)q−𝜈−1 L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = L j! Γ((r + j)q − 𝜈) j=0 ( ( )) 𝜋 × cos (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , 2 k = 0, 1, 2, … , D − 1. (11.10)

11.2 Laplace Transforms for the G-Trigonometric Functions

From tables of transforms, we have { } (t)(r+j)q−𝜈−1 1 L = (r+j)q−v , rq > v, Γ((r + j)q − 𝜈) s

(11.11)

giving as a final result ∞ ( ( )) (r)j (a)j } ∑ { 𝜋 cos (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = 2 j! s(r+j)q+v j=0

rq > v, k = 0, 1, 2, … , D − 1. (11.12) A closed form of the transform containing imaginary constants and leading to an interesting connection to the G-function may be derived as follows. Rewrite the cosine as ( ( )) 𝜋 cos (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k )2 ( )) ( ( 𝜋 𝜋 + 2𝜋k + 𝛽(rq − 𝜈 − 1) + 2𝜋k = cos ( j𝜂 + 𝜌) . = cos j(𝛼 + 𝛽q) 2 2 This may be expressed as ei( j𝜂+𝜌) + e−i( j𝜂+𝜌) , 2 ( ) ( ) where 𝜂 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k and 𝜌 = 𝛽(rq − 𝜈 − 1) 𝜋2 + 2𝜋k . Substituting this result into equation (11.12) gives cos (j𝜂 + 𝜌) =

∞ ∞ (r)j (a)j i( j𝜂+𝜌) 1 ∑ (r)j (a)j −i( j𝜂+𝜌) { } 1∑ L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = e + e , 2 j=0 j!s(r+j)q−v 2 j=0 j!s(r+j)q−v ∞ ∞ j i𝜂 j j −i𝜂 j ei𝜌 ∑ (r)j (a) e e−i𝜌 ∑ (r)j (a) e + , 2srq−v j=0 2srq−v j=0 j! sjq j! sjq ( i𝜂 )j ( −i𝜂 )j ∞ ∞ } { ei𝜌 ∑ (r)j e−i𝜌 ∑ (r)j ae ae + rq−v , L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = rq−v q 2s j! s 2s j! sq j=0 j=0

{ } L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] =

(11.13)

with rq > v, k = 0, 1, 2, … , D. Now, the following relationship is found in Ref. [116], p. 150: ∞ [ ] ∑ tn (x)n . (1 − t)−x = exp −x ln(1 − t) = n! n=0

Thus, equation (11.13) may be written as ( )−r ( )−r ei𝜌 e−i𝜌 aei𝜂 ae−i𝜂 + rq−v 1 − q , 1− q 2srq−v s 2s s )r ( )r { } ei𝜌 sv ( sq sq e−i𝜌 sv L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = rq q + . 2s 2srq sq − ae−i𝜂 s − aei𝜂

} { L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] =

Now, using equations (3.124) and (3.125), we have { } i𝛽(rq−v−1) i−𝛽(rq−v−1) sv sv L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = + , 2 2 (sq − aei𝜂 )r (sq − ae−i𝜂 )r rq > v, k = 0, 1, 2, … , D − 1.

(11.14)

231

232

11 Further Generalized Fractional Trigonometries

The ratios are recognized as Laplace transforms of G-functions, thus relating the GCos transform to the G-function transforms as } i𝛽(rq−v−1) { } i−𝛽(rq−v−1) { } { L GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] = L Gq,v,r (aei𝜂 , t + L Gq,v,r (ae−i𝜂 , t , 2 2 rq > v, k = 0, 1, 2, … , D − 1. (11.15) Inverse transforming gives the time-domain relation GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] =

i𝛽(rq−v−1) i−𝛽(rq−v−1) Gq,v,r (aei𝜂 , t) + Gq,v,r (ae−i𝜂 , t) , 2 2 rq > v, k = 0, 1, 2, … , D − 1.

(11.16)

After some algebra, this may be written as GCosq,𝜈,r [a, 𝛼, 𝛽, k, t] =

Gq,v,r (ai𝛼 , i𝛽 t) + Gq,v,r (ai−𝛼 , i−𝛽 t)

, 2 rq > v, k = 0, 1, 2, … , D − 1, t > 0.

(11.17)

From this result, we can easily obtain a closed form for the Laplace transform of GCosq,v,r [a, 𝛼, 𝛽, k, t], yielding a closed form for equation (11.12). Here, we see the G-function acting as a generalized exponential or generalization of the R-function. The development of the Laplace transform for the GSinq,𝜈,r [a, 𝛼, 𝛽, k, t]-function, is similar to that for GCosq,𝜈,r [a, 𝛼, 𝛽, k, t]; then, { } L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] { } ∞ ( ( )) ∑ (r)j (a)j (t)(r+j)q−𝜈−1 𝜋 L sin (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , = j! Γ((r + j)q − 𝜈) 2 j=0 k = 0, 1, … , D − 1.

(11.18)

Applying equation (11.11), we have as a final result } { L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] ∞ ( ( )) ∑ (r)j (a)j 𝜋 sin (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) + 2𝜋k , rq > v, k = 0, 1, 2, … , D − 1. = 2 j!s(r+j)q+v j=0 (11.19) Rewrite the sine as )) ( ( 𝜋 + 2𝜋k sin (𝛼 j + 𝛽((r + j)q − 𝜈 − 1)) ) 2 ( )) ( ( 𝜋 𝜋 + 2𝜋k + 𝛽(rq − 𝜈 − 1) + 2𝜋k = sin (j𝜂 + 𝜌), = sin j(𝛼 + 𝛽q) 2 2 and this may be expressed as ei( j𝜂+𝜌) − e−i( j𝜂+𝜌) , 2i ) ( ) ( where 𝜂 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k and 𝜌 = 𝛽(rq − 𝜈 − 1) 𝜋2 + 2𝜋k .Substituting this result into equation (11.19) gives sin (j𝜂 + 𝜌) =

∞ ∞ j j { } 1 ∑ (r)j (a) i( j𝜂+𝜌) 1 ∑ (r)j (a) −i( j𝜂+𝜌) L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] = e − e , 2i j=0 j!s(r+j)q−v 2i j=0 j!s(r+j)q−v

11.2 Laplace Transforms for the G-Trigonometric Functions ∞ ∞ j i𝜂 j j −i𝜂 j ei𝜌 ∑ (r)j (a) e e−i𝜌 ∑ (r)j (a) e − , 2isrq−v j=0 2isrq−v j=0 j! sjq j! sjq ( i𝜂 )j ( −i𝜂 )j ∞ ∞ } { ei𝜌 ∑ (r)j e−i𝜌 ∑ (r)j ae ae − , L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] = 2isrq−v j=0 j! sq 2isrq−v j=0 j ! sq

{ } L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] =

(11.20)

with rq > v, k = 0, 1, 2, … , D − 1. Now, applying equation (11.14), equation (11.20) may be written as ( )−r ( )−r { } ei𝜌 e−i𝜌 aei𝜂 ae−i𝜂 L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] = − rq−v 1 − q , 1− q 2isrq−v s 2is s )r ( )r { } ei𝜌 sv ( sq sq e−i𝜌 sv L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] = − , (11.21) rq rq 2is sq − aei𝜂 2is sq − ae−i𝜂 { } ei𝜌 e−i𝜌 sv sv L GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] = − , 2i (sq − aei𝜂 )r 2i (sq − ae−i𝜂 )r rq > v, k = 0, 1, 2, … , D − 1. (11.22) Inverse transforming gives GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] =

ei𝜌 e−i𝜌 Gq,v,r (aei𝜂 , t) − G (ae−i𝜂 , t) , 2i 2i q,v,r rq > v, k = 0, 1, 2, … , D − 1.

(11.23)

Now, using equations (3.124) and (3.125), we have GSinq,𝜈,r [a, 𝛼, 𝛽, k, t] =

i𝛽(rq−r−1) i−𝛽(rq−r−1) Gq,v,r (aei𝜂 , t) − Gq,v,r (ae−i𝜂 , t) , 2i 2i rq > v, k = 0, 1, 2, … , D − 1, t > 0.

(11.24)

This may also be written as GSinq,𝜈,r [a, 𝛼, 𝛽, t] =

Gq,v,r (ai𝛼 , i𝛽 t) − Gq,v,r (ai−𝛼 , i−𝛽 t)

, rq > v, t > 0. (11.25) 2i From this result, we can easily obtain a closed form for the Laplace transform of GSinq,v,r [a, 𝛼, 𝛽, k, t], yielding a closed form for equation (11.19). Here, we see the G-function playing the role for the GSinq,𝜈,r similar to that played by the fractional exponential function for the fractional Cosq,v . Some properties of this trigonometry follow. Squaring equations (11.17) and (11.25) gives [ ] 2 𝛼 𝛽 2 −𝛼 −𝛽 1 Gq,𝜈,r (ai , i t) + Gq,𝜈,r (ai , i t) 2 GCosq,𝜈,r [a, 𝛼, 𝛽, t] = , rq > v, (11.26) 2 2 4 +2Gq,𝜈,r (ai𝛼 , i𝛽 t)Gq,𝜈,r (ai−𝛼 , i−𝛽 t) [ ] 2 𝛼 𝛽 2 −𝛼 −𝛽 (ai , i t) + G (ai , i t) G q,𝜈,r q,𝜈,r 1 , rq > v. (11.27) GSin2q,𝜈,r [a, 𝛼, 𝛽, t] = − 2 2 4 (ai𝛼 , i𝛽 t)Gq,𝜈,r (ai−𝛼 , i−𝛽 t) −2Gq,𝜈,r Adding equations (11.26) and (11.27) gives the identity GCos2q,𝜈,r [a, 𝛼, 𝛽, t] + GSin2q,𝜈,r [a, 𝛼, 𝛽, t] 2 2 (ai𝛼 , i𝛽 t)Gq,𝜈,r (ai−𝛼 , i−𝛽 t) , t > 0, rq > v. = Gq,𝜈,r

(11.28)

Subtracting gives the companion identity GCos2q,𝜈,r [a, 𝛼, 𝛽, t] − GSin2q,𝜈,r [a, 𝛼, 𝛽, t] ] 1[ 2 2 Gq,𝜈,r (ai𝛼 , i𝛽 t) + Gq,𝜈,r = (ai−𝛼 , i−𝛽 t) , t > 0, rq > v. 2

(11.29)

233

234

11 Further Generalized Fractional Trigonometries

11.3 The H-Function-Based Trigonometry The H-function was introduced in Chapter 3. Its usefulness is in the solution of fractional differential equations with solutions containing repeated complex fractional roots. The H-function is defined as Hq,v,r (c0 , c1 , t) =

j ∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j cn−j 1 c0

n! j!

n=0 j=0

t q(2r+n+j)−v−1 , t > 0, Γ(q(2r + n + j) − v)

(11.30)

where (r)n and (n − j + 1)j are Pochhammer polynomials. Its Laplace transform is given as } { L Hq,v,r (c0 , c1 , t) =

(s2q

s𝜈 , + c1 sq + c0 )r

| | c1 | + c0 | < 1, | sq s2q | | |

| c0 | | | | c sq | < 1, 2qr > v. | 1 |

(11.31)

The H-function-based trigonometry is based on Hq,v,r (c0 i𝛼 , c1 i𝛿 , i𝛽 t), that is, Hq,v,r (c0 i𝛼 , c1 i𝛿 , i𝛽 t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 i𝛿 )n−j (c0 i𝛼 )j

n! j!

n=0 j=0

(i𝛽 t)q(2r+n+j)−v−1 , Γ(q(2r + n + j) − v)

t > 0,

Hq,v,r (c0 i𝛼 , c1 i𝛿 , i𝛽 t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j (t)q(2r+n+j)−v−1 i𝛽(q(2r+n+j)−v−1)+𝛿(n−j)+𝛼 j , n! j! Γ(q(2r + n + j) − v) n=0 j=0

t > 0.

Under the assumption of rational variables, the exponent of i may be written as M = 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j D = 2𝛽qr + 𝛽qn + 𝛽qj − 𝛽v − 𝛽 + 𝛿n − 𝛿j + 𝛼j =

2N𝛽 Nq Nr Dv D𝛼 D𝛿 + nN𝛽 Nq Dv Dr D𝛼 D𝛿 + jN𝛽 Nq Dv Dr D𝛼 D𝛿 − N𝛽 Nv Dq Dr D𝛼 D𝛿 Dq Dv Dr D𝛼 D𝛿 D𝛽 +

−N𝛽 Dq Dv Dr D𝛼 D𝛿 + nN𝛿 Dq Dv Dr D𝛼 D𝛽 − jN𝛿 Dq Dv Dr D𝛼 D𝛽 + jN𝛼 Dq Dv Dr D𝛿 D𝛽 Dq Dv Dr D𝛼 D𝛿 D𝛽

.

Then, we have Hq,v,r (c0 i𝛼 , c1 i𝛿 , i𝛽 t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j n=0 j=0

n! j!

( ( )) (t)q(2r+n+j)−v−1 M 𝜋 cis + 2𝜋k , Γ(q(2r + n + j) − v) D 2 t > 0, k = 0, 1, 2, … , D − 1, (11.32)

where D = Dq Dv Dr D𝛼 D𝛿 D𝛽 , and M/D is rational and in minimal form. From this form, the H-fractional trigonometric functions are defined as the real and imaginary parts of equation (11.32): HCosq,v,r (c0 , c1 , k, t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j

(t)q(2r+n+j)−v−1 n! j! Γ(q(2r + n + j) − v) n=0 j=0 ( ( )) 𝜋 × cos (𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j) + 2𝜋k , 2 t > 0, k = 0, 1, 2, · · · D − 1, (11.33)

11.4 Laplace Transforms for the H-Trigonometric Functions

and HSinq,v,r (c0 , c1 , k, t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j

(t)q(2r+n+j)−v−1 n! j! Γ(q(2r + n + j) − v) n=0 j=0 (( )) ) (𝜋 × sin 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j + 2𝜋k , 2 t > 0, k = 0, 1, 2, … , D − 1, (11.34)

where D = Dq Dv Dr D𝛼 D𝛿 D𝛽 and is in minimal form. The principal H-fractional trigonometric functions, k = 0 and with t > 0, are HCosq,v,r (c0 , c1 , t) =

∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j

(t)q(2r+n+j)−v−1 n! j! Γ(q(2r + n + j) − v) n=0 j=0 (( ) ( 𝜋 )) , × cos 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j 2

(11.35)

and ∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j

(t)q(2r+n+j)−v−1 n! j! Γ(q(2r + n + j) − v) n=0 j=0 (( ) ( 𝜋 )) . (11.36) × sin 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j 2

HSinq,v,r (c0 , c1 , t) =

11.4 Laplace Transforms for the H-Trigonometric Functions ( ) ∞ ∞ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j } ∑ (t)q(2r+n+j)−v−1 L HCosq,v,r (c0 , c1 , k, t) = L n! j! Γ(q(2r + n + j) − v) n=0 j=0 (( )) ) (𝜋 × cos 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j + 2𝜋k , 2 t > 0, k = 0, 1, 2, · · · D − 1, (11.37) {

or ∞ ∞ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j } ∑ { L HCosq,v,r (c0 , c1 , k, t) = n! j! sq(2r+n+j)−v n=0 j=0 (( )) ) (𝜋 × cos 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j + 2𝜋k , 2 2qr > v, k = 0, 1, 2, … , D − 1, (11.38)

where D = Dq Dv Dr D𝛼 D𝛿 D𝛽 , and is in minimal form. A closed form of the transform containing imaginary constants and leading to a connection to the H-function may be derived as follows. Rewrite the cosine as (( )) ) (𝜋 cos 𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j + 2𝜋k 2 ( ) )) ( ( 𝜋 𝜋 + 2𝜋k + {𝛽(q(2r + n) − v − 1) + 𝛿n} + 2𝜋k = cos j(𝛽q − 𝛿 + 𝛼) 2 2 = cos (j𝜂 + 𝜌) .

235

236

11 Further Generalized Fractional Trigonometries

This may be expressed as ei( jn+p) + e−i( jn+p) cos( jn + p) = , 2 ) ( ) ( where 𝜂 = (𝛽q − 𝛿 + 𝛼) 𝜋2 + 2𝜋k and 𝜌 = {𝛽(q(2r + n) − v − 1) + 𝛿n} 𝜋2 + 2𝜋k . Therefore, { i(j𝜂+𝜌) } ∞ ∞ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j { } ∑ e + e−i(j𝜂+𝜌) L HCosq,v,r (c0 , c1 , k, t) = , , 2 n! j! sq(2r+n+j)−v n=0 j=0 2qr > v, k = 0, 1, 2, … , D − 1, ( )j c0 ∞ (n − j + 1) ∞ ei𝜂j n n v i𝜌 ∑ (−1) (r)n (c1 ) s e ∑ j c 1

{ } L HCosq,v,r (c0 , c1 , k, t) =

2 n! sq(2r+n)

n=0

j! sq j

j=0

( )j +

∞ ∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌 ∑ (n − j + 1)j

2 n! sq(2r+n)

n=0

c0 c1

e−i𝜂j

j! sq j

j=0

,

2qr > v, k = 0, 1, 2, … , D − 1. Now, from Ref. [116], p. 151, we have (m − p + 1)p = (−m)p (−1)p , } ∑ (−1) (r)n (c1 ) s e L HCosq,v,r (c0 , c1 , k, t) = 2 n! sq(2r+n) n=0 {

∞

+

∞ n v i𝜌 ∑

n

(−1)j (−n)j

( )j c0 c1

j! sq j

j=0

j ∞ ∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌 ∑ (−1) (−n)j n=0

2 n! sq(2r+n)

ei𝜂j

( )j c0 c1

e−i𝜂j

j! sq j

j=0

,

2qr > v, k = 0, 1, 2, … , D − 1, ∞ ∞ } ∑ (−1)n (r)n (c1 )n sv ei𝜌 ∑ (−n)j L HCosq,v,r (c0 , c1 , k, t) = j! 2 n! sq(2r+n) n=0 j=0

{

+

(

−c0 ei𝜂 c1 sq

∞ ∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌 ∑ (−n)j n=0

2 n! sq(2r+n)

j=0

(

j!

)j

−c0 e−i𝜂 c1 sq

)j ,

2qr > v, k = 0, 1, 2, … , D − 1. Now, applying equation (11.14) { } L HCosq,v,r (c0 , c1 , k, t) =

∞ ∑ (−1)n (r)n (c1 )n sv ei𝜌 n=0

{

2 n! sq(2r+n)

} L HCosq,v,r (c0 , c1 , k, t)

(

c ei𝜂 1+ 0 q c1 s

)n +

∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌 n=0

2 n! sq(2r+n)

(

c e−i𝜂 1+ 0 q c1 s

)n

2qr > v, k = 0, 1, 2, … , D − 1,

,

11.4 Laplace Transforms for the H-Trigonometric Functions ∞ n i𝜌 sv ∑ (−1) (r)n e = 2qr s n=0 2 n! sqn

(

c1 sq + c0 ei𝜂 c1 sq

)n

∞ n −i𝜌 sv ∑ (−1) (r)n e + 2qr s n=0 2 n! sqn

(

c1 sq + c0 e−i𝜂 c1 sq

)n ,

2qr > v, k = 0, 1, 2, … , D − 1. i𝜌 i{𝛽(q(2r+n)−v−1)+𝛿n}(𝜋∕2+2𝜋k) We now write = i{𝛽(q(2r+n)−v−1)+𝛿n} = i(𝛽q+𝛿)n i𝛽(2qr−v−1) ( )e = e 𝜋 (𝛽q−𝛿+𝛼) +2𝜋k 2 ei𝜂 = e = i𝛽q−𝛿+𝛼 . Thus, we have

and

( ( ) )n ∞ i(𝛽q+𝛿) c1 sq + c0 i𝛽q−𝛿+𝛼 { } sv i𝛽(2qr−v−1) ∑ (r)n L HCosq,v,r (c0 , c1 , k, t) = − 2s2qr n! c1 s2q n=0 ( ( ) )n ∞ i−(𝛽q+𝛿) c1 sq + c0 i−(𝛽q−𝛿+𝛼) sv i−𝛽(2qr−v−1) ∑ (r)n + , − 2s2qr n! c1 s2q n=0 2qr > v. Applying equation (11.14) again gives ( )−r } sv i𝛽(2qr−v−1) i(𝛽q+𝛿) (c1 sq + c0 i𝛽q−𝛿+𝛼 ) 1+ L HCosq,v,r (c0 , c1 , k, t) = 2s2qr c1 s2q ( )−r i−(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) sv i−𝛽(2qr−v−1) + , 1 + 2s2qr c1 s2q 2qr > v. ( 2q (𝛽q+𝛿) )−r q 𝛽q−𝛿+𝛼 v 𝛽(2qr−v−1) } si { c1 s + i (c1 s + c0 i ) L HCosq,v,r (c0 , c1 , k, t) = 2s2qr c1 s2q ( )−r 2q −(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) sv i−𝛽(2qr−v−1) c1 s + i + , 2s2qr c1 s2q 2qr > v. r 𝛽(2qr−v−1) v { } 1 c1 i s L HCosq,v,r (c0 , c1 , k, t) = 2q (𝛽q+𝛿) q 2 (c1 s + i (c1 s + c0 i𝛽q−𝛿+𝛼 ))r c1 r i−𝛽(2qr−v−1) sv 1 + , 2qr > v. 2 (c1 s2q + i−(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ))r } 1 { i𝛽(2qr−v−1) sv L HCosq,v,r (c0 , c1 , k, t) = ( ( )) 2 2q (𝛽q+𝛿) q c0 𝛽q−𝛿+𝛼 r s +i s +ci {

1

i−𝛽(2qr−v−1) sv ( ))r , 2qr > v. c s2q + i−(𝛽q+𝛿) sq + c0 i−(𝛽q−𝛿+𝛼)

1 + ( 2

1

} 1 { L HCosq,v,r (c0 , c1 , k, t) = ( 2

i𝛽(2qr−v−1) sv s2q + i(𝛽q+𝛿) sq +

+

c0 2𝛽q+𝛼 i c1

)r

i−𝛽(2qr−v−1) sv

1 ( 2

s2q + i−(𝛽q+𝛿) sq +

c0 −(2𝛽q+𝛼) i c1

)r , 2qr > v.

(11.39)

237

238

11 Further Generalized Fractional Trigonometries

The ratios are recognized as Laplace transforms of H-functions; see equation (3.99). Inverse transforming gives ) ( c0 2𝛽q+𝛼 𝛽q+𝛿 i𝛽(2qr−v−1) i ,i ,t HCosq,v,r (c0 , c1 , t) = Hq,v,r 2 c1 ) ( c0 −(2𝛽q+𝛼) −(𝛽q+𝛿) i−𝛽(2qr−v−1) + i ,i , t , 2qr > v, t > 0. (11.40) Hq,v,r 2 c1 This form is analogous to equation (11.16) for the G-function. Similarly, for the HSin } { L HSinq,v,r (c0 , c1 , k, t) ( ) ∞ ∞ ∑ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j (t)q(2r+n+j)−v−1 L = n! j! Γ(q(2r + n + j) − v) n=0 j=0 )) ( ( 𝜋 + 2𝜋k , × sin (𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j) 2 t > 0, k = 0, 1, 2, … , D − 1,

(11.41)

or ∞ ∞ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j { } ∑ L HSinq,v,r (c0 , c1 , k, t) = n! j! sq(2r+n+j)−v n=0 j=0 ( ( )) 𝜋 × sin (𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j) + 2𝜋k , 2 2qr > v, k = 0, 1, 2, … , D − 1,

where D = Dq Dv Dr D𝛼 D𝛿 D𝛽 and is in minimal form. A closed form of the transform containing imaginary constants and leading to a connection to the H-function may be derived as follows. Rewrite the sine as )) ( ( 𝜋 + 2𝜋k sin (𝛽(q(2r + n + j) − v − 1) + 𝛿(n − j) + 𝛼 j) 2 ( ) )) ( ( 𝜋 𝜋 + 2𝜋k + {𝛽(q(2r + n) − v − 1) + 𝛿n} + 2𝜋k = sin j(𝛽q − 𝛿 + 𝛼) 2 2 = sin (j𝜂 + 𝜌) . This may be expressed as ei( j𝜂+𝜌) − e−i( j𝜂+𝜌) , sin (j𝜂 + 𝜌) = 2i ( ) ( ) where 𝜂 = (𝛽q − 𝛿 + 𝛼) 𝜋2 + 2𝜋k and 𝜌 = {𝛽(q(2r + n) − v − 1) + 𝛿n} 𝜋2 + 2𝜋k . Therefore, ∞ ∞ ∑ (−1)n (r)n (n − j + 1)j (c1 )n−j (c0 )j { } ∑ L HSinq,v,r (c0 , c1 , k, t) = n! j! sq(2r+n+j)−v n=0 j=0 { i(j𝜂+𝜌) } e − e−i(j𝜂+𝜌) × , 2qr > v, k = 0, 1, 2, … , D − 1, 2i ( )j c0 ∞ (n − j + 1) ∞ ei𝜂j n n v i𝜌 ∑ ∑ j } { (−1) (r)n (c1 ) s e c1 L HSinq,v,r (c0 , c1 , k, t) = i j! sq j 2 n! sq(2r+n) n=0 j=0

(11.42)

11.4 Laplace Transforms for the H-Trigonometric Functions

( )j ∑ (−1) (r)n (c1 ) s e − 2 n! sq(2r+n) n=0 ∞

∞ n v −i𝜌 ∑

n

(n − j + 1)j

c0 c1

e−i𝜂j

i j! sq j

j=0

,

2qr > v, k = 0, 1, 2, · · · D − 1. Again using the relation (m − p + 1)p = (−m)p (−1)p , we have } L HSinq,v,r (c0 , c1 , k, t) = {

j ∞ ∞ ∑ (−1)n (r)n (c1 )n sv ei𝜌 ∑ (−1) (−n)j

2 n! sq(2r+n)

n=0

∞ n v −i𝜌 ∑

n

c0 c1

ei𝜂j

i j! sq j

j=0

∑ (−1) (r)n (c1 ) s e − 2 n! sq(2r+n) n=0 ∞

( )j

(−1)j (−n)j

( )j c0 c1

e−i𝜂j

i j! sq j

j=0

,

2qr > v, k = 0, 1, 2, … , D − 1, ( )j ∞ ∞ { } ∑ (−1)n (r)n (c1 )n sv ei𝜌 ∑ (−n)j −c0 ei𝜂 L HSinq,v,r (c0 , c1 , k, t) = i j! c1 sq 2 n! sq(2r+n) n=0 j=0 −

∞ ∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌 ∑ (−n)j n=0

2 n! sq(2r+n)

j=0

(

i j!

−c0 e−i𝜂 c1 sq

)j ,

2qr > v, k = 0, 1, 2, … , D − 1. Now, applying equation (11.14) } { L HSinq,v,r (c0 , c1 , k, t) =

∞ ∑ (−1)n (r)n (c1 )n sv ei𝜌 n=0

2 i n! sq(2r+n)

} L HSinq,v,r (c0 , c1 , k, t)

( 1+

c0 ei𝜂 c1 sq

)n −

n=0

∞ n i𝜌 sv ∑ (−1) (r)n e 2qr qn s n=0 2 i n! s

2 i n! sq(2r+n)

( 1+

c0 e−i𝜂 c1 sq

)n ,

2qr > v, k = 0, 1, 2, … , D − 1.

{

=

∞ ∑ (−1)n (r)n (c1 )n sv e−i𝜌

(

c1 sq + c0 ei𝜂 c1 sq

)n −

∞ n −i𝜌 sv ∑ (−1) (r)n e 2qr qn s n=0 2 i n! s

(

c1 sq + c0 e−i𝜂 c1 sq

)n ,

2qr > v, k = 0, 1, 2, … , D − 1. We now( write) ei𝜌 = ei{𝛽(q(2r+n)−v−1)+𝛿n}(𝜋∕2+2𝜋k) = i{𝛽(q(2r+n)−v−1)+𝛿n} = i(𝛽q+𝛿)n i𝛽(2qr−v−1) and ei𝜂 = (𝛽q−𝛿+𝛼) 𝜋2 +2𝜋k e = i𝛽q−𝛿+𝛼 . Thus, we have ( (𝛽q+𝛿) )n ∞ } sv i𝛽(2qr−v−1) ∑ { i (r)n (c1 sq + c0 i𝛽q−𝛿+𝛼 ) − L HSinq,v,r (c0 , c1 , k, t) = 2 i s2qr n=0 n! c1 s2q ( −(𝛽q+𝛿) )n ∞ i (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) sv i−𝛽(2qr−v−1) ∑ (r)n − , 2qr > v. − 2 i s2qr n=0 n! c1 s2q Applying equation (11.14) again gives

( )−r } sv i𝛽(2qr−v−1) i(𝛽q+𝛿) (c1 sq + c0 i𝛽q−𝛿+𝛼 ) L HSinq,v,r (c0 , c1 , k, t) = 1+ 2 i s2qr c1 s2q {

239

240

11 Further Generalized Fractional Trigonometries

( )−r i−(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) sv i−𝛽(2qr−v−1) − , 2qr > v. 1+ 2 i s2qr c1 s2q ( )−r { } sv i𝛽(2qr−v−1) c1 s2q + i(𝛽q+𝛿) (c1 sq + c0 i𝛽q−𝛿+𝛼 ) L HSinq,v,r (c0 , c1 , k, t) = 2 i s2qr c1 s2q ( )−r 2q −(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) sv i−𝛽(2qr−v−1) c1 s + i − , 2qr > v. 2 i s2qr c1 s2q ( )r } sv i𝛽(2qr−v−1) { c1 s2q L HSinq,v,r (c0 , c1 , k, t) = 2 i s2qr c1 s2q + i(𝛽q+𝛿) (c1 sq + c0 i𝛽q−𝛿+𝛼 ) ( )r c1 s2q sv i−𝛽(2qr−v−1) − , 2qr > v. 2 i s2qr c1 s2q + i−(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ) { } c1 i𝛽(2qr−v−1) sv 1 L HSinq,v,r (c0 , c1 , k, t) = 2i (c1 s2q + i(𝛽q+𝛿) (c1 sq + c0 i𝛽q−𝛿+𝛼 ))r −

c1 i−𝛽(2qr−v−1) sv 1 , 2qr > v. 2i (c1 s2q + i−(𝛽q+𝛿) (c1 sq + c0 i−(𝛽q−𝛿+𝛼) ))r

} { 1 L HSinq,v,r (c0 , c1 , k, t) = ( 2i

i𝛽(2qr−v−1) sv s2q + i(𝛽q+𝛿) sq +

−

1 ( 2i

c0 2𝛽q+𝛼 i c1

)r

i−𝛽(2qr−v−1) sv s2q + i−(𝛽q+𝛿) sq +

c0 −(2𝛽q+𝛼) i c1

)r , 2qr > v.

(11.43)

The ratios are recognized as Laplace transforms of H-functions. Inverse transforming gives ) ( c0 2𝛽q+𝛼 𝛽q+𝛿 i𝛽(2qr−v−1) HSinq,v,r (c0 , c1 , t) = i ,i ,t Hq,v,r 2i c1 ) ( c0 −(2𝛽q+𝛼) −(𝛽q+𝛿) i−𝛽(2qr−v−1) − i ,i , t , 2qr > v. (11.44) Hq,v,r 2i c1 This is analogous to equation (11.24) for the G-function. The H-function-based trigonometry is the most general of the fractional trigonometries studied and clearly the most difficult to work with. It requires a double summation of functions containing two Pochhammer polynomials. Because the H-fractional trigonometry directly covers the cases of unrepeated roots, repeated real roots, and repeated complex roots when applied to commensurate constant-coefficient linear fractional differential equations, further generalized trigonometries will likely not be beneficial relative to this problem. On examining their Laplace transforms, it is clear that by taking r = 1 in the G-function it becomes an R-function. Furthermore, by taking c1 = 0 in the H-function, it takes the form of the G-function.

241

Introduction to Applications The primary and by far the most important application of the fractional exponential and fractional trigonometric functions is the solution of linear fractional differential equations. In Chapter 12, these functions were used for the solution of linear commensurate-order fractional differential equations with constant coefficients. Analytical tools for such differential equations with unrepeated roots, repeated real roots, and repeated complex roots were presented. Spiral behavior occurs in many natural phenomena. Galactic spirals, sea shells, whirlpools, and weather patterns are only a few such occurrences seen in the sciences. We have seen numerous examples of fractional spirals as the result of our development of the fractional trigonometries. It is natural to ask if and how the fractional spirals and associated fractional differential equations might apply to these scientific areas. Our goal is to show the application potential to such areas and not necessarily the solution to detailed problems. Specific areas to be considered in Chapters 13–19 are fractional oscillators, morphology of sea shells, in particular the Nautilus pompilius, hurricanes, and low-pressure system cloud morphology, morphological classification of spiral and ring galaxies, tornado profiles, and whirlpool morphology. While in most cases these potential application areas provide spatial definition of the phenomena, it is expected that the development of compatible cylindrical fractional field equations, considered briefly in Chapter 20, will allow full definition and possible temporal solution. There is much to be done in the development of the fractional trigonometry; Chapter 20 presents some related challenges.

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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12 The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry This chapter develops methods for the solution of linear constant-coefficient fractional differential equations of any commensurate order. Classes of fractional differential equations with unrepeated roots, repeated real roots, and repeated complex conjugate roots are considered. The solutions for the case of unrepeated roots are implemented using simplified Laplace transforms based on the fractional meta-trigonometric functions and the R-function as developed in Chapter 3. The solution of fractional differential equations with repeated roots, additionally require application of the G- and H-functions, which were studied in Chapter 3.

12.1 Fractional Differential Equations Driven by our need to understand and codify the physical world, the major objective of the fractional calculus is the formation and solution of fractional integral and, more importantly, fractional differential equations. There has been considerable effort toward the solution of fractional differential equations going back to the work of Heavyside [51], Davis [26], and others. More recently, the works of Oldham and Spanier [104], Miller and Ross [95], Samko et al. [114], West et al. [121], and Oustaloup [105] have contributed to the development of the fractional calculus as well as to the solution of fractional differential equations. Furthermore, Podlubny [109] has written an important book dedicated to the solution of fractional differential equations. These works and others present various dedicated approaches to the solution of specialized and general fractional differential equations. The development of the fractional trigonometries of Chapters 5–9 may be applied to the solution of fractional differential equations. In particular, the fractional meta-trigonometric functions developed in Chapter 9 applies toward more general classes of equations. Application of the fractional trigonometric functions to the solutions of linear fractional differential equations has been treated by Lorenzo, Hartley, and Malti [87, 88] and is the basis of this chapter. Fractional differential equations of the form nq 0 Dt x(t)

3q

2q

q

+ · · · + c3 {0 Dt x(t)} + c2 {0 Dt x(t)} + c1 {0 Dt x(t)} + c0 x(t) = f (t)

(12.1)

are known as commensurate order due to the constraining relationship of the order of the derivatives. Our goal is to find the solution of such linear fractional differential equations of

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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

any commensurate order when the coefficients ci are constants. The related Laplace transform for equation (12.1) with zero initial conditions is the transfer function L(x(t)) 1 . (12.2) = nq Lf (t) s + · · · + c3 s3q + c2 s2q + c1 sq + c0 The method for solution of equation (12.1) is similar to that required for ordinary linear differential equations (i.e., equation (12.1) with q = 1), with constant coefficients. The procedure for ordinary linear differential equations with constant coefficients is to factor the related transfer function into sums of first- and second-order subsystems; that is, into sums of the forms ks/(s + a) or k/(s + a) for the first-order subsystem and k/(s2 + bs + c), ks/(s2 + bs + c) or ks2 /(s2 + bs + c) for the second-order subsystems. Inverse transforming the subsystems yields solutions that are sums of time functions, which are combinations of real exponential functions and the classical sine and cosine functions. The approach developed here for the solution for fractional differential equations of the form of equation (12.1) is similar to that used for ordinary linear differential equations. That is, the transfer function (12.2) is decomposed (factored) into the sum of subsystems elements with denominators of the qth and 2qth order. These subsystems, known as elementary transfer functions of the first and second kinds, are of the respective forms k sv k sv and 2q . (12.3) q c1 s + c0 s + c1 sq + c0 It is important to note that Podlubny [109] has solved fractional differential equations that generalize the form of equation (12.1) using infinite summations of generalized Mittag-Leffler functions. The solutions to be developed here will be in terms of the simpler fractional trigonometric functions. Linear fractional differential equations of the form D𝛼 x(t) + 𝜆x(t) = u(t),

t > 0, 𝛼, 𝜆 ∈ ℜ

(12.4)

have been referred to as the fundamental fractional differential equation, as discussed in Chapter 2. The solutions of this equation may be expressed in terms of R-functions with real arguments. The more general form D2𝛼 x(t) + c1 D𝛼 x(t) + c2 x(t) = u(t), 0 < 𝛼 ≤ 2,

t > 0, c1 , c2 ∈ ℜ

(12.5)

is shown to have solutions that can be represented as fractional trigonometric functions. As we have seen, fractional trigonometric functions are real functions based on the R-function with complex arguments, namely Rq,v (ai𝛼 , i𝛽 t),

t > 0, q, v, a, 𝛼, 𝛽 ∈ ℜ .

(12.6)

The fractional trigonometries (of Chapters 5–9) define a wide variety of oscillatory and non-oscillatory fractional functions that are solutions or components of solutions to such linear fractional differential equations. q In equation (12.1), we use the notation 0 Dt x(t) to indicate the initialized fractional derivative q q q of the qth order. Then, 0 Dt x(t) ≡ 0 dt x(t) + 𝜓(t), where 0 dt x(t) is the uninitialized derivative and 𝜓(t) is the time-varying initialization function (see, e.g., [68, 71, 78]). The effect of the initialization terms is to add time-varying terms to the right-hand side of equation (12.1). These terms contribute the initialization response of the solution. Inclusion of the initialization terms will distract from the objectives of this chapter and for simplicity of presentation, only the forced response is considered. Therefore, it is our goal to solve uninitialized fractional differential equations of the form nq 0 dt x(t)

3q

2q

q

+ · · · + c3 {0 dt x(t)} + c2 {0 dt x(t)} + c1 {0 dt x(t)} + c0 x(t) = f (t).

(12.7)

12.2 Fundamental Fractional Differential Equations of the First Kind

A secondary goal is to create specialized versions of the Laplace transforms of the fractional meta-trigonometric functions to allow easy application to the solutions of equation (12.7). The following sections discuss the solutions for the elementary functions of the first and second types.

12.2 Fundamental Fractional Differential Equations of the First Kind Elementary transfer functions of the first kind (equation (12.4)) derive from the fundamental fractional differential equation q (12.8) 0 dt x(t) + c0 x(t) = f (t), which was introduced in Chapter 2. Solutions to this equation are based on the F-function detailed in Chapter 2. The more general case q 0 dt x(t)

+ c0 x(t) = 0 dtv f (t)

(12.9)

was considered in Chapter 3. Under quiescent initial conditions, the Laplace transform was shown to be sv L{ f (t)} L{x(t)} = q . (12.10) s + c0 When f (t) = 𝛿(t), a unit impulse, the solution to equation (12.8) may be written in terms of the R-function. Its Laplace transform was shown to be L{Rq,v (a, t)} =

sv , Re(q − v) > 0, Re(s) > 0. sq − c0

(12.11)

Then, x(t) = Rq,v (−c0 , t),

t>0

provides the solution to equation (12.9). Less conveniently, some of the Mittag-Leffler-based functions of Table 3.1 may be used in some instances for solution to equation (12.9). When f (t) ≠ 𝛿(t), the solution may usually be obtained by application of the convolution theorem to equation (12.10). Differential equations of this type may also be solved using the following transform pairs, for c0 > 0, { } { } L R1 Cosq,v (a, t) = L Covibq,v (a, 1, 0, 0, t) =

sv+q , q > v, + a2 } { } { asv , 2q > v, L R1 Sinq,v (a, t) = L Flutq,v (a, 1, 0, 0, t) = 2q s + a2 s2q

(12.12) (12.13)

and for c0 < 0 { } { } L R1 Coshq,v (a, t) = L Covibq,v (a, 0, 0, 0, t) =

sv+q , q > v, s2q − a2 } { } { asv , 2q > v. L R1 Sinhq,v (a, t) = L Cof lq,v (a, 0, 0, 0, t) = 2q s − a2

(12.14) (12.15)

Furthermore, in some cases, it may be useful to specialize the fractional meta-trigonometric functions (Table 9.2) for the solution of equation (12.9).

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12.3 Fundamental Fractional Differential Equations of the Second Kind Elementary transfer functions of the second kind, first studied by Malti et al. [91], relate to fractional differential equations of the form 2q 0 dt x(t)

q

+ c1 {0 dt x(t)} + c0 x(t) = 0 dtu f (t),

(12.16)

with c0 and c1 real. The Laplace transform associated with this equation is given by L{x(t)} =

su L{ f (t)} . + c1 sq + c0

s2q

(12.17)

In general, fractional differential equations of the second kind and their transforms are needed for the solution of higher-order fractional differential equations with complex roots in the characteristic equation. The denominator of equation (12.17) is quadratic in sq and its behavior is dependent on the discriminant, c21 − 4c0 , that is, for c21 − 4c0 < 0, the roots are complex conjugates, the case of interest for fractional differential equations of the second kind. The fractional trigonometric functions of Chapters 6–8 provide solutions to particular cases of equation (12.16). The Laplace transforms for the meta-trigonometric functions (Table 9.2) more broadly apply. Solutions related to equation (12.16) are shown using either the fractional cosine or fractional sine functions in the following sections.

12.4 Preliminaries – Laplace Transforms The Laplace transforms developed in Chapter 9 are simplified for more direct application to the solution of fractional differential equations. In particular, simplified Laplace transform pairs based on those previously determined for the fractional trigonometric functions are developed for the solution of equation (12.16). This section is adapted from Lorenzo et al. [87, 88]. Note that the mathematical results in Ref. [87] have systemic errors and should not be used. Errors in previous versions have been corrected here. Proceeding with permission of ASME and the Royal Society: 12.4.1

Fractional Cosine Function

The Laplace transform of the fractional cosine function is given as [ q ] } { v s cos(𝜆) − a cos(𝜆 − 𝜎) , q > v, L Cosq,v (a, 𝛼, 𝛽, k, t) = s s2q − 2a cos(𝜎)sq + a2 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k . To reduce the number of parameters in this equation and in other equations to follow, we consider only the principal functions. That is, we take k = 0; thus, [ q ] { } v s cos(𝜆) − a cos(𝜆 − 𝜎) L Cosq,v (a, 𝛼, 𝛽, t) = s , q > v, s2q − 2a cos(𝜎)sq + a2 where 𝜎 = (𝛼 + 𝛽q)𝜋∕2 , 𝜆 = 𝛽 [q − 1 − v]𝜋∕2. This Laplace transform is now specialized for application to equation (12.17). For f (t) = 𝛿(t), a unit impulse, we take

12.4 Preliminaries – Laplace Transforms

v = u, a2 = c0 , −2a cos(𝜎) = c1 , cos(𝜆) = 0, and let −a cos(𝜆 − 𝜎) = K ∗ . From these conditions, the following requirements are determined: √ a2 = c0 ⇒ a = c0 , cos(𝜆) = 0 ⇒ cos(𝛽 [q − 1 − u]𝜋∕2) = 0 , thus 𝜆∗ = m𝜋∕2,

m = ±1, ±3, ±5, … .

For simplicity of results, take m = 1 𝛽 ∗ (q − 1 − u)𝜋∕2 = 𝜋∕2, 𝛽∗ =

1 . (q − 1 − u)

(12.18)

√ Now, we also require −2a cos(𝜎) = c1 ⇒ −1 ≤ (c1 ∕2a) ≤ 1. When −1 < (c1 ∕2 c0 ) < 1, the roots are complex conjugates. Then, √ 𝜎 ∗ = arccos(−c1 ∕2 c0 ). But we must also have that 𝜎 ∗ = (𝛼 + 𝛽 ∗ q)𝜋∕2 .

(12.19)

Solving for 𝛼 𝛼= 𝛼 = ∗

2𝜎 ∗ − 𝛽 ∗q , 𝜋 √ 2 arccos(−c1 ∕2 c0 )

Now, we can write

𝜋

−𝛽 q= ∗

√ 2 arccos(−c1 ∕2 c0 ) 𝜋

−

q . q−1−v

(12.20)

) ( √ √ 𝜋 K ∗ = − c0 cos(𝜆∗ − 𝜎 ∗ ) = − c0 cos − 𝜎∗ 2 √ √ = − c0 sin(𝜎 ∗ ), ⇒ −1 < K • ∕ c0 < 1, √ √ √ 1 K ∗ = − c0 sin(arccos(−c1 ∕2 c0 )) = − 4c0 − c21 . 2

These results give the useful transform ⎫ [ ⎧ ] (√ )⎪ ⎪ su −2 ∗ ∗ Cosq,u c0 , 𝛼 , 𝛽 , t ⎬ = 2q L ⎨√ , q > u, 4c0 > c1 2 , q +c s + c s 1 0 ⎪ ⎪ 4c0 − c21 ⎭ ⎩

(12.21)

where 𝛼 ∗ , 𝛽 ∗ , and 𝜎 ∗ are given by equations (12.20), (12.18), and (12.19), respectively, and the star superscripts are introduced to indicate that the values apply only to the formulation of this section. Thus, the forced response solution for equation (12.16), with f (t) a unit impulse, is given by (√ ) 1 c0 , 𝛼 ∗ , 𝛽 ∗ , t , q > 0. (12.22) x(t) = ∗ Cosq,u K When f (t) is not the unit impulse, the convolution theorem may be applied to evaluate the solution. Because the following transform pairs are derived in a similar manner, and apply under different constraint conditions, only the results are given.

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12.4.2

Fractional Sine Function

The principal fractional sine function may also be the basis of a simplified Laplace transform. We start from the transform pair [ q ] { } s sin(𝜆) − a sin(𝜆 − 𝜎) L Sinq,v (a, 𝛼, 𝛽, t) = sv 2q , q > v, (12.23) s − 2 a cos(𝜎)sq + a2 where 𝜎 = (𝛼 + 𝛽q)𝜋∕2 and 𝜆 = 𝛽[q − 1 − v]𝜋∕2. The Sinq,v (a, 𝛼, 𝛽, t)-based transform is given by ⎧ ⎫ ] (√ )⎪ [ ⎪ 2 sv ∗ L ⎨√ Sinq,v c0 , 𝛼 , 0, t ⎬ = 2q , q > v , 4c0 > c1 2 . s + c1 sq + c0 ⎪ 4c0 − c21 ⎪ ⎩ ⎭

(12.24)

The star superscripts indicate that the values apply only to this formulation, and 𝛼 ∗ is given by ( √ ) 2 arccos −c1 ∕2 c0 . (12.25) 𝛼∗ = 𝜋 12.4.3

Higher-Order Numerator Dynamics

12.4.3.1

Fractional Cosine Function

We also require Laplace transforms of the form L{x(t)} =

sq+v L{ f (t)} . + c1 sq + c0

(12.26)

s2q

This form is not covered by either equation (12.21) or (12.24) because of the limiting constraints on those equations. The Laplace transform of the fractional cosine function is [ q ] { } v s cos(𝜆) − a cos(𝜆 − 𝜎) L Cosq,v (a, 𝛼, 𝛽, t) = s , q > v, s2q − 2a cos(𝜎)sq + a2 where 𝜎 = (𝛼 + 𝛽q)𝜋∕2 , and 𝜆 = 𝛽[q − 1 − v]𝜋∕2. The result for the fractional cosine function is ⎫ ⎧ (√ )⎪ ⎪ sq+v −1 ∗ ∗ Cosq,v c0 , 𝛼 , 𝛽 , t ⎬ = 2q , q > v, 4c0 > c21 , L ⎨√ q +c s + c s 2 1 0 ⎪ ⎪ 1 − c1 ∕4c0 ⎭ ⎩ where 𝛼 ∗ and 𝛽 ∗ are given by √ ) 𝛼 ∗ = 2 arccos −c1 ∕2 c0 ∕𝜋 − 𝛽 ∗ q , (

12.4.3.2

and 𝛽 ∗ =

(12.27)

( √ ) 1 + 2 arccos −c1 ∕2 c0 ∕𝜋 [q − 1 − v]

.

(12.28)

Fractional Sine Function

A Laplace transform with a higher order of the numerator term may be based on the fractional sine function [88] [ q ] } { v s sin(𝜆) − a sin(𝜆 − 𝜎) L Sinq,v (a, 𝛼, 𝛽, t) = s , q > v, s2q − 2 a cos(𝜎)sq + a2

12.4 Preliminaries – Laplace Transforms

where 𝜎 = (𝛼 + 𝛽q)(𝜋∕2), 𝜆 = 𝛽[q − 1 − v](𝜋∕2) follows. The transform pair is given by ⎫ [ ⎧ ] (√ )⎪ ⎪ sq+v 1 ∗ ∗ Sinq,v c0 , 𝛼 , 𝛽 , t ⎬ = 2q , q > v, 4c0 > c21 , L ⎨√ q +c s + c s 2 1 0 ⎪ ⎪ 1 − c1 ∕4c0 ⎭ ⎩ where 𝛼 ∗ and 𝛽 ∗ are 𝛼∗ =

( √ ) 2 arccos −c1 ∕2 c0 (−1 − v) 𝜋 (q − 1 − v)

,

𝛽∗ =

(12.29)

( √ ) 2 arccos −c1 ∕2 c0 𝜋 (q − 1 − v)

.

(12.30)

12.4.4 Parity Functions – The Flutter Function

The parity functions may also be used to solve the fundamental fractional differential equation of the second kind. Here, only the Flutter function is considered, and its Laplace transform is given as [ 2q ] 2 { } v s sin(𝜆 + 𝜎) − a sin(𝜆 − 𝜎) L Flutq,v (a, 𝛼, 𝛽, t) = a s , 2q > v, (12.31) s4q − 2a2 cos(2𝜎)s2q + a4 where 𝜎 = (𝛼 + 𝛽q)𝜋∕2 , 𝜆 = 𝛽 [q − 1 − v]𝜋∕2. Letting q = w/2, we have [ w ] } { s sin(𝜆w + 𝜎w ) − a2 sin(𝜆w − 𝜎w ) v , w > v, L Flutw∕2,v (a, 𝛼, 𝛽, t) = a s s2w − 2a2 cos(2𝜎w )sw + a4 where 𝜎w = (𝛼 + 𝛽w∕2)𝜋∕2 , 𝜆w = 𝛽 [(w∕2) − 1 − v]𝜋∕2. −2a2 cos(2𝜎w ) = c1 , −a3 sin(𝜆w − 𝜎w ) = 0, and a4 = c0 . Then, a = c0 1∕4 and from −a3 sin(𝜆w − 𝜎w ) = 0, we have

Now,

let

𝜆w = 𝜎w ⇒ 𝛼 ∗ = −𝛽 ∗ (v + 1).

(12.32)

a sin(𝜆w + 𝜎w ) = K ∗ ,

(12.33)

From −2a2 cos(2𝜎w ) = c1 ,

) ( √ 𝜋 = −c1 ∕2 c0 , cos 2(𝛼 + 𝛽w∕2) 2 ( √ ) (𝛼 + 𝛽w∕2)𝜋 = arccos −c1 ∕2 c0 , ( √ ) arccos −c1 ∕2 c0 𝛽∗ = . 𝜋(w∕2 − 1 − v)

(12.34)

Using K ∗ = a sin(𝜆w + 𝜎w ), we have K ∗ = a sin(𝜆w + 𝜎w ) = c0 1∕4 sin(2𝜆w ), ( ( √ )) K ∗ = c0 1∕4 sin(2𝜆w ) = c0 1∕4 sin arccos −c1 ∕2 c0 , √ K ∗ = c0 1∕4 1 − c1 2 ∕4c0 . A useful transform pair based on the fractional meta-trigonometric parity function (equation (12.32) is given as { } [ ] ( ) 1 sv+w 1∕4 ∗ ∗ Flutw∕2, u c0 , 𝛼 , 𝛽 , t L , w > v, 4c0 > c21 , = 2w √ s + c1 sw + c0 c0 1∕4 1 − c1 2 ∕4c0 (12.35) where 𝛼 ∗ and 𝛽 ∗ are given by equations (12.33) and (12.34), respectively.

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12.4.5

Additional Transform Pairs

There is considerable interest in elementary transfer functions of the second kind, defined by Malti et al. [91]. The Laplace transform of interest is (𝜔0 )2u , s2u + 2𝜁 (𝜔0 )u su + (𝜔0 )2u

(12.36)

where 𝜔0 is a fractional generalization of natural frequency and 𝜁 has been named the pseudo-damping factor. Here, we consider the related form K∗ . s2u + 2𝜁 (𝜔0 )u su + (𝜔0 )2u

(12.37)

Both the fractional sine and cosine functions may be used to achieve such transforms. Here, the sine function (equation (12.23)) is considered with 𝜎 = (𝛼 + 𝛽q)𝜋∕2 and 𝜆 = 𝛽 [q − 1 − v]𝜋∕2. Thus, we have the Laplace transform as { } [ ] ( u ∗ ) 1 sv Sinu,v 𝜔0 , 𝛼 , 0, t , u > v, L = 2u √ s + 2𝜁 (𝜔0 )u su + (𝜔0 )2u 𝜔u 1 − 𝜁 2 0

4(𝜔0 )2u > (2𝜁 (𝜔0 )u )2 ,

(12.38)

where 𝛼 ∗ is given by

2 arccos(−𝜁 ) . (12.39) 𝜋 The fractional cosine function may also be used to obtain transform pairs related to equation (12.4.5.2). For this case, we have the result { } [ ] ( u ∗ ∗ ) 1 sv Cosu,v 𝜔0 , 𝛼 , 𝛽 , t , u > v, L = 2u √ s + 2𝜁 (𝜔0 )u su + (𝜔0 )2u 𝜔u0 1 − 𝜁 2 𝛼∗ =

4(𝜔0 )2u > (2𝜁 (𝜔0 )u )2 ,

(12.40)

where 𝜁 2 ≤ 1, and 𝛼∗ =

2 arccos(−𝜁 ) − 𝛽 ∗u 𝜋

and 𝛽 ∗ =

1 . (u − 1 − v)

(12.41)

Note that in the previous simplified transform pairs the constraint 4c0 > c1 2 (or 4(𝜔0 )2u > (2𝜁 (𝜔0 )u )2 ) was needed to assure complex conjugates roots and the applicability of the fractional trigonometric functions. In the case where c1 2 ≥ 4c0 , the roots are real. In this case, the factorization process should yield the real roots, and the resulting inverse transforms takes the form of R-functions or fractional hyperbolic functions, that is, equations (12.14) or (12.15). The latter case, of course, results in unstable behavior. Stability behavior for the elementary transfer functions is discussed in Section 2.5 and in Sections 13.6–13.8.

12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots The Laplace transforms presented in the previous sections together with the R-function may now be used to solve any linear constant-coefficient fractional differential equation of commensurate order, that is, equations of the form nq 0 dt x(t)

3q

2q

q

+ · · · + c3 {0 dt x(t)} + c2 {0 dt x(t)} + c1 {0 dt x(t)} + c0 x(t) = f (t),

(12.42)

12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots

so long as there are no repeated roots in the characteristic equation. We restrict our attention to the forced response only, avoiding the initialization response to simplify the presentation. The Laplace transform of equation (12.36) without initialization terms, and with f (t) = 𝛿(t), is given by L{𝛿(t)} L{x(t)} = nq . (12.43) s + · · · + c3 s3q + c2 s2q + c1 sq + c0 The general procedure for the solution for equations of this type is first to factor the denominator polynomial in sq . Following factorization, the method of partial fractions is used to reduce the problem to multiple solutions of elementary fractional transfer functions of the first and second kinds. These simplified transforms are those presented in the earlier sections. Finally, the simplified inverse transforms are summed to assemble the solution to the higher-order original problem. The process is best illustrated by example. Here, we wish to determine the forced response for 3q 0 dt x(t)

2q

q

− 4{0 dt x(t)} + {0 dt x(t)} + 26{x(t)} = 𝛿(t).

(12.44)

Taking the Laplace transform, we have 1 . s3q − 4s2q + sq + 26 Factoring the denominator the roots are found to be −2, 3 ± 2i. Application of the method of partial fractions gives A Bsq + C L{x(t)} = q + 2q . s + 2 s − 6sq + 13 The solution for the coefficients A, B, and C yields L{x(t)} =

L{x(t)} =

(−sq + 8)∕29 1∕29 + 2q q s + 2 s − 6sq + 13

or

1∕29 sq ∕29 8∕29 − 2q + 2q . q q s + 2 s − 6s + 13 s − 6sq + 13 Using equation (12.11) to inverse transform the first term, we have } { 1∕29 1 L−1 = R (−2, t). sq + 2 29 q,0 L{x(t)} =

(12.45)

(12.46)

Equation (12.29) with c0 = 13, c1 = −6, and v = 0, is used to inverse transform the second term. Then, we have ( √ ) 2 arccos −c1 ∕2 c0 (−1 − v) 𝛼∗ = , 𝜋(q − 1 − v) ) ( √ 2 arccos 6∕2 13 (−1) −0.3743 = , (12.47) 𝛼∗ = 𝜋(q − 1) q−1 ) ( √ 2 arccos −c1 ∕2 c0 0.3743 = , 𝛽∗ = 𝜋(q − 1) q−1 √ 1 − c21 ∕4c0 = 0.5547. For the second term, we have } ) { (√ sq ∕29 −0.3743 0.3743 −1 −1 13, = Sin , ,t . − 2q L s − 6sq + 13 0.5547•29 q,0 q−1 q−1

(12.48)

251

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

Using equation (12.24) to inverse transform the third term of equation (12.45), with c0 = 13, c1 = −6, and v = 0, we have √ 2 arccos(6∕2 13) ∗ 𝛼 = = 0.3743 𝜋 and

√ 1 4c0 − c21 = 2 } { 8∕29 −1 L = s2q − 6sq + 13

1√ • 4 13 − 36 = 2. 2 (√ ) 8 13, 0.3743, 0, t . Sinq,0 2•29

(12.49)

Summing the results of solutions (12.46), (12.48) and (12.49) yields the forced response for t > 0 as ) (√ 1 −0.3743 0.3743 1 13, x(t) = Rq,0 (−2, t) − Sinq,0 , ,t 29 0.5547•29 q−1 q−1 √ 8 + (12.50) Sin ( 13, 0.3743, 0, t), t > 0. 2•29 q,0

12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots 12.6.1 Repeated Real Fractional Roots

Special transforms are required for the solution of fractional differential equations containing repeated real fractional-order roots. The G-function was developed in Section 3.9.1. It provides the Laplace transform needed for the solution of fractional differential equations containing repeated real fractional roots. While the G-function was derived by inverse transforming its Laplace transform, it may be defined in the time domain as { } ∞ ∑ (−r)(−1 − r) · · · (1 − j − r) (−a)j t (r+j)q−𝜈−1 Gq,𝜈,r [a, t] ≡ , t > 0, (12.51) j! Γ((r + j)q − 𝜈) j=0 or in terms of the Pochhammer polynomial [14] Gq,𝜈,r [a, t] =

∞ ∑ (r)j (a)j t (r+j)q−𝜈−1 j=0

| a| , Re(qr − v) > 0, Re(s) > 0, || q || < 1 , t > 0 , j! Γ((r + j)q − 𝜈) |s |

(12.52)

where 𝜈, q, and r are real but not constrained to be integers. The Laplace transform basis for the G-function is L{Gq,𝜈,r [a, t]} =

(sq

s𝜈 , − a)r

| a| Re(qr − v) > 0, Re(s) > 0, || q || > 0 , |s |

(12.53)

where again 𝜈, q, and r are not constrained to be integers. It is also clear that taking r = 1 specializes the G-function into the R-function. In a similar manner, relationships of increasing generality may be determined. Podlubny [109] presents a form that is a special case of the G-function where r is constrained to be an integer.

12.7 Fractional Differential Equations Containing Repeated Roots

12.6.2 Repeated Complex Fractional Roots

Fractional differential equations containing repeated complex roots may be analyzed using the H-function and its Laplace transform. The development of this function and its transform is given in Section 3.9.2. The H-function (equation (3.106)) is defined in the time domain as Hq,v,r (c0 , c1 , t) =

∞ ∞ k q(2r+n+k)−v−1 ∑ ∑ (−1)n (r)n (n − k + 1)k cn−k 1 c0 t

n! k! Γ(q(2r + n + k) − v)

n=0 k=0

, 2qr > v, n ≥ k,

(12.54)

where again (r)n is the Pochhammer polynomial. The Laplace transform for the H-function is given by L{Hq,v,r (c0 , c1 , t)} =

s𝜈 , (s2q + c1 sq + c0 )r

| | c1 | + c0 | < 1, | sq s2q | | |

| c0 | | | | c sq | < 1, 2qr > v, t > 0, | 1 |

(12.55)

where Hq,v,r (c0 , c1 , t) is given by equation (12.54). Now, with the fractional meta-trigonometric functions together with the R-, G-, and H-functions and their respective Laplace transforms, we have all the tools required to solve linear constant-coefficient fractional differential equations of any commensurate order. When r = 1, the H-function reduces to the form of the fractional trigonometric functions.

12.7 Fractional Differential Equations Containing Repeated Roots Our interest here is the solution of linear fractional differential equations of commensurate order, nq, and with constant coefficients c0 , c1 , c2 ,…, nq 0 Dt x(t)

3q

2q

q

+ · · · + c3 {0 Dt x(t)} + c2 {0 Dt x(t)} + c1 {0 Dt x(t)} + c0 x(t) = f (t)

(12.56)

and containing repeated roots. Previously, the fractional meta-trigonometric functions and the R-function were sufficient to solve such equations with unrepeated fractional roots. Here, the G- and H-functions are used to deal with repeated real and complex fractional roots, respectively. Thus, our objective here is to obtain the forced response solution of uninitialized fractional differential equations of the form of equation (12.56) containing repeated roots. To demonstrate the solution of such equations with repeated roots, we consider the forced response of the following uninitialized fractional differential equation 6q 0 dt x(t)

5q

4q

3q

+ 6{0 dt x(t)} + 19{0 dt x(t)} + 36{0 dt x(t)} 2q

q

+43{0 dt x(t)} + 30{0 dt x(t)} + 9{x(t)} = 𝛿(t).

(12.57)

The Laplace transform of equation (12.57) (with initialization terms = 0) is 1 . s6q + 6s5q + 19s4q + 36s3q + 43s2q + 30sq + 9 The denominator may be factored into the form L{x(t)} =

L{x(t)} =

1 . (sq + 1)2 (s2q + 2sq + 3)2

(12.58)

(12.59)

The denominator has two real roots at −1 and two pairs of complex roots at −1 ± Application of partial fractions gives L{x(t)} =

4(sq

1 1 1 − . − 2 2q q 2q + 1) 4(s + 2s + 3) 2(s + 2sq + 3)2

√

2 i.

(12.60)

253

254

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

The first term on the right-hand side of (12.60) results from the repeated real roots at −1; its inverse transform is determined from equation (12.53) as { } 1 1 −1 (12.61) L = Gq,0,2 [−1, t], Re(2q) > 0. 4(sq + 1)2 4 For the second term of equation (12.60) the Laplace transform (equation (12.24)) is applied. Then, with c0 = 3, c1 = 2, and v = 0, we have ( ( √ ) √ ) 2 arccos −c1 ∕2 c0 2 arccos −2∕2 3 = = 1.3918, 𝛼∗ = 𝜋 𝜋 } { (√ ) 1 1 3, 1.3918, 0, t , =− ( √ L−1 − 2q ) Sinq,0 q 4(s + 2s + 3) 4 12 4 • 3 − 22 } { (√ ) 1 1 −1 Sin L 3, 1.3918, 0, t . (12.62) = − − 2q √ q,0 4(s + 2sq + 3) 2 8 The third term of equation (12.60) is inverse transformed based on the H-function using the Laplace transform pair (12.55) } { 1 1 (12.63) = − Hq,0,2 (3, 2, t), 4q > 0, t > 0. L − 2q 2(s + 2sq + 3)2 2 Then, the forced response solution to equation (12.57) is sum of equations (12.61)–(12.63): (√ ) 1 1 1 3, 1.3918, 0, t − Hq,0,2 (3, 2, t), q > 0, t > 0. (12.64) x(t) = Gq,0,2 [−1, t] − √ Sinq,0 4 2 2 8

12.8 Fractional Differential Equations of Non-Commensurate Order Occasionally, a system may be found that cannot be (exactly) expressed as a commensurateorder fractional differential equation. The system may be solvable using the fractional exponential and/or fractional trigonometric functions if the non-commensurate terms can be arranged into separate additive terms. Consider the forced response of a system defined by the transfer function 𝜃o (s) 1 1 = 2q + , (12.65) 𝜃i (s) s + sq + 2 sw + 2 where, for example, w = 𝜋∕4 and q is non-commensurate with w. In unfactored form, the transfer function has the expanded form 𝜃o (s) sw + 2 + s2q + sq + 2 = 2q+w . 𝜃i (s) s + sq+w + 2sw + 2s2q + 2sq + 4

(12.66)

Furthermore, the related fractional differential equation is 2q+w q+w 𝜃o (t)+0 dt 𝜃o (t) 0 dt

2q

q

+ 20 dtw 𝜃o (t) + 20 dt 𝜃o (t) + 20 dt 𝜃o (t) + 4𝜃o (t) 2q

q

= 0 dtw 𝜃i (t)+0 dt 𝜃i (t)+0 dt 𝜃i (t) + 4𝜃i (t), t > 0.

(12.67)

12.9 Indexed Fractional Differential Equations: Multiple Solutions

For 𝜃i (t) a unit impulse, the forced response solution may be directly written from equation (12.65) using equation (12.21) with w = 0, and related constants as (√ ) 1 2, 𝛼 ∗ , 𝛽 ∗ , t + Rw,0 (−2, t), t > 0, (12.68) 𝜃o (t) = ∗ Cosq,0 K √ √ 1 , K ∗ = − 12 4c0 − c21 = − 12 7, and where 𝛽 ∗ = q−1 ( ( √ ) √ ) 2 2 arccos −c1 ∕2 c0 2 arccos −1∕2 q q q 𝛼∗ = − = − = 1.2300 − . 𝜋 q−1 𝜋 q−1 q−1

12.9 Indexed Fractional Differential Equations: Multiple Solutions In the development of the fractional trigonometric functions, we have been careful to achieve the most general mathematical forms for the trigonometric functions. Specifically, when series terms such as (i𝛽 t) have been raised to fractional powers we have included the nonprincipal function forms along with the principal function. These functions (see, e.g., equations (9.6) and (9.7)) have been expressed in indexed form, based on the index k. To maintain the generality discussed earlier, the presence of the k index in the fractional trigonometric functions has also been extended to the determination of the related Laplace transforms (see, e.g., Section 9.4 and Table 9.2). This indexing has also been extended to the fractional calculus-based differintegrals of the fractional trigonometric functions in Section 9.6. Thus, it is clear that differential equations containing fractional-order operators can (mathematically) yield multiple solutions. Consider the following indexed fractional-order differential equation: ( ( } ){ 2 𝜋 4∕3 2∕3 D x(t) − 2 cos D x(t) + x(t) = + 2𝜋k 0 t 0 t 3) {2 ( } ( 1 𝜋 2∕3 (12.69) + 2𝜋k cos − 0 Dt u(t) , t > 0, k = 0, 1, 2, 3 2 where our interest is the forced response, and thus the equation is not initialized. This gives rise to the following three nonindexed fractional-order differential equations: 4∕3 0 Dt x(t) 4∕3 0 Dt x(t) 4∕3 0 Dt x(t)

2∕3

2∕3

− 0 Dt x(t) + x(t) = 0.86600 Dt u(t), t > 0, k = 0, + −

2∕3 0 Dt x(t) 2∕3 0 Dt x(t)

+ x(t) = + x(t) =

2∕3 0.50000 Dt u(t), t > 0, k = 1, 2∕3 −0.86600 Dt u(t), t > 0, k = 2.

(12.70) (12.71) (12.72)

With u(t) = 𝛿(t), a unit impulse function, the solution to equation (12.69) and by inference (12.70)–(12.72) is given by x(t) = Cos2∕3,0 (1, 0, 1, k, t),

t > 0, k = 0, 1, 2.

(12.73)

Clearly, equation (12.69) has associated with its multiple solutions represented by equation (12.73) and by the individual equations (12.70)–(12.72). ( ) The three solutions are shown graphically in Figure 12.1, where 𝜎 = 23 𝜋2 + 2𝜋k and 𝜆 = ( ) − 13 𝜋2 + 2𝜋k . The responses shown in Figure 12.1 are for equations (12.70)–(12.72), also for all three values of k from equation (12.73). We can see that the non-zero responses occur in a symmetric pair: k = 0, k = 1. Clearly, all provide legitimate solutions for equation (12.69). The issue of multiple solutions for fractional-order differential equations has been studied by other investigators, for example, Ibrahim and Momani [55].

255

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

2 k=0 1.5 1 Cos2/3,0 (1,0,1,k,t)

256

0.5 k=2

0 –0.5 –1 –1.5 –2

k=1 0

2

4

6

8

10

t-Time

Figure 12.1 Multiple solutions, x(t),)for equation given by equations (12.70, k = 0), (12.71, k = 1), and ( ( (12.69), )

(12.72, k = 2). a = 1, 𝜎 =

2 3

𝜋 2

+ 2𝜋k , 𝜆 = − −1 3

𝜋 2

+ 2𝜋k , and k = 0, 1, 2.

As a practical matter, when the possibilities of multiple solutions arises in nearly all applications of physical modeling of nature, the combination of boundary and initial conditions provides the path to unique solutions. An interesting question here is: are there cases in nature where more than a single solution is required to properly describe the physics? One such possibility might be the entanglement of photons emitted as a result of a UV pulse passing through a nonlinear crystal [106]. Such experiments yield responses in which the behavior of the individual elements are symmetric and appear to be “entangled” but may possibly be explained as a multiple (or related) responses sourced by a fractional-order defined field. Multiple paths through nonlinear crystals [31] may also be such a possibility.

12.10 Discussion This chapter has presented a method for the solution of linear constant-coefficient fractional differential equations of commensurate order. The possible cases of unrepeated, repeated real, and repeated complex poles in the denominator of the characteristic equation were studied and example problems were solved. The methodology is based on the Laplace transforms of the fractional meta-trigonometric functions together with the R-, G-, and H-functions. The G-function provides transforms applicable to fractional differential equations with real fractional repeated poles, while the H-function applies to FDEs with complex fractional repeated poles. The method parallels the method used for the solution of linear ordinary differential equations. Current methods with equations of this type require infinite summations of Mittag-Leffler functions [109], a considerably more difficult approach. A comparison of the two approaches may be found in Ref. [88].

12.10 Discussion

The fractional trigonometric approach provides solutions in the form of real functions as opposed to fractional exponential functions with complex arguments. This allows a direct connection to the physical world. The fractional meta-trigonometric forms provide important insights and connections to the natural world as is discussed in the application chapters. Finally, because of the indexed nature of the fractional trigonometric functions, the interesting possibility of multiple solutions to physical problems may be investigated.

257

259

13 Fractional Trigonometric Systems This chapter considers the R-function and the fractional trigonometric functions as system impulse responses. That is, the Laplace transforms of the R-function and the trigonometric functions are treated as system transfer functions. The properties of these transfer functions reflect intrinsic characteristics of the underlying trigonometric functions. Specifically, the stability of the fractional trigonometric-based transfer functions indicates the growth/decay properties of the function. Chapters 6–12 have basically been studies of the R-function, Rq,v (a, t), with complex parameters for a and t. Here, we study a form of the R-function where a is selected to generalize the time constant and natural frequency of integer-order linear dynamic systems.

13.1 The R-Function as a Linear System The two building blocks of linear integer-order dynamics are commonly known as first- and second-order systems. A first-order system is defined by the ordinary differential equation: d𝜃o (13.1) + 𝜃o = 𝜃i , dt where 𝜃i (t) is the system input, 𝜃o (t) is the system output, and 𝜏 is known as the system time constant. The Laplace transform of this system, with zero initialization, is 𝜏

Θo (s) 1 = . Θi (s) 𝜏s + 1

(13.2)

A second-order system is defined by the ordinary differential equation: 2 1 d 𝜃o 2𝛿 d𝜃o + + 𝜃o = 𝜃i , 2 dt 2 𝜔n dt 𝜔n

(13.3)

where 𝜔n is the system natural frequency and 𝛿 is known as the damping ratio. The second-order system is also known as the harmonic oscillator when 𝛿 = 0. The Laplace transform of a second-order system, with zero initialization, is given by 𝜔2n Θo (s) = = Θi (s) s2 + 2𝛿𝜔n s + 𝜔2n

1 s2 𝜔2n

+

2𝛿 s 𝜔n

+1

.

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(13.4)

260

13 Fractional Trigonometric Systems

Many properties of these foundation systems have been studied in controls and linear system theory (see, e.g., Kuo [60], Gabel and Roberts [36]) as well as physics textbooks. The fractional calculus allows the following generalization of these fundamental systems. Consider the fractional differential equation q 0 Dt 𝜃o (t)

q

+ Ωn 𝜃o (t) = 0 Dvt 𝜃i (t).

(13.5)

The system transfer function associated with this equation is Θo (s) sv = q. Θi (s) sq + Ωn

(13.6)

The related Laplace transform of the R-function may be written as { sv q } L Rq,v (−Ωn , t = q > v. q, sq + Ωn

(13.7)

Clearly, an undamped second-order system and a first-order system may be represented by equations (13.5)–(13.7). For the first-order system with q = 1 and v = 0, we have Ωn = 1∕𝜏. For the undamped second-order system with q = 2 and v = 0, we have Ωn = 𝜔n as the natural frequency. Thus, Ωn is seen as a fundamental property of linear dynamic systems, it has been named the natural quency; see Appendix E. The following sections present typical time and frequency responses associated with these generalized systems.

13.2 R-System Time Responses Two input time responses are evaluated when the R-function Laplace transform is considered as a physical system transfer function. These are the unit impulse and the unit step function. When the input is a unit impulse function 𝜃i (t) ⇔ Θi (s) = 1; thus, 1 q Θo (s) = t > 0. (13.8) q ⇔ 𝜃o (t) = Rq,0 (−Ωn , t), q s + Ωn For a unit step function input, 𝜃i (t) ⇔ Θi (s) = 1∕s and 1 q Θo (s) = q ⇔ 𝜃o (t) = Rq,−1 (−Ωn , t), s(sq + Ωn )

t > 0.

(13.9)

Figure 13.1 shows the system impulse response, 𝜃o (t), for various values of q with Ωn = 1. The step response over the same range is shown in Figure 13.2.

13.3 R-Function-Based Frequency Responses The frequency responses associated with the R-function, that is, equations (13.8) and (13.9), are determined by the substitution of s = i𝜔 in equation (13.6) Θo (i𝜔) (i𝜔)v = q Θi (i𝜔) (i𝜔)q + Ωn

(13.10)

and then computing the magnitude ratio and phase angle. A low-pass response is shown in Figure 13.3 with Ωn = 1, 𝜈 = 0, and the symmetrical band-pass case is shown in Figure 13.4 with Ωn = 1, 𝜈 = q∕2. In Figure 13.3a, the asymptotic slope of the high-frequency response is seen to be 20q db/decade, while in Figure 13.3b, the overall phase shift is 90q degrees or q𝜋∕2 rad. The stability properties of the F-function and the Rq,0 -function interpreted as system impulse responses have been studied in Chapter 2.

13.4 Meta-Trigonometric Function-Based Frequency Responses

θo(t) Fractional system impulse response

1.5

1 q = 2.00 0.5

0

q = 0.25

–0.5

–1 0

2

4 t

6

8

Figure 13.1 System impulse response, 𝜃o (t), equation (13.8), with q = 0.25–2.0 in steps of 0.25, Ωn = 1.

θo(t) Fractional system step response

2.5

2 q = 2.00 1.5

1 q = 0.25

0.5

0 –0.5

0

2

4 t

6

8

Figure 13.2 System step response, 𝜃o (t), equation (13.9), with q = 0.25–2.0 in steps of 0.25, Ωn = 1.

13.4 Meta-Trigonometric Function-Based Frequency Responses In the followings sections, the stability (growth and decay) of the fractional meta-trigonometric functions are studied based on the fractional transfer functions of the second kind and the Laplace transforms of the fractional meta-trigonometric functions. Linear fractional differential equations of the form D𝛼 x(t) + 𝜆x(t) = u(t), 1 < 𝛼 ≤ 2, t > 0,

𝜆∈ℜ

(13.11)

261

13 Fractional Trigonometric Systems

Magnitude ratio Θo(ω) / Θi(ω), (db)

20 10 0 q = 0.25

–10 –20 q = 2.00

–30 –40 –50 –4 10

10–2

100 f - Frequency (Hz) (a)

0

102

104

q = 0.25

–20 Phase angle Θo(ω) / Θi(ω), (°)

262

–40 –60 –80 –100 –120 –140 –160 –180 10–4

q = 2.00 10

–2

100 f - Frequency (Hz) (b)

102

104

Θo (i𝜔) 1 with = Θi (i𝜔) (i𝜔)q + Ωqn q = 0.25–2.0 in steps of 0.25, Ωn = 1. (a) Magnitude ratio in decibel. (b) Phase angle in degrees.

Figure 13.3 System frequency response for equation (13.10) with v = 0, that is

can be taken as the basis of fractional oscillation and have been referred to as fractional oscillators. The solutions of this equation may be expressed in terms of R-functions with real arguments. We consider the more general form D2𝛼 x(t) + c1 D𝛼 x(t) + c2 x(t) = u(t), 0 < 𝛼 ≤ 2,

t > 0, c1 ∈ ℜ and c2 ∈ ℜ,

(13.12)

which may also have oscillatory solutions. These solutions are based on R-functions with complex arguments, and can also be the basis of fractional oscillation. The fractional trigonometries, defined in previous chapters, define a wide variety of oscillatory fractional functions that, as we have seen in Chapter 12, are the solutions or components of solutions to such linear fractional differential equations.

13.4 Meta-Trigonometric Function-Based Frequency Responses

Magnitude ratio Θo(ω) / Θi(ω), (db)

20 10 0 q = 0.25

–10 –20 –30

q = 2.00

–40 –50 10–4

10–2

100 f - Frequency (Hz) (a)

102

104

100

Phase angle Θo(ω) / Θi (ω), (°)

q = 2.00 50

0

q = 0.25

–50

–100 10–4

10–2

100 f - Frequency (Hz) (b)

Figure 13.4 System frequency response for

102

104

Θo (i𝜔) (i𝜔)q∕2 , with q = 0.25–2.0 in steps of 0.25, Ωn = 1. = Θi (i𝜔) (i𝜔)q + Ωqn

The character of the frequency responses associated with the fractional meta-sine and meta-cosine functions is best considered based on the behavior of their denominators. Thus, we consider simplified transforms associated with equations (12.21) and (12.24), namely for the fractional meta-cosine ⎧ ⎫ [ ] (√ )⎪ ⎪ −2 su ∗ ∗ L ⎨√ Cosq,u c0 , 𝛼 , 𝛽 , t ⎬ = 2q , s + c1 sq + c0 ⎪ 4c0 − c21 ⎪ ⎩ ⎭

q > u, 4c0 > c1 2 .

263

264

13 Fractional Trigonometric Systems

The frequency responses based on the transfer function

or in terms of i𝜔

Θo (s) 1 , = 2q Θi (s) s + c1 sq + c0

(13.13)

Θo (i𝜔) 1 , = Θi (i𝜔) (i𝜔)2q + c1 (i𝜔)q + c0

(13.14)

best exposes the frequency domain behavior. Figure 13.5 shows the frequency response for this function with c0 = 1 and c1 = 3.0. Immediately obvious in this plot is the presence of a second resonance at q = 2.0 and the inflection in the magnitude when q = 1.75. Furthermore, because of the order 2q term, the high-frequency amplitude asymptotes fall off at twice the rate as those seen in Figure 13.3a. Also note that the high-frequency asymptotic phase shifts are double those of Figure 13.3b. This chapter explores the growth and decay of the fractional meta-trigonometric functions based on the stability of their inferred transfer functions. It is noted that Green’s functions for equations such as equations (13.11) and (13.12) have also been determined by Podlubny [109].

13.5 Fractional Meta-Trigonometry The fractional meta-trigonometries are an infinite set of fractional trigonometries and hyperboletries that are based on the R-function, with fully complex arguments, namely √ Rq,v (ai𝛼 , i𝛽 t), t > 0, i = −1, q, v, a, 𝛼, 𝛽 ∈ ℜ (13.15) and are based on the definition Rq,v (a, t) ≡

∞ ∑ an t (n+1)q−1−v , Γ((n + 1)q − v) n=0

t > 0.

(13.16)

The meta-trigonometric functions based on complexity and parity are defined for t > 0 in Table 9.2. For convenience the Laplace transforms for these functions are repeated here: [ q ] v s sin(𝜆) − a sin(𝜆 − 𝜎) L{Sinq,v (a, 𝛼, 𝛽, k, t)} = s , q > v, (13.17) s2q − 2 a cos(𝜎)sq + a2 [ q ] s cos(𝜆) − a cos(𝜆 − 𝜎) , q > v, (13.18) L{Cosq,v (a, 𝛼, 𝛽, k, t)} = sv s2q − 2a cos(𝜎)sq + a2 [ 2q ] s sin(𝜆) − a2 sin(𝜆 − 2𝜎) L{Vibq,v (a, 𝛼, 𝛽, k, t)} = sq+v 4q , q > v, (13.19) s − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s cos(𝜆) − a2 cos(𝜆 − 2𝜎) , q > v, (13.20) L{Covibq,v (a, 𝛼, 𝛽, k, t)} = sq+v s4q − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s sin(𝜆 + 𝜎) − a2 sin(𝜆 − 𝜎) L{Flutq,v (a, 𝛼, 𝛽, k, t)} = a sv , 2q > v, (13.21) s4q − 2a2 cos(2𝜎)s2q + a4 [ 2q ] s cos(𝜆 + 𝜎) − a2 cos(𝜆 − 𝜎) L{Cof lq,v (a, 𝛼, 𝛽, k, t)} = a sv , 2q > v, (13.22) s4q − 2a2 cos(2𝜎)s2q + a4 ) ( ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , 𝜆 = 𝛽[q − 1 − v] 𝜋2 + 2𝜋k . In the following sections, these functions are considered as system transfer functions and their stability analyzed.

13.5 Fractional Meta-Trigonometry

20 q = 2.00

Magnitude ratio Θo(ω) / Θi(ω), (db)

10

0 –10 –20

q = 0.25

–30

q = 2.00

–40 –50 10–4

10–2

100 f - Frequency (Hz)

102

104

(a) 0 q = 0.25

Phase angle Θo(ω) / Θi(ω), (°)

–50 –100 –150 –200 –250 –300 –350 –400 10–4

q = 2.00 10–2

100

102

104

f - Frequency (Hz) (b)

Θo (i𝜔) 1 , with q = 0.25–2.0 in steps of 0.25. = Θi (i𝜔) (i𝜔)2q + 3(i𝜔)q + Ωqn (a) Magnitude ratio in decibel. (b) Phase angle in degrees. Figure 13.5 System frequency response for

265

266

13 Fractional Trigonometric Systems

13.6 Elementary Fractional Transfer Functions Elementary transfer functions of the first kind, studied in Chapter 2 and by Hartley and Lorenzo [45, 48], and applied in Chapter 12 to the solution of fractional differential equations, are given by ̃ ̃ = K , F(s) (13.23) su + b̃ with real-valued coefficients K̃ ∈ ℜ, b̃ ∈ ℜ, and u ∈ ℜ+∗ . Elementary transfer functions of the second kind studied by Malti et al. [90, 91] and Lorenzo et al. [84] are given by F(s) =

K(𝜔0 )2u , s2u + 2𝜁 (𝜔0 )u su + (𝜔0 )2u

(13.24)

with real-valued coefficients K ∈ ℜ, 𝜁 ∈ ℜ, and 𝜔0 ∈ ℜ+∗ . The parameter 𝜁 does not have the same meaning as in integer-order systems: it is the classical damping factor only if u = 1. It is referred to as a pseudo-damping factor (Malti et al. [91]). As we have seen in Chapter 12, elementary transfer functions of the first and second types are required to reduce higher-order fractional differential equations to analytically manageable elements. Linear fractional system transfer functions may be represented by B ∑

T(su ) H(s) = = R(su )

k=0

1+

bk su k

A ∑ i=0

, ai

(13.25)

su i

with real-valued coefficients (ai , bk ) ∈ ℜ2 . Stability conditions for such fractional transfer functions have been studied by Matignon and independently by Hartley and Lorenzo [45] and in Chapter 2. The following section studies the stability of such transfer functions.

13.7 Stability Theorem The following theorem originally proved by Matignon [92], and revisited by Malti [91], applies to (13.25). From Malti et al. [91], with permission of Automatica: “A commensurable transfer function with a commensurable order u, as in (13.25) with T and R two coprime polynomials, is stable if and only if 𝜋 0 < u < 2 and ∀p ∈ C where R( p) = 0, ||arg(p)|| > u , 2 u where p = s .◽” Applying the stability theorem to the elementary transfer function of the first kind (equation (13.23)) with real-valued coefficients yields the following stability conditions: b̃ > 0

and 0 < u < 2.

(13.26)

Moreover, the commensurate-order transfer function of the first kind is neutrally stable if b̃ > 0

and u = 2.

(13.27)

13.8 Stability of Elementary Fractional Transfer Functions

Applying the stability theorem on the elementary transfer function of the second kind (equation (13.24)) requires computing both su poles: ( ) √ (13.28) su1,2 = 𝜔u0 −𝜁 ± 𝜁 2 − 1 , and finding all 𝜁 satisfying the following inequalities: 𝜋 | | u < |arg(su1,2 )| < 𝜋. | 2 |

(13.29)

13.8 Stability of Elementary Fractional Transfer Functions The stability conditions for elementary transfer functions of the second kind in terms of the pseudo-damping factor 𝜁 and the commensurable order u, and are summarized in the following corollary to the Stability theorem. Continuing from Malti et al. [91]: Elementary transfer functions of the first kind 1 F1 (s) = ( )v s 𝜔n

+1

with 𝜔n > 0 are stable iff 0 < v < 2. The transfer function of the second kind (13.6.2) is stable iff: ( ) 𝜋 < 𝜁 and 0 < u < 2. ◽ − cos u 2

(13.30)

Refer to Ref. [90] for the proof of this corollary. It is well known from integer-order system theory that the second-order rational transfer function, equation (13.24) with u = 1, is stable if and only if 𝜁 > 0. (13.31) From the preceding development (see also Ref. [84]), it can be seen that the commensurateorder transfer function of the second kind equation (13.24) is neutrally stable for 𝜔0 > 0, if ( ) u𝜋 . (13.32) ∀u ∈ (0, 2], there exists a 𝜁 such that 𝜁 = − cos 2 The work of Malti et al. [91] is summarized graphically in Figure 13.6 in which the behavior of fractional elementary transfer functions of the second kind (equation (13.24)) are determined for various values of u, 𝜁 . Our particular interest here is the stability of the transfer function; however, the figure also contains information pertaining to the system resonances. Most importantly, the transfer function equation (13.24) is stable for all values of u, 𝜁 above area 4, that is, those values that are greater than the negative cosine function and located in areas 1, 2, and 3, and with 0 < u < 2. The responses for values of u, 𝜁 in the four areas are as follows: Area 1, Area 2, Area 3, Area 4,

– – – –

stable and non-resonant, stable with a single resonance, stable with two resonances, unstable.

The stability results are specialized for application to the Laplace transforms of fractional meta-trigonometric functions in the following section.

267

13 Fractional Trigonometric Systems

3 No resonant frequency A single resonant frequency Two resonant frequency Unstable system

2.5

1.5

3

2

Stability limit

2 Pseudo-damping factor ζ

268

1

1 0.5 0 4

–0.5 –1

0.2

0.4

0.6

0.8 1.0 1.2 1.4 Commensurable order u

1.6

1.8

Figure 13.6 Stability and resonance regions of the elementary transfer function of the second kind in the 𝜁 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. Source: Malti, Moreau, and Khemane 2011 [91]. Reproduced with permission of Elsevier. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

13.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions The general theoretical results on the stability of fractional transfer functions of the second kind developed in the previous sections can provide insight into the behavior of the fractional meta-trigonometric functions. In this section, these results are applied to the meta-trigonometric Laplace transforms. The complexity functions are considered first. 13.9.1 Complexity Function Stability

The stability of the complexity-based meta-trigonometric functions is determined from their respective Laplace transforms equations (13.17) and (13.18). When expressed in minimum form, only the denominator terms influence the stability for these transforms. The form [ ] 𝜃o (s) K = 2q , (13.33) 𝜃i (s) s − 2 a cos(𝜎)sq + a2 ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k then applies to both transforms. The substitution of a2 = 𝜔0 2q yields the mathematical form [ ] 𝜃o (s) K = 2q , (13.34) 𝜃i (s) s − 2 (±𝜔0 q ) cos(𝜎)sq + 𝜔0 2q which may be compared with equation (13.24). Relating equation (13.34) to equation (13.24), the pseudo-damping factor, 𝜁 , for the complexity functions is found to be 𝜁 = −sgn(a) cos(𝜎) . (13.35)

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

It is seen from Section 13.8 (equation (13.30)) that the complexity function will be growing (unstable) when 𝜁 < − cos(q𝜋∕2). From equation (13.35), for the complexity functions, we have that 𝜁 = −sgn(a) cos(𝜎). Therefore, the Laplace transform of the fractional complexity meta-trigonometric functions represented by equation (13.33) will be decaying (stable) when − cos(q𝜋∕2) < 𝜁 = −sgn(a) cos(𝜎) or when cos(q𝜋∕2) − sgn(a) cos(𝜎) > 0 and 0 < q < 2, giving the general result that ⎧> 0 function is stable, decaying ⎪ cos(q𝜋∕2) − sgn(a) cos(𝜎) ⎨= 0 function neutrally stable , (13.36) ⎪< 0 function is unstable, growing ⎩ ) ( where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k , and, 0 < q ≤ 2. Insight into the resonance behavior of the functions in the stable domain is given in Section 13.8 and in considerable detail in Ref. [90]. 13.9.2 Parity Function Stability

The Laplace transforms of the parity functions (equations (13.19)–(13.22)) may also be considered as system transfer functions and analyzed for their stability. As in the complexity function case, the numerator terms of these functions do not influence the stability. Thus, for these transforms, we consider the form [ ] 𝜃o (s) K = 4q , (13.37) 𝜃i (s) s − 2a2 cos(2𝜎)s2q + a4 to apply to all four fractional trigonometric functions (equations (13.19)–(13.22)). For this ) ( 𝜋 equation, we also have 𝜎 = (𝛼 + 𝛽q) 2 + 2𝜋k . For a real, substituting a4 = 𝜔0 4q , and letting q = w∕2 gives [ ] 𝜃o (s) K = 2w , (13.38) 𝜃i (s) s − 2 𝜔0 w cos(2𝜎w )sw + 𝜔0 2w ( ) with 𝜎w = (𝛼 + 𝛽w∕2) 𝜋2 + 2𝜋k . This transform now may be directly compared with equation (13.24). Comparing equations (13.38) with (13.24), the pseudo-damping factor, 𝜁 , for the parity functions is given by 𝜁 = − cos(2𝜎w ) ,

0 < w ≤ 2,

(13.39)

𝜁 = − cos(2𝜎) ,

0 < q ≤ 1.

(13.40)

or in terms of q Thus, the pseudo-damping factor for the parity functions is 𝜁 = − cos(2𝜎). Then, the Laplace transform of the fractional parity meta-trigonometric functions represented by equation (13.38) will be stable when 𝜁 = − cos(2𝜎) > − cos(q𝜋) and 0 < q < 1, or when cos(q𝜋) − cos(2𝜎) > 0 and 0 < q < 1. This gives the general result for the parity functions: ⎧> 0 function is stable, (decaying) ⎪ cos(q𝜋) − cos(2𝜎) ⎨= 0 function is neutrally stable , (13.41) ⎪< 0 function is unstable, (growing) ⎩ ) ( 𝜋 where 𝜎 = (𝛼 + 𝛽q) 2 + 2𝜋k , and, 0 < q ≤ 1. This behavior is validated in Figure 13.7, where 𝛽 + 𝜉 = 0.95, 1.0, and 1.05, for which cos(q𝜋) − cos(2𝜎) = −0.0446, 0.0, and 0.0460, respectively. In turn, this yields unstable, neutrally stable, and stable responses for the Flutter function, respectively. These stability results apply to all of the parity functions (equations (13.19)–(13.22)).

269

13 Fractional Trigonometric Systems

3

ξ = –0.05

2

Flutq,v (a,α,β,k,t)

270

1 ξ = +0.05

0 –1

–2

–3

0

5

10

15

t-Time

Figure 13.7 Neutrally stable, underdamped stable, and oscillatory unstable transfer functions of the second kind. Flutq,0 (1, 0, 𝛽 + 𝜉, k, t) with q = 1/3, v = 0, a = 1.0, 𝜉 = ±0.05, 0.0, 𝛼 = 0, and 𝛽 = 1.0.

13.10 Discussion In this chapter, we have considered the Laplace transforms of the fractional trigonometric functions as impulses responses of transfer functions describing physical systems and have applied linear system theory to help further characterize these functions. This process defines the combinations of variables for which the fractional meta-trigonometric functions will exhibit sustained oscillations (neutrally stable), growing responses (unstable), and decaying response (stable). In Chapter 16, these results will be applied to the study of fractional oscillators.

271

14 Numerical Issues and Approximations in the Fractional Trigonometry The practical application of the fractional trigonometry requires the ability to evaluate the various fractional trigonometric functions as well as the R-function over a wide range of the order variables, q and v, and over a wide range of the arguments, a and t. For the fractional trigonometric functions, the range must further include the variations of the 𝛼 and 𝛽 parameters. Because all of the functions are defined by infinite series and are basically expanded around zero, the function evaluations fail for large values of a, t, and q, or more specifically, for large values of at q . As a preliminary, we establish the convergence of the series definitions for the R-functions and fractional trigonometric functions. Following this, some numerical issues and approximations are considered.

14.1 R-Function Convergence The R-function is defined by the infinite series ∞ ∑ an t (n+1)q−1−v Rq,v (a, t) ≡ , Γ((n + 1)q − v) n=0

0 < t < ∞.

(14.1)

The problem considered here is to show convergence for the R-function. The ratio test is applied to the R-function written in the following form: Rq,v (a, t) = t q−1−v

∞ ∑ n=0

an t nq , Γ((n + 1)q − v)

0 < t < ∞,

(14.2)

with a > 0 and q > 0. Thus, we consider | an+1 t(n+1)q | | | | un+1 | | | = lim || Γ((n+2)q−v) lim || | , 0 < t < ∞, n nq a t | n→∞ | u || n→∞ | n | Γ((n+1)q−v) | | | | un+1 | | at q Γ((n + 1)q − v) | | = lim | | , 0 < t < ∞. lim | n→∞ || u || n→∞ || Γ((n + 2)q − v) || n From previous considerations, we know that the effect of v is to differentiate Rq,0 , v > 0, or to integrate Rq,0 , v < 0. Therefore, we may take v = 0 without loss of generality, giving | | un+1 | | | = lim | at q Γ((n + 1)q) | , 0 < t < ∞. (14.3) lim || | | n→∞ | u | n→∞ | Γ((n + 2)q) || n Now, it is well known that for x > 2, Γ(x) increases monotonically as x → ∞. Thus, for q > 0, and finite it may be seen that, after a finite number of terms n = Nfinite , as n → ∞ the ratio The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

272

14 Numerical Issues and Approximations in the Fractional Trigonometry

of gamma functions → 0, albeit slowly for small values of q. Therefore, we conclude that the summation converges absolutely over 0 < t < ∞. When a < 0 equation (14.2) becomes an alternating series. From Apostol [7], p. 358, we have ∑∞ the following definition and theorem. If bn > 0 for each n, the series n=1 (−1)n+1 bn is called an alternating series. Theorem 14.1 If {bn } is a decreasing sequence converging to 0, the alternating series ∑∞ n+1 bn converges. n=1 (−1) Then, writing for a > 0, and finite, (−a)n ⇒ (−1)n an and equation (14.1) becomes Rq,v (a, t) = t q−1−v

∞ ∑ n=0

(−1n )(at q )n , Γ((n + 1)q − v)

0 < t < ∞.

Again, we may take v = 0 without loss of generality, giving for the related sequence { } (at q )n . {bn } = Γ((n + 1)q) Again, after a finite number of terms, the gamma function in the denominator grows more quickly than the numerator; thus, } { (at q )n = 0. (14.4) lim {bn } = lim n→∞ n→∞ Γ((n + 1)q) Therefore, the R-function converges for both positive and negative values of a. The bases for the fractional meta-trigonometric functions is Rq,v (ai𝛼 , i𝛽 t) ,

0 < t < ∞.

(14.5)

We may consider the convergence of this function by letting a ⇒ ai𝛼 and t ⇒ i𝛽 t in equation (14.3); this yields | un+1 | | | | = lim | ai𝛼+𝛽q t q Γ((n + 1)q) | , lim || | | n→∞ | u | n→∞ | Γ((n + 2)q) || n

0 < t < ∞.

(14.6)

Clearly, the effect of i𝛼+𝛽q is only a rotation in the complex plane and the magnitude is not changed. Therefore, as with equation (14.3), we conclude that the summation converges absolutely. Similar arguments may be made relative to equation (14.4). Thus, Rq,v (ai𝛼 , i𝛽 t) is convergent for all choices of 𝛼 and 𝛽.

14.2 The Meta-Trigonometric Function Convergence The convergence of the meta-trigonometric functions may be related to that of the R-function. Consider Cosq,v (a, 𝛼, 𝛽, t), from Section 9.5, we express Cosq,v (a, 𝛼, 𝛽, t), in terms of the R-function as Rq,v (ai𝛼 , i𝛽 t) + Rq,v (ai−𝛼 , i−𝛽 t) Cosq,v (a, 𝛼, 𝛽, t) = , t > 0. (9.82) 2 Because we have shown the convergence of the R-functions, it is clear that the Cosq,v (a, 𝛼, 𝛽, t) as the sum of two convergent functions is also convergent. Similar arguments may be made for the remaining principal fractional trigonometric functions that also are expressible in terms of R-functions (see Section 9.5).

14.3 Uniform Convergence

14.3 Uniform Convergence The topic of uniform convergence is important to our studies as it is required for operations with infinite series such as term-by-term integration. A definition for this property is given as follows. An infinite series of real functions S(x) = lim

N ∑

N→∞

fn (x)

n=0

convergent on the interval P is uniformly convergent to S(x) if, with Sm (x) = N = N(𝜀) can be determined for each value of 𝜀, such that

m ∑ n=1

fn (x), a number

|S(x) − Sm (x)| < 𝜀, for all n > N and for all x ∈ P. For our considerations for the R-function series (equation (14.1)), we extend the development of Churchill [22], pp. 139–140 that proves the uniformity of convergence for power series to the fractional case. Let us write for our series of interest S(t) = lim

N→∞

N ∑

cn t

(n+1)q−1−v

=

n=0

∞ ∑

cn t (n+1)q−1−v ,

0 < t < ∞,

(14.7)

n=0

where cn = an ∕Γ((n + 1)q − v) for the R-function. The remainder after N terms is N

(t ) = S (t ) −

N −1

∑c t n

( n +1) q −1− v

= lim

m →∞

n =0

m

∑c t n

( n +1) q −1− v

, 0 < t < ∞.

(14.8)

n= N

When t ≤ t1 and 0 < t1 < ∞, we can write m m m |∑ | ∑ ∑ | | cn t (n+1)q−1−v | ≤ |cn t (n+1)q−1−v | ≤ |cn ||t1 |(n+1)q−1−v . | | | n=N |n=N | n=N The limit of the last summation m ∑ 𝜌N = lim |cn ||t1 |(n+1)q−1−v m→∞

(14.9)

(14.10)

n=N

is the remainder of series (14.7) with absolute values of cn and t1 . As we have shown in Section 14.1, this equation is absolutely convergent for 0 < t1 < ∞; thus, the remainder (equation (14.10)) approaches zero as N → ∞. In other words, for any positive number, 𝜀, an integer N𝜀 exists such that 𝜌N < 𝜀 whenever N ≥ N𝜀 . The terms of the last series are all non-negative real numbers, then for every integer m > N m ∑

|cn ||t1 |(n+1)q−1−v ≤ 𝜌N .

n=N

Then, by equation (14.9)

m |∑ | | | cn t (n+1)q−1−v | ≤ 𝜌N , | | | |n=N | for every integer m greater than N and, therefore, the remainder (equation (14.8)) satisfies N (t )

≤ ρ N 0.

This may also be written as Rq,v (a, t) = t q−1−v

∞ ∑ n=0

(at q )n , Γ((n + 1)q − v)

t > 0.

Examining the summation, we see that, as the index n becomes large, both the numerator and the denominator of this ratio will eventually grow to exceed the maximum number size associated with typical computer software. This is clearly indicated in Figure 14.1 where the general term in the summation (14.4.2), namely Term =

(at q )n , Γ((n + 1)q − v)

(14.12)

is evaluated for three values of at q . When q = 1 and v = 0, this becomes Term = (at q )n ∕n!. As at q changes from 10 to 40, the value on the largest term grows by approximately 12 orders of 3

× 105

20

× 109

atq = 10

2.5

2.5

atq = 20

15 2 Term value

274

× 1019

atq = 40

2 1.5

10

1.5

1 1

5 0.5

0.5 0

0

0 –0.5

0

50 100 n-index term

–0.5

0

50 100 n-index term

–0.5

0

50 100 n-index term

Figure 14.1 Size of the nth term versus n, with q = 1.0 and atq = 10, 20, 40.

14.5 The R2 Cos- and R2 Sin-Function Asymptotic Behavior

magnitude! Thus, we can see that the numerator and denominator terms separately grow very rapidly. Because the gamma function cannot be written as a polynomial, progress cannot be made by cancellation of common factors of the numerator and the denominator. The issues are the same for the fractional trigonometric functions because of the similar forms for the numerators and the denominators. The problem is further compounded for large values of a, t, and q.

14.5 The R2 Cos- and R2 Sin-Function Asymptotic Behavior The principal R2 Cos-function is defined as ∞ ∑ an t (n+1)q−1−v R2 Cosq,v (a, t) = cos{((n + 1)q − 1 − v)(𝜋∕2)}, Γ((n + 1)q − v) n=0

t > 0.

As we have observed in Chapter 7, the phase plane plots R2 Cosq,0 (1, t) versus R2 Sinq,0 (1, t) are attracted to (asymptotic to) circles of radius 1∕q when 0 < q < 2. Thus, for example, for the R2 Cosq,0 (1, t), we have lim {R2 Cosq,0 (1, t)} →

t→∞

1 cos(t) , q

0 < q < 2,

t > 0.

We now need to determine the radius when a ≠ 1. Thus, we write R2 Cosq,0 (a , t) = a q

−(q−1)

∞ ∑ (at)(n+1)q−1 cos{((n + 1)q − 1)(𝜋∕2)}, Γ((n + 1)q) n=0

t > 0,

and we have R2 Cosq,0 (aq , t) = a−(q−1) R2 Cosq,0 (1, at),

0 < q < 2,

t > 0.

(14.13)

0 < q < 2,

t > 0.

(14.14)

In a similar manner, for R2 Sin, we have R2 Sinq,0 (aq , t) = a−(q−1) R2 Sinq,0 (1, at),

Thus, if R2 Cosq,0 (1, t) versus R2 Sinq,0 (1, t) are attracted to circles of radius 1∕q, then R2 Cosq,0 (aq , t) and R2 Sinq,0 (aq , t) will be attracted to circles of radius ( ) 1 r= . (14.15) qaq−1 In other words, R2 Cosq,0 (aq , t) is asymptotic to (1∕qaq−1 ) cos(at). Setting b = aq , we have the desired results R2 Cosq,0 (b, t) is asymptotic to (1∕qb(q−1)∕q ) cos(b1∕q t),

0 < q < 2, t > 0,

(14.16)

R2 Sinq,0 (b, t) is asymptotic to (1∕qb(q−1)∕q ) sin(b1∕q t),

0 < q < 2, t > 0.

(14.17)

and

The quality of the asymptotic approximation is indicated in Figure 14.2. Note the function numerical failure near t = 4 for q = 0.4 and at t = 10 for q = 0.6; however, the asymptotic approximations, of course, successfully continue beyond these points. In Chapter 16, the space of neutrally stable fractional oscillators based on the fractional trigonometric functions is studied. These functions are also candidates to be continued asymptotically.

275

14 Numerical Issues and Approximations in the Fractional Trigonometry

8

R2Cosq,0(a,t) and cos approximation

276

q = 0.6

q = 0.4

6 4 2 0 –2 –4 –6 –8

0

5

10 t-Time

15

Figure 14.2 R2 Cosq,0 (a, t) and the approximation (1∕(qa(q−1)∕q ) cos(a1∕q t) versus t-Time. Solid line is R2 Cosq,0 (a, t) and dashed is approximation, q = 0.4 and 0.6, v = 0.0, a = 1.75.

14.6 R-Function Approximations Some approximations for the R-function have been developed in Ref. [70] and are presented in this section: As suggested in Section 4.2, the principal value in equation (4.26) provides the basis of such an approximation. The result (equation (4.27)) is generalized to Rq,0 (1, t) ≈

1 t e, q

t > q > 0.5 .

(14.18)

The approximation is shown in Figure 14.3. The following are improved approximations when t < 1 and 0 < q ≤ 1 ( 1.5 ) (1 − q)(1∕(q )) 1 t , t > 0, 0 < q ≤ 1 . Rq,0 (1, t) ≈ e 1 + q t 1−q ( ) (1 − q)2.5−q 1 , t > 0, 0 < q < 1 . Rq,0 (1, t) ≈ et q t 1−q

(14.19) (14.20)

This approximation is shown graphically in Figure 14.4. These approximations (equations (14.18) and (14.19)) may be extended to include Rq,0 (a, t) in the following manner. By a simple replacement of the t variable with at, we may rewrite equation (14.18) as Rq,0 (1, at) ≈

1 at e , q

a > 0,

1 ≥ q > 0,

at > 1.

Now, using equation (4.9) along with equation (14.20), we infer the approximation Rq,0 (a, t) ≈

1 (1−q)∕q a1∕q t e , a q

a > 0,

q > 0,

at > 1.

(14.21)

14.6 R-Function Approximations

R-function Rq,0(1,t) and approximation

10

q = 0.25

q = 0.50

0.75

1.00

8

6

q = 2.00

4

2

0 0

0.5

1

1.5

2

2.5

3

t-Time

Figure 14.3 Approximation of Rq,0 (1, t) by q1 et . Source: Lorenzo and Hartley 2000a [70]. Public domain.

R-function Rq,0(1,t) and approximation

30

25

20

q = 0.25

15

q = 1.00

10

5

0 0

0.5

1

1.5

2

2.5

3

t-Time

Figure 14.4 Approximation of Rq,0 (1, t) by equation (14.19). Source: Lorenzo and Hartley 2000a [70]. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

When the a parameter of the R-function becomes negative, a different set of approximations is required. The following approximation works well for 1 ≤ q ≤ 2 Rq,0 (−1, t) ≈ e−(2−q)

1.25

t

cos(t − 𝜋∕2) ,

This approximation is shown in Figure 14.5.

1 ≤ q ≤ 2,

t > 𝜋 ∕2.

(14.22)

277

14 Numerical Issues and Approximations in the Fractional Trigonometry

1

R-function Rq,0(–1,t) and approximation

278

0.5 q = 1.75

q = 2.00

0 q = 1.00 q = 1.50 –0.5

–1

2

4

6 t-Time

8

10

12

Figure 14.5 Approximation of Rq,0 (−1, t) by equation (14.22), q = 1.0–2.0 in steps of 0.25. Approximations are dashed and overlap at q = 2.0. Source: Lorenzo and Hartley 2000a [70]. Public domain. Please see www.wiley .com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

Improved approximations may be determined for particular values of q by optimizing the A, B, C, and D constants in the following equation: Rq,0 (−1, t) ≈ A e−B t cos(C(D t − 𝜋∕2)) ,

1≤q≤2.

(14.23)

When the a parameter in equation (14.21) takes on values other than −1 the following approximation works reasonably well for values of a not too large Rq,0 (−a, t) ≈ a−1∕q e−(2−q)

1.25 1∕q

a

t

cos(a1∕q t − 𝜋∕2) ,

t > 1,

1 ≤ q ≤ 2.

(14.24)

Continuing from Ref. [70]: Figure 14.6 graphically shows this approximation for a = −4. Approximations for values of q > 2 with parameter a negative, require positive values for the argument of the real exponential in the approximation. For example, with 2 ≤ q ≤ 3, the following approximation 2

Rq,0 (−1, t) ≈ e(−0.2q +1.45q−2.1) t cos((−0.2q + 1.4) t − 𝜋∕2) ,

2 ≤ q ≤ 3,

t>1

(14.25)

is presented in Figure 14.7. An additional approximation of Rq,0 (1, t) based on the equation ( ) et b 2 Rq,0 (1, t) ≈ + 1−q , where b = (1 − q), q t 3

t > 0.1,

(14.26)

is shown in Figure 14.8. Compare this approximation with that of Figure 14.4. Clearly, the approximations presented in this section only hint at the possibilities, and much more is possible.

14.7 The Near-Order Effect

R-function Rq,0(–4,t) and approximation

0.6

q = 2.00

0.4

0.2 q = 1.75 0 q = 1.50 –0.2 –0.4

2

1

3 t-Time

4

5

Figure 14.6 Approximation of Rq,0 (−4, t) 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. Source: Lorenzo and Hartley 2000a [70]. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

R-function Rq,0(–1,t) and approximation

15 10 5 q = 2.00

0 –5

q = 2.50 q = 2.75

–10

q = 3.00 –15 –20

2

4

6

8

10

t-Time

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. Source: Lorenzo and Hartley 2000a [70]. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

14.7 The Near-Order Effect There is an interesting phenomenon that occurs for the fractional trigonometric functions when the order variables are not integer. As we have seen in previous chapters, the presence of fractional exponents of t in the defining series of the trigonometric functions yields a total

279

14 Numerical Issues and Approximations in the Fractional Trigonometry

R-function Rq,0(1,t) and approximation

25

20

q = 0.20

15

q = 1.00 10

5

0

0

0.5

1

1.5 t-Time

2

2.5

3

Figure 14.8 Approximation of Rq,0 (1, t) 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 t >≈ 0.3. Please see www.wiley.com/go/Lorenzo/Fractional_ Trigonometry for a color version of this figure.

of Dq Dv many functions, where the Ds are the denominators of the rational q and v order variables. That is, there are Dq Dv − 1 functions in addition to the principal function. Thus, for example, with q = 1/n and v = 0∕1, there will be a total of n, R2 Sin1∕n,0 -functions. The number of R2 Sinq,0 -functions versus the trigonometric order q (with v = 0), for 0 < q < 1 is plotted in Figure 14.9 for denominators Dq ≤ 50. When the size of the denominators is allowed to increase, the density of the points increases but the lower envelope remains the same. 50

Number of R2 Sin function

280

40

30

20

10

0

0

0.2

0.4 0.6 0.8 Trignometric function order q

1

Figure 14.9 Number of R2 Sinq,0 (a, t) functions versus Order-q for q denominators Dq ≤ 50. Source: Lorenzo and Hartley 2004b [74]. Reproduced with permission of Springer.

Reciprocal of number of R2 Sin function

14.8 High-Precision Software

0.5

0.4

0.3

0.2

0.1

0 0

0.2

0.4 0.6 0.8 Trignometric function order q

1

Figure 14.10 Reciprocal of Number of R2 Sinq,0 (a, t) functions versus Order-q for q denominators Dq ≤ 500. Source: Lorenzo and Hartley 2004b [74]. Reproduced with permission of Springer.

Note that as the order q approaches 1 (or 0), the number of functions approaches infinity. We have called this the “Near-Order Effect” [73, 74]. Surprisingly, at the limiting case q = 1, only a single function occurs! This likely reflects the infinite number of revolutions that occur on the unit circle, as t → ∞. That is, we are “spiraling” in or out in the phase plane as t increases but with a spiral radius change of zero (see Chapter 15). In Figure 14.9, the bounding curves, together with those above them, are seen to be segments of hyperbolic curves. Note that near-order behavior is also observed in the neighborhood of the major fractions, 12 , 13 , 23 , 14 , 34 , …. A plot of the reciprocal of the number of functions is presented in Figure 14.10; for this plot, we have Dq ≤ 500. Straight lines and triangular structures are easily seen in the figure and validate the presence of hyperbolic curves in Figure 14.9. Note that Dq becomes infinite at all irrational values of q.

14.8 High-Precision Software Application of asymptotic methods requires individual study of many functions with a variety of parameter choices. As such it does not represent a satisfactory general approach to the problem of function evaluation for large values of at q . Our experience has shown that the application of high-precision symbolic software such as Maple or Mathematica has been helpful at extending the range of values of at q that may be evaluated. A Maple program for the evaluation of a fractional trigonometric function is given in Appendix B.

®

®

®

281

283

15 The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry In Chapter 9, we have seen a number of spirals that resulted from the fractional metatrigonometry. Spiral behavior is observed in many areas of physics and nature. Some examples are turbulence, vortical flow, weather formations, tornadic and hurricane structures, the shape of galaxies, spiral capture of light in a gravitational well (black hole), impact traces in bubble chambers, and Nautilus shells. A particularly interesting instance is the Belousov–Zhabotinsky (B-Z) chemical reaction where chemical rotors spontaneously exhibit rotating spiral behavior; see Winfree [124]. Many spiral functions occur in mathematics: the logarithmic or logistic spiral, the hyperbolic, parabolic, and Cornu spirals, and so on. The mathematical relationships between these spirals and the fractional spirals described here have not yet been studied. Because of their frequent occurrence in nature, there have been several books that have studied spirals and their occurrences; see Gazale [37], Davis [27], Hargittai [42, 43] and Hartmann and Mislin [50]. In this chapter, we study more deeply this important new class, the fractional spirals, and compare them with some of the classical spirals. Later chapters examine some of the applications of the fractional trigonometric spirals.

15.1 The Fractional Spiral Functions In the ordinary trigonometry, it is well known that when x = cos(t) is plotted against y = sin(t), the result is a circle of radius one in the x–y plane. Here, we study some of the phase-plane behavior of the fractional trigonometric functions. We also note that a few of figures presented in this section do not qualify as a spiral under the definitions given later in this chapter; however, they are still of interest in that they help characterize the fractional trigonometric functions. In Figure 15.1, R1 Sinq,v (1, 0, t) is plotted as a function R1 Cosq,v (1, 0, t) for various values of the order, q, between zero and one. Note that when q = 1, v = 0, we obtain the well-known circular functions. However, when q ≠ 1, we obtain a more general set of plots, the R1 -Spiral Functions. Parametrically reducing q in Figure 15.1 from q = 1 to q = 0.1 generalizes the circular functions to the set of fractional spirals that traverse from infinity to the origin. The fractional trigonometry releases us from the radius constraint of the unit circle, which is the basis of the classical trigonometry! The R1 -spiral functions contain the circular functions as a special case. Note that, in this figure, the parameter t-time increases approaching origin. Figure 15.2 is the same plot but with 1 ≤ q ≤ 2. Here, the spirals move outward from the origin. Note that the growth rate increases as q increases. The morphology of the R1 -spiral functions in the range of 1 ≤ q ≤ 2 is not seen in the classical spirals and may be unique. This morphology is denoted as a “barred spiral” in astronomy. We consider the application of fractional barred spirals to galactic morphology in Chapter 18. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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

4

R1Sinq,0(1,t)

q = 0.10 0.2

0.30 0.40

2

0.50 0.60 0

q = 0.90 q = 1.0

–2 –2

0

2

4 6 R1Cosq,0(1,t)

8

10

Figure 15.1 Phase plane plot, R1 Cosq,v (1, 0, t) versus R1 Sinq,v (1, 0, t) for q = 0.1–1 in steps of 0.1. With a = 1, v = 0, t = 0–7.4. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

5

q = 2.00

1.80 1.70 1.60

R1Sinq,0(1,t)

284

1.50 0 q = 1.0 q = 1.10 q = 1.20 q = 1.40 –5 –5

q = 1.30 0 R1Cosq,0(1,t)

5

Figure 15.2 Phase plane plot, R1 Cosq,v (1, t) versus R1 Sinq,v (1, t). 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. Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

The R2 -trigonometric functions also have a very special spiral-like behavior in the phase plane. With v = 0, when the order variable q is between zero and two, that is, 0 < q < 2, the phase plane loci are always attracted to circles. In Figure 15.3a, for 0.2 ≤ q < 1, the loci arrive from infinity and are attracted to circles of radius 1/q. In Figure 15.3b, when 1 < q < 2, the loci emanate from the origin and are also attracted to circles of radius 1/q. Figure 15.3c shows the loci spiraling outward toward infinity when q>2. A generalization of this behavior is discussed in Section 14.5.

15.1 The Fractional Spiral Functions

6 4 2

1 1.0

–2

0.6

0.80

0.4

–4

0.60 q = 0.40

–6 –8

0.2 0

1.2

–0.2

q = 2.00

–0.4 –0.6

–10

–0.8

q = 0.20

–12 –14

q = 1.0

0.8 q = 0.80 R2Sinq,0(1,t)

R2Sinq,0(1,t)

0

–5

0

5

–1 –0.5

–1

10

0

R2Cosq,0(1,t)

R2Cosq,0(1,t)

(a)

(b)

0.5

1

10 8

q = 3.0

q = 2.8

6 q = 2.6

R2Sinq,0(1,t)

4 2

q = 2.2

0 2.0

–2 –4

q = 2.4

–6 –8 –10 –10

–5

0 R2Cosq,0(1,t)

5

10

(c)

Figure 15.3 Phase plane plots, R2 Cosq,v (1, 0, t) versus R2 Sinq,v (1, 0, t) 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) 0.2 ≤ q ≤ 1.0, (b) 1.2 ≤ q ≤ 2.0, (c) 2.0 ≤ q ≤ 3.0

Phase plane plots for R3 Cosq,v (1, 0, t) and R3 Sinq,v (1, 0, t) for values of q between 1.0 and 5.0 are shown in Figures 15.4 and 15.5. The spiral growth rate is seen to rapidly increase with increases in q. The effect of v on selected R1 -functions is shown in Figure 15.6. The three-dimensional loci R2 Cos38∕39,0 (1, k, h) versus R2 Sin38∕39,0 (1, k, h) versus h are shown in Figure 15.7. Four views along the major axes of the figure show the effect of variation of the index variable, k. Because the order variable q has been selected to be q = 38/39, which is near

285

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

2 q = 2.0

1.5

R3Sinq,0(1,t)

1 q = 1.8

0.5 q = 1.0 0 1.2 –0.5

q = 1.4 0.8

–1 –1.5

q = 1.6

0.2 0.6 0.4

–2 –1

0

1 R3Cosq,0(1,t)

2

3

Figure 15.4 Phase plane plot for R3 Cosq,0 (1, t) versus R3 Sinq,0 (1, t) for q = 0.2–2 in steps of 0.2, v = 0, a = 1, k = 0, t = 0–10.

2 q = 2.50

q = 2.25

q = 2.0

1.5 1

R3Sinq,0(1,t)

286

0.5 0

q = 3.00

q = 5.0

–0.5 –1

q = 3.50

q = 0.50

–1.5 q = 4.00 –2 –2

–1

0

1

2

R3Cosq,0(1,t)

Figure 15.5 Phase plane plot for R3 Cosq,0 (1, t) versus R3 Sinq,0 (1, t) for q = 2–5 in steps of 0.25, v = 0, a = 1, k = 0, t = 0–4.

15.1 The Fractional Spiral Functions

v = 0.00

6 v = 0.15

R1Sinq,v(0.25,t)

4

2

0

–2

–4 –4

–2

0 2 R1Cosq,v(0.25,t)

4

6

Figure 15.6 Effect of v on R1 Cos1.20,v (0.25, t) versus R1 Sin1.20,v (0.25, t), phase plane plot with q = 1.20, v = 0–0.15 in steps of 0.05, a = 0.25, k = 0, t = 0–28.

unity, we have an example of the “near-order effect” discussed in Section 14.6. Strong symmetry may be observed when the Cos plane is viewed from the Sin axis (Figure 15.7b) and in the Cos versus Sin phase plane (Figure 15.7d). This figure is strongly suggestive of tornadic or cyclonic weather forms. Figure 15.8 is another four part view. Panels a–d show four views of R2 Coflut38∕39,0 (1, k, h) versus R2 Covib38∕39,0 (1, k, h) versus h, for k = 0–38. The unique aspect of this figure is the apparent symmetry seen from all three axes. Symmetry seen in Figure 15.8b–d and when viewed along the diagonals (view not shown) is stunning. The fractional meta-trigonometric functions are rich with spiral behavior. A small selection of these functions is shown in Figures 15.9–15.13. Figure 15.9 shows the effect of 𝛼 on the phase plane plot for Sin0.8,−1 (1, 𝛼, 0.5, t) versus Cos0.8,−1 (1, 𝛼, 0.5, t). The loci for this set are unusual for the fractional trigonometric functions in that they do not terminate at infinity, the origin, or in a circular orbit. Figures 15.10 and 15.11 are phase plane plots for the complexity trigonometric functions Cosq,v (a, 𝛼, 𝛽, k, t) versus Sinq,v (a, 𝛼, 𝛽, k, t) and −Cosq,v (a, 𝛼, 𝛽, k, t) versus −Sinq,v (a, 𝛼, 𝛽, k, t) (dashed). These are typical fractional spirals found in various domains of the fractional trigonometry. In Figure 15.10, for q 1, the loci spiral outward. Figures 15.12 and 15.13 show some spiral functions found for the parity fractional trigonometric functions. Observe how the 𝛽 parameter controls the exit angle as the loci leave the end of the bars. The loci in Figure 15.14 are not spirals; however, the figure is of interest because the loci leave the origin with “double-barred” behavior, a very unusual phenomenon.

287

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

5 4.5

5

3.5

4

3

h-Height

h-Height

4

3 2 1

2.5 2 1.5

0

1 5

5

0 R2Sin38/39,0(1,k,h)

–5

–5

0.5

0

0

–5

R2Cos38/39,0(1,k,h)

(a)

0 R2Cos38/39,0(1,k,h)

5

(b)

5

8

4.5

6

4 4 R2Sin38/39,0(1,k,h)

3.5 h-Height

288

3 2.5 2 1.5

2 0 –2 –4

1 –6

0.5 0

–5

–8

–5

R2Sin38/39,0(1,k,h)

0 R2Cos38/39,0(1,k,h)

(c)

(d)

0

5

5

Figure 15.7 Four views of R2 Cos38∕39,0 (1, k, t) versus R2 Sin38∕39,0 (1, k, t) versus h for k = 0–38.

15.2 Analysis of Spirals In this section, a general consideration of spiral behavior is pursued. The objective is the development of a general analysis capability for spirals. This will allow comparison of the fractional spirals with some of the classical spirals and enable further exploration of the relationships of the fractional spiral functions to the circular functions. 15.2.1

Descriptions of Spirals

There are several ways in which spirals, and other plane curves, may be described. The interested reader is referred to the books by Lawrence [64], Lockwood [67], and other similar books

15.2 Analysis of Spirals

5 4.5 4

5

3.5 h-Height

h-Height

4 3 2

3 2.5 2 1.5

1

1 5

0

0.5

0

5

0

R2Cov38/39,0(1,k,h)

–5

–5

R2Cofl38/39,0

0

–5

0 R2Cov38/39,0(1,k,h)

(a)

(b)

5

4

4.5

3

4

2 R2Cov38/39,0(1,k,h)

3.5 h-Height

5

3 2.5 2 1.5

1 0 –1 –2

1 –3

0.5 0

–5

0

5

–4 –4

–2

0

R2Cofl38/39,0(1,k,h)

R2Cofl38/39,0(1,k,h)

(c)

(d)

2

4

Figure 15.8 (a–d). Four views of R2 Coflut38∕39,0 (1, k, h) versus R2 Covib38∕39,0 (1, k, h) versus h, for k = 0–38. With a = 1.0, and v = 0.

referenced therein. We shall consider the polar and parametric descriptions. The parametric descriptions are of particular interest since the fractional spirals that have been shown in this and previous chapters are in that form. 15.2.1.1 Polar Description

The polar representation of a spiral, or more generally of a plane curve, is given by r = f (𝜃) or equivalently, 𝜃 = g(r),

(15.1)

where r is a radial distance from the origin, or other point of reference, and 𝜃 is an angle of rotation that is positive in the counterclockwise direction; see Figure 15.15.

289

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

1.6 α = 0.6

1.4

Sin0.8,–1(1,α,0.5,0,t)

α = 0.7 1.2 α = 0.8

1 0.8 0.6 0.4 α = 0.5

0.2 0 –1.5

–1

–0.5 Cos0.8,–1(1,α,0.5,0,t)

0

Figure 15.9 Phase plane plot for Sin0.8,−1 (1, 𝛼, 0.5, t) versus Cos0.8,−1 (1, 𝛼, 0.5, t) for q = 0.8, v = −1, a = 1.0, 𝛼 = 0.5–0.8 in steps of 0.1, 𝛽 = 0.5, k = 0 and t = 0–20.

Figure 15.10 Phase plane plot for Cosq,v (a, 𝛼, 𝛽, k, t) versus Sinq,v (a, 𝛼, 𝛽, k, t) and −Cosq,v (a, 𝛼, 𝛽, k, t) versus −Sinq,v (a, 𝛼, 𝛽, k, t) (dashed) for q = 0.5, v = 1, a = 1.0, 𝛼 = 0.15, 𝛽 = 0.75, k = 0 and t = 0–22.

4 3 2 Sin0.5,1(1,0.15,0.75,0,t)

290

1 0 1 –2 –3 –4 –3

15.2.1.2

–2

1 2 0 –1 Cos0.5,1(1,0.15,0.75,0,t)

3

Parametric Description

The parametric description of a spiral, or a plane curve, expresses the rectangular (or polar) coordinates of the curve in terms of a parametric value, for example, t, thus x = x(t) and y = y(t), or r = r(t) and 𝜃 = 𝜃(t).

(15.2)

15.2 Analysis of Spirals

25 20 15

Sin1.2,–0.8(1,1,0,0,t)

10 5 0

+/+

–/–

–5 –10 –15 –20 –25

–20

–10

0 10 Cos1.2,–0.8(1,1,0,0,t)

20

Figure 15.11 Phase plane plot for Cosq,v (a, 𝛼, 𝛽, k, t) versus Sinq,v (a, 𝛼, 𝛽, k, t) and −Cosq,v (a, 𝛼, 𝛽, k, t) versus −Sinq,v (a, 𝛼, 𝛽, k, t) (dashed) for q = 1.2, v = −0.8, a = 1.0, 𝛼 = 1.0, 𝛽 = 0, k = 0 and t = 0–14.

1 0.8

Flut0.5,0(1,0.16,0.8,0,t)

0.6 0.4 0.2 0

+/+

–/–

–0.2 –0.4 –0.6 –0.8 –1 –1

–0.5

0

0.5

1

Cofl0.5,0(1,0.16,0.8,0,t)

Figure 15.12 Phase plane plot for Flutq,v (a, 𝛼, 𝛽, k, t) versus Cof lq,v (a, 𝛼, 𝛽, k, t) and −Flutq,v (a, 𝛼, 𝛽, k, t) versus −Cof lq,v (a, 𝛼, 𝛽, k, t) (dashed) for q = 0.5, v = 0, a = 1.0, 𝛼 = 0.16, 𝛽 = 0.80, k = 0 and t = 0–22.

291

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

10 β = 1.25

8

β = 1.20

6

Vibq,v(a,α,β,k,t)

4 2 β = 1.30

0 –2 –4

β = 1.35

–6 –8

β = 1.40

–10 –10

–5

0 Covibq,v(a,α,β,k,t)

5

10

Figure 15.13 Phase plane plot for Vibq,v (a, 𝛼, 𝛽, k, t) versus Covibq,v (a, 𝛼, 𝛽, k, t) and −Vibq,v (a, 𝛼, 𝛽, k, t) versus −Covibq,v (a, 𝛼, 𝛽, k, t) (dotted) for q = 1.07, v = −0.070, a = 1.0, 𝛼 = 0.02, 𝛽 = 1.2–1.4 in steps of 0.05, k = 0 and t = 0–10. 1.5

β = 1.40

1

0.5 Coflq,v(a,α,β,k,t)

292

β = 1.20

0

–0.5

–1

–1.5 –1.5

–1

0.5 0 0.5 Covibq,v(a,α,β,k,t)

1

1.5

Figure 15.14 Phase plane plot for Cof lq,v (a, 𝛼, 𝛽, k, t) versus Covibq,v (a, 𝛼, 𝛽, k, t) and −Cof lq,v (a, 𝛼, 𝛽, k, t) versus −Covibq,v (a, 𝛼, 𝛽, k, t) (dotted) for q = 1.06, v = −0.01, a = 1.0, 𝛼 = 0.05, 𝛽 = 1.2–1.4 in steps of 0.05, k = 0 and t = 0–2.5.

15.2 Analysis of Spirals

y

Figure 15.15 Coordinate systems for both polar and parametric forms.

(x1,y1) or (r1,θ1) r

θ

x

t-Parameter

8 6 4 (x,y,t) or (r,θ,t)

2

0 10 5 0 y = Sinq,v(a,α,β,t)

(x,y,0) or (r,θ,0) –5

0 –10

10 5 x = Cosq,v(a,α,β,t)

Figure 15.16 Parametric spiral interpreted as a three-dimensional space curve.

The conversion between the polar and rectangular forms is described by the well-known relationships: (y) √ or (15.3) r = x2 + y2 and 𝜃 = tan−1 x x = r cos(𝜃) and y = r sin(𝜃). (15.4) The parametric description may also be considered as a three-dimensional space curve in three dimensions x, y, and t. This is illustrated in Figure 15.16 for the spiral defined by the equations x = R1 Cos1.3,0 (1, 0, t) and y = R1 Sin1.3,0 (1, 0, t), (15.5) with t = 0–7.9. 15.2.1.3

Definitions

In his book on spirals, Davis [27] describes a search for a meaningful and mathematically useful definition for spirals. Specialized definitions for a particular purpose have been used; however, few generalized definitions are found. One such definition is found in his notes p. 205. It is attributed to G. Fouret, “r = f (𝜃)∕(𝜃 + g(𝜃)) where f and g are complex trigonometric functions

293

294

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

of 𝜃.” The next section proposes some definitions that may serve as a basis for further study of spirals. 15.2.1.4 Alternate Definitions

For simplicity, here we consider only the origin as the basis of rotation, and only consider plane spirals. Let R be a radius vector in a polar coordinate system and 𝜃 the polar angle measured from the x-axis. Then, a simple ccw (counter clockwise) spiral is defined by the function r(𝜃), where r(𝜃) is a continuous non-negative monotonic function with 𝜃 ≥ 0 and 𝜃̇ > 0. A general, ccw, spiral is defined as a function R(𝜃), where r1 (𝜃) ≤ R(𝜃) ≤ r2 (𝜃),

∀𝜃 ≥ 0,

(15.6)

where r1 (𝜃) and r2 (𝜃) are simple ccw spirals, typically with no common points other than the origin (or other point of reference). Note that R(𝜃) may be a discontinuous and/or multivalued function with only the constraint that it bounded by the simple spirals. Typically, we could also constrain R(𝜃) to be a continuous function. This definition allows considerable latitude in the types of functions considered to be spiral while maintaining the general spirit of what a spiral is. For spirals that wrap in a clockwise (cw) manner, the definition is as follows: A simple cw (clockwise) spiral is defined by the function r(𝜃), where r(𝜃) is a continuous nonpositive monotonic function with 𝜃 ≤ 0 and 𝜃̇ < 0. A general, cw, spiral then is defined as a function R(𝜃), where r1 (𝜃) ≤ R(𝜃) ≤ r2 (𝜃),

∀𝜃 ≤ 0,

(15.7)

where r1 (𝜃) and r2 (𝜃) are simple, cw, spirals, typically, with no common points other than possibly the origin (or other point of reference). Furthermore, R(𝜃) may be a discontinuous and/or multivalued function with only the constraint that it is bounded by the simple spirals. From these definitions, a mathematical theory of spirals, that is, addition, multiplication, and so on, is possible. 15.2.1.5

Examples

Consider the three curves defined by r1 = 0.5 exp(0.3 𝜃), r2 = 1.0 exp(0.3 𝜃), and r3 = (r +r ) 0.2 exp(0.3 𝜃) sin(5𝜃) + 1 2 2 . This spiral is shown in Figure 15.17a. For the spiral in Figure 15.17a, the r1 and r2 curves are simple ccw spirals, and the r3 curve is seen to be a continuous general ccw spiral (over the range shown). Figure 15.17b shows a general ccw spiral composed of a random sequence of points bounded by the same r1 and r2 simple ccw spirals. 15.2.2 15.2.2.1

Spiral Length and Growth/Decay Rates Spiral Length

The length of an arc is well known from the classical calculus. It is given in rectangular coordinates by the equation x2

L=

∫ x1

√ 1+

(

dy dx

)2

y2

dx =

∫ y1

√ 1+

(

dx dy

)2 dy,

(15.8)

15.2 Analysis of Spirals

90

90

4

120

60

60

3

3 150

4

120

150

30

2

2

30

1

1 180

0

180

0

r1

210

r1

330

330

r2

300

240

210

300

240

270 (a)

r2

270 (b)

Figure 15.17 Polar plot showing two simple counterclockwise spirals bounding (a) a general continuous and (b) a general discontinuous counterclockwise spiral.

where it is assumed that the arc length is given as

dy dx

or

dx dy

is a continuous bounded function. In polar coordinates, 𝜃2

L=

√ r2 +

∫

𝜃1

(

dr d𝜃

)2 d𝜃 ,

dr d𝜃

where again it is assumed that is a continuous bounded function. The parametric form will be most useful for application to the fractional spirals, that is, t2 √( )2 ( )2 dy dx L= + dt , (15.9) ∫ dt dt t1

where again the derivatives are continuous bounded functions. For application to an R1 -spiral, for example, the relationships dtd R1 Cosq,v (a, t) = R1 Cosq,v+1 (a, t) and dtd R1 Sinq,v (a, t) = R1 Sinq,v+1 (a, t) are applied to an R1 -spiral to provide the length from the origin to tfinal as tmf

L(t) =

∫

((

)2

R1 Cosq,v+1 (a, t)

+ (R1 Sinq,v+1 (a, t)2

)1∕2

dt ,

t > 0.

(15.10)

tmi =0+

15.2.2.2

Spiral Growth Rates for ccw Spirals

The rate of growth or decay of spirals described using the polar representation (equation (15.1)) is given by dr , (15.11) Radial Growth Rate = d𝜃 where a positive value indicates spiral growth and a negative value indicates decay for ccw spirals. When the spiral is described by the parametric form, the growth rate can also be set in

295

296

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

those terms. Thus, based on equation (15.3), dr(t) = d(x2 (t) + y2 (t))1∕2 =

x(t)dx(t) + y(t)dy(t) , (x2 (t) + y2 (t))1∕2

(15.12)

and the time rate of growth of the radius is

For the spiral angle

x(t)x′ (t) + y(t)y′ (t) dr . = √ dt (x2 (t) + y2 (t))

(15.13)

( )) ( y(t) x(t)dy(t) − y(t)dx(t) d𝜃(t) = d tan−1 = . x(t) x2 (t) + y2 (t)

(15.14)

Similarly, the time rate of growth of the angle is given as d𝜃(t) x(t)y′ (t) − y(t)x′ (t) = . dt x2 (t) + y2 (t)

(15.15)

Combining the results of equations (15.12) and (15.14) gives (x(t)dx(t) + y(t)dy(t)) (x2 (t) + y2 (t))1∕2 dr(t) = . d𝜃(t) x(t)dy(t) − y(t)dx(t)

(15.16)

Similarly, from the time rates, equations (15.13) and (15.15), we have (x(t)x′ (t) + y(t)y′ (t))(x2 (t) + y2 (t))1∕2 dr∕dt dr = = . d𝜃 d𝜃∕dt x(t)y′ (t) − y(t)x′ (t)

(15.17)

15.2.2.3 Component Growth Rates

It is necessary in some analysis to use the component growth rates; these are obtained as follows. Differentiating x = r cos(𝜃) and y = r sin(𝜃) gives dx = cos(𝜃) dr − r sin(𝜃)d𝜃, dy = sin(𝜃)dr + r cos(𝜃)d𝜃.

(15.18)

Now, r = f (𝜃) ⇒ dr = f ′ (𝜃)d𝜃 in the aforementioned equations dy = sin(𝜃)f ′ (𝜃) + r cos(𝜃) or d𝜃 dy dx = cos(𝜃)f ′ (𝜃) − f (𝜃) sin(𝜃), = sin(𝜃)f ′ (𝜃) + f (𝜃) cos(𝜃). d𝜃 d𝜃 dx = cos(𝜃)f ′ (𝜃) − r sin(𝜃), d𝜃

(15.19)

Alternatively, 𝜃 = g(r) ⇒ d𝜃 = g ′ (r) dr in equation (15.18) gives dy = sin(𝜃) + rg ′ (r) cos(𝜃) or dr dy dx = cos(g(r)) − rg ′ (r) sin(g(r)), = sin(g(r)) + rg ′ (r) cos(g(r)). dr dr dx = cos(𝜃) − rg ′ (r) sin(𝜃), dr

(15.20)

15.2.3 Scaling of Spirals

In this section, we consider the process of scaling spiral functions; consideration shall be done in both polar and rectangular forms. The parameters of the original (reference) spiral are subscripted as xo (t) and yo (t), while the scaled versions are given as xs (t) and ys (t), and so on.

15.2 Analysis of Spirals

15.2.3.1

Uniform Rectangular Scaling

In uniform rectangular scaling, the parametric or rectangular values defining the spiral are multiplied by the same scale factor, k. Thus, xs = kxo and ys = kyo . The polar variables are rs =

(15.21)

√ √ (kxo )2 + (kyo )2 = k xo 2 + yo 2 or

(15.22)

rs = kro and tan 𝜃s =

kyo = tan 𝜃o ⇒ 𝜃s = 𝜃o . kxo

(15.23)

Since drs = kdro and d𝜃s = d𝜃o , we have dr drs =k o. d𝜃s d𝜃o

(15.24)

Thus, we see that the growth rate of the uniformly scaled spiral increases or decreases proportional to the value of the scale factor, k. 15.2.3.2

Nonuniform Rectangular Scaling

In nonuniform rectangular scaling, different factors are used to multiply the parametric or rectangular values of the defining the spiral. Thus, xs = kx xo and ys = ky yo , and the polar variables are √ rs = (kx xo )2 + (ky yo )2 = kx √ rs = kx xo

(

1+

√( rs = ky yo

ky yo

(

xo 2 + √

)2

kx xo

kx xo ky yo

√

√ + 1 = ky yo

(

= ky

kx ky

kx xo ky

)2 + yo 2 ,

)2 tan2 (𝜃o ) ,

kx (

1+

ky

√(

)2

kx

1+

= kx xo

)2

ky yo

(15.25)

)2 cot2 (𝜃o ) .

(15.26)

Solving for kx xo and ky yo from these equations, we may write √

xo =

1+

kx √

yo = ky

rs (

1+

ky

)2

(

kx ky

(15.27)

,

(15.28)

tan2 (𝜃o )

kx rs

,

)2 cot2 (𝜃o )

297

298

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

r0 =

√

√ (xo )2 + (yo )2 =

rs = √

kx2

+

rs2 2 ky tan2 (𝜃o )

+

r0

ky2

+

rs2 , 2 kx cot2 (𝜃o )

,

(15.29)

tan(𝜃o ),

(15.30)

) tan(𝜃o ) .

(15.31)

1 1 + kx2 + ky2 tan2 (𝜃o ) ky2 + kx2 cot2 (𝜃o )

and tan(𝜃s ) =

ky yo kx xo

ky

=

kx

( 𝜃s = tan−1

ky kx

Therefore, the nonuniform scaling distorts the x, y relationship, it scales the radius and rotates the spiral. It is noted that if kx = ky = k in equations (15.28) and (15.29) the equations reduce to the uniform scaling equations (15.22) and (15.23). 15.2.3.3

Polar Scaling

In the following sections, we apply scaling factors to the polar variables and examine the effect on the rectangular variables. 15.2.3.4

Radial Scaling

In radial scaling, we scale the radius while keeping the polar angle fixed, that is, rs = kr ro and 𝜃s = 𝜃o , then √ √ √ rs = kr xo 2 + yo 2 = (kr xo )2 + (kr yo )2 = xs 2 + ys 2 .

(15.32)

One possible relationship pair to satisfy equation (15.32) is xs = kr xo ,

y s = kr y o ,

(15.33)

that is, the condition resulting from uniform rectangular scaling. Thus, we see radial scaling is equivalent to uniform rectangular scaling. 15.2.3.5

Angular Scaling

In angular scaling, the radius is fixed while the angle is scaled; thus, 𝜃s = ka 𝜃o ,

rs = ro .

(15.34)

Now, xs = rs cos(𝜃s ) ,

ys = rs sin(𝜃s ),

xo = ro cos(𝜃o ) ,

yo = ro sin(𝜃o ).

and Since rs = ro ,

( xs =

cos(ka 𝜃o ) cos(𝜃o )

(

) xo ,

ys =

sin(ka 𝜃o ) sin(𝜃o )

) yo .

Thus, we see that the scaling of the x and y coordinate varies with 𝜃s .

(15.35)

15.2 Analysis of Spirals

Spiral

1

Vr

Vy

Vn

0.8

y

π–θ–ϕ ϕ

0.6

Vrad Vx

0.4

θ

0.2

θ 0 0

0.2

0.4

0.6 x

0.8

1

1.2

Figure 15.18 Velocity components.

15.2.4

Spiral Velocities

If a spiral, such as that shown in Figure 15.18, is considered as a trajectory, then velocities along the trajectory may be determined when the t parameter is treated as time. Our goal here is treatment of the fractional spiral functions; however, we start with a general treatment for spirals expressed rectangular parametric form. Based on the parametric definition (equation (15.2)), the x and y components of velocity are defined as } { x(t + 𝛿 t) − x(t) dx(t) = , (15.36) Vx (t) = lim 𝛿 t→0 𝛿t dt } { y(t + 𝛿 t) − y(t) dy(t) Vy (t) = lim = . (15.37) 𝛿 t→0 𝛿t dt Thus, the resultant velocity is given by Vr (t) =

√ Vx 2 (t) + Vy 2 (t),

(15.38)

and the associated direction is expressed as ( 𝜑(t) = tan

−1

Vy (t) Vx (t)

) .

(15.39)

Now, the slope of the spiral at time equal t is M = dy(t)∕dx(t). The angle associated with this slope is given by ) ) ( ( ′ dy(t) y (t)dt = tan−1 = 𝜑(t), 𝛽 = tan−1 dx(t) x′ (t)dt and we see that the resultant velocity is tangent to the spiral at time t.

299

300

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

It is also of interest to determine the radial and normal components of the resultant velocity relative to the vector from the origin to the spiral at time t. If 𝜃 is the angle from the positive x axis to the point x(t), y(t); see Figure 15.18. Then, the radial and normal components, Vrad (t) and Vn , respectively, are determined as follows: Vrad (t) = Vr (t) cos(𝜋 − 𝜃 − 𝜙) = −Vr (t) cos(𝜃 + 𝜙),

(15.40)

where 𝜙 is defined by equation (15.39). Applying the sum of angles formula for the cosine function from the ordinary trigonometry, Vrad (t) = −Vr (t){ cos(𝜃) cos(𝜙) − sin(𝜃) sin(𝜙)}, ( )) ( ( V )) ⎫ ( ⎧ y(t) cos tan−1 Vy −⎪ ⎪ cos tan−1 x(t) x ⎪ ⎪ Vrad (t) = −Vr (t) ⎨ , ( )) ( ( V )) ⎬ ( ⎪ ⎪ y −1 y(t) −1 sin tan ⎪ sin tan ⎪ x(t) Vx ⎩ ⎭ ⎫ ⎧ Vy Vx y(t) ⎪ ⎪ x(t) −√ Vrad (t) = −Vr (t) ⎨ √ √ √ ⎬, 2 2 2 2 x +y ⎪ x +y Vx2 + Vy2 Vx2 + Vy2 ⎪ ⎭ ⎩ ⎧ ⎫ Vx (t)x(t) − Vy (t)y(t) ⎪ ⎪ Vrad (t) = −Vr (t) ⎨ √ ⎬. √ ⎪ Vx 2 (t) + Vy 2 (t) x2 (t) + y2 (t) ⎪ ⎩ ⎭

(15.41)

√ x2 + y2 = r, the radial distance from the origin to the point x(t), y(t), and √But Vx2 + Vy2 = Vr , giving Vy (t)y(t) − Vx (t)x(t) Vrad (t) = . (15.42) r The normal velocity is determined in similar manner, that is, Vn (t) = Vr (t) sin(𝜋 − 𝜃 − 𝜙) = Vr (t) sin(𝜃 + 𝜙) .

(15.43)

Applying the sum of angles formula for the sine from the ordinary trigonometry Vn (t) = Vr (t) {sin(𝜃) cos(𝜙) + cos(𝜃) sin(𝜙)}, { ( )) ( )) ( ( Vy (t) y(t) cos tan−1 Vn (t) = Vr (t) sin tan−1 x(t) Vx (t) ( )) ( ))} ( ( Vy (t) y(t) + cos tan−1 sin tan−1 , x(t) Vx (t) ⎫ ⎧ Vy Vx ⎪ ⎪ y(t) x(t) Vn (t) = Vr (t) ⎨ √ +√ √ √ ⎬, 2 2 2 2 2 2 2 2⎪ x + y ⎪ x +y Vx + Vy Vx + Vy ⎭ ⎩ { } Vx (t)y(t) + Vy (t)x(t) Vx (t)y(t) + Vy (t)x(t) Vn (t) = Vr (t) . = rVr r

(15.44)

(15.45)

15.2 Analysis of Spirals

When applied to say, the R1 -spiral functions, we have x(t) = R1 Cosq,v (a, t), y(t) = R1 Sinq,v (a, t), { Vx (t) = lim

R1 Cosq,v (a, t + 𝛿 t) − R1 Cosq,v (a, t)

}

𝛿t

𝛿 t→0

= 0 D1t R1 Cosq,v (a, t) = R1 Cosq,v+1 (a, t), { Vy (t) = lim

R1 Sinq,v (a, t + 𝛿 t) − R1 Sinq,v (a, t)

𝛿 t→0

(15.46) }

𝛿t

= 0 D1t R1 Sinq,v (a, t) = R1 Sinq,v+1 (a, t),

(15.47)

where the differentiation rules (equations (6.36) and (6.40)) have been applied. The velocities for the R1 -spiral functions then may be obtained by substitution of these values into equations (15.33), (15.42), and (15.44); thus, √ Vr (t) = R1 Cos2q,v+1 (a, t) + R1 Sin2q,v+1 (a, t) , (15.48) Vrad (t) =

R1 Sinq,v+1 (a, t)R1 Sinq,v (a, t) − R1 Cosq,v+1 (a, t)R1 Cosq,v (a, t) , √ R1 Cos2q,v (a, t) + R1 Sin2q,v (a, t)

(15.49)

Vn (t) =

R1 Cosq,v+1 (a, t)R1 Sinq,v (a, t) + R1 Sinq,v+1 (a, t)R1 Cosq,v (a, t) . √ R1 Cos2q,v (a, t) + R1 Sin2q,v (a, t)

(15.50)

These results may be readily extended to the meta-trigonometric spirals. 15.2.5

Referenced Spirals: Retardation

It is useful at times to compare spirals either from the same family or of different families. It is of particular interest to compare with a special spiral, the zero growth spiral, that is, the unit circle. The unit circle is, of course, the basis of the circular functions and is the geometric locus for the q = 1, v = 0, R1 -trigonometry. As a spiral, it is obtained when x(t) = cos(t) and y(t) = sin(t), where typically t is considered to be the time variable. We observe that t = tan−1 (y(t)∕x(t)), for 0 ≤ t ≤ 2𝜋, that is, for the first revolution. Thus, for this situation, t represents both t-time the parameter and t the geometric angle. This is not the typical case for the R1 - and R2 -spiral functions, or the meta-trigonometric spirals. Note, the results and definitions in this section apply only to positive evolving or counterclockwise developing spirals. Obvious changes are required for clockwise evolving spirals. In Figure 15.19, we compare spirals that have evolved to the same time t. The angles, 𝛼, associated with these spirals are in general different. When, for the same value of the parametric variable t, the angle of a test spiral is smaller than that for the reference spiral to which it is being compared, we consider it to be “retarded,” and if larger it is considered to be “advanced.” We define Angle of Retardation 𝜆r (t) ≡ 𝛼ref (t) − 𝛼test (t).

(15.51)

301

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

1.2 1

Common times, t = 1.25

Test spiral, q = 1.1

y(t)

0.8 0.6 0.4

Reference spiral, q = 1.0

0.2 0 –0.2 –1.5

–1

–0.5

0.5

0

1

x(t)

Figure 15.19 Angular retardation demonstrated for spiral defined by R1 Trigq,v (a, t) with q = 1.1 for test spiral and q = 1.0 for reference spiral; for both spirals, a = 1, k = 0, and v = 0, x(t) = R1 Cosq,0 (a, t) and y(t) = R1 Sinq,0 (a, t).

If the unit circle is considered to be the reference spiral in this instance, then the retardation angle 𝜆r is given by 𝜆r (t) = t − 𝛼test (t). (15.52) Spirals may also be compared based on the value of the parameter t (time) for spirals that have evolved to a common angle. Thus, with reference to Figure 15.20, the retardation time is defined as Retardation Time = 𝛿 tr (𝛼) ≡ tref (𝛼) − ttest (𝛼). (15.53) Again if the unit circle is considered to be the reference spiral, then the retardation time 𝛿 tr is given by 𝛿 tr (𝛼) = t − ttest (𝛼). (15.54) When the retardation time is positive, 𝛿 tr (𝛼) < 0, the test spiral is “late or retarded” relative to the reference spiral, and conversely it is “early or advanced” when 𝛿 tr (𝛼) > 0.

1.2 1

t = ttest

Test spiral, q = 1.1

0.8 y(t)

302

t = tref

0.6 0.4

Reference spiral, q = 1.0

0.2

Common angles

0 –0.2 –1.5

–1

–0.5

0

0.5

1

x(t)

Figure 15.20 Time retardation demonstrated for spiral defined by R1 Trigq,v (a, t) with q = 1.1 for test spiral and q = 1.0 for reference spiral; for both spirals, a = 1, k = 0, and v = 0, x(t) = R1 Cosq,0 (a, t) and y(t) = R1 Sinq,0 (a, t).

15.3 Relation to the Classical Spirals

We now consider, as an example, the R1 family of spirals. The retardation angle is given by ( ( ( )) ) 𝜆r = 𝛼ref − 𝛼test = tan−1 R1 Tanqref,vref aref , t − tan−1 R1 Tanqtest,vtest (atest , t) , where the reference spiral is taken to be a general member of the R1 family. If the reference spiral is taken to be the unit circle, this becomes 𝜆r = 𝛼ref − 𝛼test = t − tan−1 (R1 Tanqtest,vtest (atest , t)).

(15.55)

For the retardation time of the R1 family of spirals, we need a common reference angle 𝛼 on which to base the comparison. For the test and reference spirals, these are given by tan(𝛼) = R1 Tanqtest,vtest (atest , ttest ) = R1 Tanqref,vref (aref , tref ), and the related times are −1 ttest = R1 Tan−1 qtest,vtest (atest ) and tref = R1 Tanqref,vref (aref ).

(15.56)

Note that explicit forms for the R1 Tan−1 q,v are not available; thus, numerical computations are required. The retardation time 𝛿tr (𝛼) is then determined as 𝛿tr (𝛼) ≡ tref (𝛼) − ttest (𝛼) = R1 Tan−1 (a ) − R1 Tan−1 qtest,vtest (atest ). qref,vref ref

(15.57)

If the reference spiral is taken to be the unit circle, the retardation time becomes 𝛿tr (𝛼) ≡ tref (𝛼) − ttest (𝛼) = t − R1 Tan−1 qtest,vtest (atest ).

(15.58)

15.3 Relation to the Classical Spirals In this section, a comparison is made between some of the classical spirals and the fractional spirals. 15.3.1

Classical Spirals

Here, we examine the behavior of a few of the classical spirals of mathematics. Perhaps the most famous is the spiral of Archimedes; this is defined by the equation r = b𝛼.

(15.59)

The general family of Archimedean-like spirals is given by r m = bm 𝛼 ,

m = … , −2, −1, 1, 2, … ,

(15.60)

where m = 1 is equation (15.3.1) and m = 2 defines Fermat’s spiral. The Hyperbolic spiral is defined as r𝛼 = b . (15.61) The Logistic or Logarithmic spiral is well known for its fractal behavior, that is, it looks the same at all scales. Its equation is log r = b𝛼 or r = eb𝛼 .

(15.62)

Finally, the parabolic spiral is given by r2 = b𝛼, another special case of equation (15.60).

(15.63)

303

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

Table 15.1 The classical spirals in polar, rectangular, and parametric forms. Spiral

Polar

Rectangular

Parametric

Archimedes

r = b𝛼

x2 + y2 = b2 (arctan(y∕x))2

x = b𝛼 cos(𝛼)

Fermat

r 2 = b2 𝛼

x2 + y2 = b2 arctan(y∕x)

x = b𝛼 1∕2 cos(𝛼)

General

r =b 𝛼

y = b𝛼 sin(𝛼) y = b𝛼 1∕2 sin(𝛼) m

2

m

2

2

2∕m

x + y = b (arctan(y∕x))

x = b𝛼 1∕m cos(𝛼) y = b𝛼 1∕m sin(𝛼)

Hyperbolic

2

2

2

(x + y )(arctan(y∕x)) = b

r𝛼 = b 2

Parabolic

2

2

x + y = b arctan(y∕x)

r = b𝛼

2

x=

b 𝛼

y=

b sin(𝛼) 𝛼 √

x= y=

x2 + y2 = e2 b

log r = b𝛼 or r = eb𝛼

Logarithmic

arctan(y∕x)

cos(𝛼)

b𝛼 cos(𝛼) √ b𝛼 sin(𝛼)

x = eb𝛼 cos(𝛼) y = eb𝛼 sin(𝛼)

8 Logarithmic spiral

7 6 5 r-Radius

304

m=1

4

Archimedes spiral

3

m=2 2 (Also Fermat’s spiral) 1 Hyperbolic spiral 0

0

2

4 α-Alpha

6

8

Figure 15.21 Rectangular plots for some of the classical spirals.

A summary of some of the classical spirals in the three basic forms is shown in Table 15.1. The rectangular forms are easily converted from the polar form using equation (15.3). These spirals are better differentiated from each other graphically by comparing r versus 𝛼 rectangular plots than by comparing the actual spirals. A number of these cases are shown graphically in Figure 15.21. The equivalent plot for the spirals generated from the parametric equations x = R1 Cosq,0 (1, t) and y = R1 Sinq,0 (1, t) is presented in Figure 15.22. By comparing the two figures, we can see that for q > 1 and t > 1 the spirals have the character of the Logistic spiral, and when q < 1 the spirals mimic the hyperbolic spiral.

15.3 Relation to the Classical Spirals

20

q = 2.0

18

q = 1.8

q = 1.6

t=5

16

q = 0.2

r-Radius

14 12 0.4 q = 1.4

10 t=4

8 6

t=3

4 0.6

q = 1.2

t=2

2 0

q = 1.0 0

2

1

3 α - Angle

4

5

6

Figure 15.22 Radius versus 𝛼-angle for spiral defined by R1 Cosq,0 (1, t) versus R1 Sinq,0 (1, t), 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. 11 t=2

10

t=3

t=4

t=5

t=1

q = 0.1

9 8

r-Radius

7 6 q = 0.2

5 4

q = 0.3

3

q = 0.4

2 1 0

q = 1.0 0

2

4 α -Angle

6

8

Figure 15.23 Radius versus angle for spiral defined by R2 Cosq,0 (1, t) versus R2 Sinq,0 (1, t), 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.

Figures 15.23 is the equivalent plot for the parametric equations x = R2 Cosq,0 (1, t) and y = R2 Sinq,0 (1, t). Observe for this case that t-time and 𝛼-angle are much more closely related than in the R1 -trigonometric case. For all the figures, the constant lines for the q = 1 case represents the unit circle and of course the classical trigonometry. For the R2 -trigonometric case, it may also be seen that for all q such that 0 < q < 2 the loci are attracted to radii of

305

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

4 3.5 3 q = 0.1

r-Radius

2.5 2 1.5

t=2 1

t=3

t=4

t=5

4

5

q = 1.0

0.5 t=1

0

1

0

2

3

6

α -Angle

Figure 15.24 Radius versus angle for spiral defined by Sinq,0 (1, 0.5, 0.5, t) versus Cosq,0 (1, 0.5, 0.5, t), 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. 3

3

2

2 Sinq,v(a,α,β,k,η)

Bθ1/msin(θ)

306

1 0

–/–

+/+

–1

0

+/+

–/–

–1 –2

–2 –3 –4

1

–2

0 Bθ1/mcos(θ) (a)

2

4

–3 –4

–2

0 Cosq,v(a,α,β,k,η)

2

4

(b)

Figure 15.25 Comparison of an Archimedean spiral 15-25a and a Fractional spiral 15-25b. (a) Archimedean spiral defined by rm = Bm 𝜃 with B = 1.25 and m = 1.8, 𝜃 = 0–7.2. (b) Fractional spiral with Cosq,v (a, 𝛼, 𝛽, k, t) versus Sinq,v (a, 𝛼, 𝛽, k, t) and −Cosq,v (a, 𝛼, 𝛽, k, t) versus −Sinq,v (a, 𝛼, 𝛽, k, t), dashed line, for q = 1.15, v = 0, a = 1.0, 𝛼 = 1.0, 𝛽 = 0, k = 0 and t = 0–8.

r = 1∕q. Figure 15.24 shows the fractional meta-trigonometric functions Sinq,0 (1, 0.5, 0.5, t) versus Cosq,0 (1, 0.5, 0.5, t), as rectangular plots, with 𝛼 = 0.5 and 𝛽 = 0.5 and q = 0.1–1.0. We note that the R2 -loci may not satisfy our criteria for spirals because they may not be monotonic. The panels a and b of Figure 15.25 compare the morphologies of a generalized Archimedean spiral defined by rm = Bm 𝜃 [64] to a fractional barred spiral in the vicinity of the origin. While the shapes are quite similar in this area, the growth rates are quite different as the spirals

15.4 Discussion

y(T) = imag(T) and Sinq,v(a,α,β,k,η)

6

4

2

0 –2 –4

Sprial of theodorus

–6 –6

–4

–2

0

2

4

6

x(T) = real(T) and Cosq,v(a,α,β,k,η)

Figure 15.26 Comparison of the Spiral of Theodorus, and a fractional spiral. Solid line is the Theodorus spiral with kf = 1000, b = −1–33.35 in steps of 0.05, and circles are a meta-trigonometric spiral with q = 1.0239, v = −0.22, 𝛼 = 4.2, 𝛽 = −3.204, k=0, 𝜂 = 0–9.2.

continue to develop. Furthermore, by changing v from zero, the bar of the fractional spiral may be softened to more closely match the Archimedean spiral near the origin. In his book “Spirals from Theodorus to Chaos,” Davis [26] gives considerable attention to the spiral of Theodorus. While this spiral is not generally considered to be one of the classical spirals, its apparent antiquity is of some interest. Davis dates Theodorus around 465–399 B.C. and uses the following infinite product to define the spiral in the complex plane: ( √ ) ∞ 1 + i∕ k ∏ (15.64) T(b) = ( ). √ k=1 1 + i∕ k + b In Figure 15.26, the Theodorus spiral with b = −1 to 33.35 in steps of 0.05 and k = 1 to 1000 has been approximated by a fractional meta-trigonometric spiral with the parameter set q = 1.0239, v = −0.22, a = 1.0, 𝛼 = 4.2, 𝛽 = −3.204, k = 0, 𝜂 = 0 → 9.2 with both axes scaled by a factor of 1.43 and rotated by 240∘ . The fit is fairly good over the range considered, but could possibly be improved with continued search and perhaps even matched with some analytical study.

15.4 Discussion The fractional spirals are an important addition to the classical spirals. These spirals are generalizations of the circle in the same way that the fractional trigonometric functions are generalizations of the classical trigonometric functions. Importantly, based on their Laplace transforms, we may associate fractional differential equations with the morphology of these spirals.

307

309

16 Fractional Oscillators Oscillators, electrical, mechanical and natural, are found in many applications and situations in the physical world. In the past, analysis of such oscillators has been based on the methods of the integer-order calculus and the application of ordinary differential equations. With the growing number of applications of the fractional calculus and fractional differential equations, there is considerable interest in determining how these new analytical tools will impact the practical and theoretical applications of fractional-order oscillators. In a series of papers, the dynamics- and energy-based behaviors of fractional oscillators have been studied by Achar et al. [1–4]. Their work has also forwarded a concept of generalized momentum for application to the area and further studies of chained fractional oscillators [4]. Others have studied various aspects concerning fractional oscillators. Practical implementation of fractance-based oscillators and derivation of the Barkhausen conditions has been done by Radwan et al. [111]. The concept of fractional calculus-based oscillators modeled by ensembles of harmonic oscillators has been contributed by Stanislavsky [117]. Gorenflo and Mainardi [39] provide fundamental analysis of fractional oscillations and relaxation. Additional papers on the subject include Koh and Kelly [59], Eab and Lim [32], and Drodzdov [30]. Lorenzo et al. [84] have determined the space of fractional oscillators defined by the fractional trigonometry. This work is the basis of the following section.

16.1 The Space of Linear Fractional Oscillators A fractional oscillator is a physical system definable by a fractional differential equation and characterized by sustained oscillations after an initial transient. The Laplace transforms of the fractional meta-trigonometric functions when considered as system transfer functions may yield such sustained oscillations, Lorenzo et al. [84]. The objective of this section is to identify the parameters of the fractional trigonometric functions that cause their transfer functions to behave as fractional oscillators. In other words, the goal is to identify the trigonometric-based transfer functions that are oscillatory and are neutrally stable. Because the oscillators are definable as transfer functions, they will also be linear as well as fractional. Such a transfer function based on the Laplace transform L{Sinq,v (a, 𝛼, 𝛽, k, t)} is given as [ q ] 𝜃o (s) s sin(𝜆) − a sin(𝜆 − 𝜎) = sv 2q , q > v, (16.1) 𝜃i (s) s − 2 a cos(𝜎)sq + a2 and for the L{Cosq,v (a, 𝛼, 𝛽, k, t)}

[ q ] 𝜃o (s) v s cos(𝜆) − a cos(𝜆 − 𝜎) =s , 𝜃i (s) s2q − 2 a cos(𝜎)sq + a2

q > v.

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(16.2)

310

16 Fractional Oscillators

Here, 𝜃o (s) is considered to be a system output variable and 𝜃i (s) is a system input variable. Solving for the highest order output term from equation (16.1) gives s2q 𝜃o (s) = 2 a cos(𝜎)sq 𝜃o (s) − a2 𝜃o (s) + sq+v sin(𝜆)𝜃i (s) − a sin(𝜆 − 𝜎)sv 𝜃i (s), 𝜃o (s) = 2 a cos(𝜎)s 𝜃o (s) − a s −q

v−q

+s

2 −2q

q > v,

(16.3)

𝜃o (s)

sin(𝜆)𝜃i (s) − a sin(𝜆 − 𝜎)sv−2q 𝜃i (s),

q > v.

(16.4)

As a practical consideration, because q>v, 𝜃o (s) can be computed using only fractional integrations based on equation (16.4). Figure 16.1 shows a fractional integration-based block diagram for a Sinq,v (a, 𝛼, 𝛽, k, t)-based system. The objective of the following sections is to determine the arguments for the set of fractional meta-trigonometric functions that define the space of linear neutrally stable fractional oscillators for the fractional meta-trigonometry. 16.1.1 Complexity Function-Based Oscillators

The following results are adapted from Lorenzo et al. [84]: The numerator terms of equations (16.1) and (16.2) do not influence the stability for these transforms, thus the form [ ] 𝜃o (s) K = 2q , (16.5) 𝜃i (s) s − 2 a cos(𝜎)sq + a2 ( ) with 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k is applicable to both the fractional sine and cosine transforms. In Chapter 13, the conditions for neutral stability for the complexity functions were determined in Section 13.9.1 and given as equation (13.36), that is, ( q𝜋 ) , and, 0 < q ≤ 2. (16.6) cos(𝜎) = sgn(a) cos 2 Now, when a is positive, that is, sgn(a) = 1, ( ( q𝜋 )) , 𝜎 = arccos cos 2 and when a is negative, that is, sgn(a) = −1, ( ( q𝜋 )) 𝜎 = arccos cos −𝜋 . 2 Both cases are captured by the equation )) ( ( q𝜋 𝜋 𝜋 . + sgn(a) − 𝜎 = arccos cos 2 2 2 a2 S𝜈–2q θi (s)

asin (λ–σ) – +

S𝜈–q

sin (λ)

–

+

S–q

Figure 16.1 Block diagram for Sinq,v (a, 𝛼, 𝛽, k, t) interpreted as a physical system.

S–q θo (s)

2acos (σ) q>v

16.1 The Space of Linear Fractional Oscillators

For the complexity-based oscillators (equations (16.5) and (16.6)), 𝜎 = (𝛼 + 𝛽q) and with 0 < q ≤ 2 ( ) ( q𝜋 ) 𝜋 𝜋 𝜋 + sgn(a) − + 2k ′ 𝜋 , 𝜎 = (𝛼 + 𝛽q) + 2𝜋k = 2 2 2 2 or (𝛼 + 𝛽q) =

q + sgn(a) − 1 + 4k ′ , k = 0, 1, 2, … , D − 1, 1 + 4k k ′ = 0, ±1, ±2, … , 0 < q ≤ 2,

(

𝜋 2

) + 2𝜋k ,

(16.7)

(16.8)

defines the argument relationships for neutrally stable responses and hence defines the space of linear fractional oscillators for the complexity functions. For a > 0, sgn(a) = 1 and this simplifies to (𝛼 + 𝛽q) =

q + 4k ′ , 1 + 4k

k = 0, 1, 2, … , D − 1, k ′ = 0, ±1, ±2, … , a > 0, 0 < q ≤ 2,

while a < 0 gives (𝛼 + 𝛽q) =

q − 2 + 4k ′ , 1 + 4k

k = 0, 1, 2, … , D − 1, k ′ = 0, ±1, ±2, … , a < 0, 0 < q ≤ 2.

For convenience, alternative expressions are given. These simplify to q + 4k ′ q + 4k ′ 𝛼 − 𝛽q , or 𝛽 = − , 1 + 4k (1 + 4k)q q k = 0, 1, 2, … , D − 1, k ′ = 0, ±1, ±2, … , a > 0, 0 < q ≤ 2, q − 2 + 4k ′ 𝛼 q − 2 + 4k ′ − 𝛽q , or 𝛽 = − , 𝛼= 1 + 4k (1 + 4k)q q ′ k = 0, 1, 2, … , D − 1, k = 0, ±1, ±2, … , a < 0, 0 < q ≤ 2.

𝛼=

(16.9)

(16.10)

For the principal meta-trigonometric functions k = 0, and when k ′ = 0, these results simplify further. 16.1.2 Parity Function-Based Oscillators

Similar arguments apply to transfer functions of fractional oscillators associated with the parity functions (equations (13.17)–(13.22), also Section 9.2). Here again, the numerator terms of these functions do not influence the stability; thus, for these transforms, we consider the form [ ] 𝜃o (s) K = 4q , (16.11) 𝜃i (s) s − 2a2 cos(2𝜎)s2q + a4 ( ) as applicable to all four forms and again where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k . In Chapter 13, the conditions for stability for the parity functions were determined in Section 13.9.2 and given as equation (13.41), that is, the parity function-based oscillators will be neutrally stable when cos(2𝜎) = cos(q𝜋),

0 < q ≤ 1,

or 2𝜎 = arccos(cos(q𝜋)),

0 < q ≤ 1.

(16.12)

311

16 Fractional Oscillators

Thus, 2 (𝛼 + 𝛽q)

(

) 𝜋 + 2𝜋k = q𝜋 + 2k ′ 𝜋 , 2

0 < q ≤ 1,

q + 2k ′ (16.13) , k = 0, 1, 2, … , D − 1, k ′ = 0, ±1, ±2, … , 0 < q ≤ 1, 1 + 4k defines the argument relationships for neutrally stable responses and hence defines the space of linear fractional oscillators for the parity functions. Note that the order variable, q, is limited to values between 0 and 1. For convenience, alternative expressions are given: 𝛼 + 𝛽q =

𝛼=

q + 2k ′ q + 2k ′ 𝛼 − 𝛽q , or 𝛽 = − , 1 + 4k (1 + 4k)q q k = 0, 1, 2, … , D − 1, k ′ = 0, ±1, ±2, … , a > 0, 0 < q ≤ 1.

(16.14)

Figure 16.2 shows selected fractional oscillators based on the results for the complexity function Cosq,v (a, 𝛼, 𝛽, k, t). Figure 16.3 presents a set of oscillators using the parity function Vibq,v (a, 𝛼, 𝛽, k, t). The effect of variations in the index variable, k, for a set of fractional oscillators based on the Flutq,v (a, 𝛼, 𝛽, k, t)-function is presented in Figure 16.4. 16.1.3 Intrinsic Oscillator Damping

The previous sections have studied neutrally stable oscillators; here, we consider briefly damped oscillators. It is interesting to observe that the pseudo (or fractional)-damping of the oscillators associated with the fractional trigonometric functions is fixed when the oscillator parameters are chosen. That is, when q, v, 𝛼, and 𝛽 are selected for any oscillator based on a fractional trigonometric function, the pseudo-damping is determined. To see this, from Chapter 13, the pseudo-damping factor, 𝜁 , for the complexity functions is given by 𝜁 = −sgn(a) cos(𝜎) ,

(16.15)

3

2

Cosq,𝜈(a,α,β,k,t)

312

α = 0.1

1

0 –1 α = 1.0 –2 –3

0

2

4

6 t-Time

8

10

12

Figure 16.2 A set of linear fractional oscillators based on the complexity function Cosq,v (a, 𝛼, 𝛽, k, t), with k′ = 0, 𝛽 from equation (16.9), 𝛼 = 0.1 to 1.0 in steps of 0.1, q = 0.5, v = 0, a = 1.0, and k = 0.

16.1 The Space of Linear Fractional Oscillators

1.5

Vibq,𝜈(a,α,β,k,t)

1

0.5

0 –0.5 –1 –1.5

α = 0.1

0

α = 1.0

5

10

15

t-Time

Figure 16.3 A set of linear fractional oscillators based on the parity function Vibq,v (a, 𝛼, 𝛽, k, t), with 𝛽 from equation (16.14), 𝛼 = 0.1 to 1.0 in steps of 0.1, q = 0.5, v = 0, a = 1.0, k′ = 0, and k = 0. 1.5

Flutq,𝜈(a,α,β,k,t)

1

k=0

0.5

0

k=1

–0.5

–1

0

5

10

15

t-Time

Figure 16.4 Effect of variation of k, for a set of linear fractional oscillators based on the parity function Flutq,v (a, 𝛼, 𝛽, k, t), with 𝛼 from equation (16.14) q = 0.9, v = 0, a = 1.0, 𝛽 = 1∕2, k′ = 0, and k = 0 to 6, in steps of 1.

( ) where 𝜎 = (𝛼 + 𝛽q) 𝜋2 + 2𝜋k . For the oscillators associated with the parity functions, we also have from Chapter 13 𝜁 = − cos(2𝜎) , (16.16) ) ( 𝜋 where again 𝜎 = (𝛼 + 𝛽q) 2 + 2𝜋k . The pseudo-damping factor, of course, determines the length of time required for the oscillator to arrive at its long-term amplitude. The reader interested in this area is referred to [119] and the references therein.

313

314

16 Fractional Oscillators

16.2 Coupled Fractional Oscillators Coupled oscillators have found important application in the study of physical systems (cf. Achar [4)]. It is reasonable to expect that coupled fractional oscillators will similarly find important application to systems exhibiting fractional behavior. The coupling of fractional oscillators is considered here by application to the linear commensurate-order time-fractional diffusion–wave equation. Thus, we consider equations of the form 𝜕2u 𝜕 2𝜆 u 𝜕𝜆u = c + c + c3 u. 1 2 𝜕x2 𝜕t 2𝜆 𝜕t 𝜆

(16.17)

Discretizing in the x dimension (Figure 16.5) will allow this equation to be approximated by a system of (nonpartial) fractional differential equations. Then, the second spatial derivative is approximated by the following second-order central difference approximation: 𝜕 2 u(x, t) u(x + Δx, t) − 2u(x, t) + u(x − Δx, t) = . 𝜕x2 (Δx)2

(16.18)

For the nth increment, we have In = or In =

d2𝜆 u d𝜆 u 𝜕 2 u(n, t) un+1 − 2un + un−1 ≅ = c1 2𝜆n + c2 𝜆n + c3 un , 2 2 𝜕x Δx dt dt

𝜕 2 u(n, t) un+1 − 2un + un−1 𝜆 ≅ = c1 {0 D2𝜆 t un } + c2 {0 Dt un } + c3 un . 𝜕x2 Δ x2

(16.19)

In terms of the highest order derivative, we have the system of N fractional differential equations of the form 2𝜆 0 Dt un

=

1 [I − c2 {D𝜆t un } − c3 un ], c1 n

n = 1, 2, 3, … , N.

(16.20)

Each of these N equations is, of course, a fractional oscillator of (differential) order 2𝜆 and of trigonometric order 𝜆. The coupling of these equations to approximate the solution of time-fractional diffusion–wave equation is illustrated in Figure 16.6. Note that, for clarity of the figure, the oscillator prefilter shown in Figure 16.2 has been omitted. This figure, of course, illustrates only two oscillators, without consideration of driving forces or boundary and initial conditions. However, as the number of oscillators, N, approaches infinity, the size of Δx approaches zero and the approximation approaches the exact solution. The transfer function associated with these oscillators is obtained by taking the Laplace transform of equation (16.19), neglecting initialization, In (s) =

un+1 (s) − 2un (s) + un−1 (s) = c1 s2𝜆 un (s) + c2 s𝜆 un (s) + c3 un (s), Δ x2

ΔX

X

• • • un–1

Figure 16.5 Spatial discretization.

un

un+1 • • •

(16.21)

16.2 Coupled Fractional Oscillators

un–1 – +

un

–

+

un+1

–λ

–λ 0 dt

0 dt

2

Δx

2

Δx

–λ

0 dt

d 2λ t

1/c1 – – + c c3 2 In – +

–λ 0 dt

1/c1 – + c c3 2 In+1 – + –

Figure 16.6 Coupled fractional oscillators. Individual oscillators are indicated by dashed boxes.

or

un (s) 1∕c1 = c . c In (s) s2𝜆 + 2 s𝜆 + 3 c1 c1

(16.22)

These forms are readily analyzed using the fractional meta-trigonometric functions developed in Chapter 12, specifically, equations (12.21), (12.24), (12.27) (12.29) (12.35), (12.38), and (12.40). These results clearly indicate the importance of fractional oscillators and fractional-trigonometric-function-based oscillators in the analysis of physical systems. The results also lead to the expectation that the fractional meta-trigonometric functions will be components of the solution of equations of the form of the time-fractional diffusion–wave equation (16.17). The question one might ask is: what is the benefit of using a fractional oscillator over using an integer-order sinusoidal oscillator, since the fractional oscillator response is ultimately sinusoidal? The primary answer, of course, is that the local oscillation is fractional, leading to the fractional behavior defined by equation (16.17). However, it may be more important in many applications, such as those discussed in the later chapters, that the transient preceding the sustained oscillation is the focus of interest.

315

317

17 Shell Morphology and Growth In Chapter 15, detailed studies of the parametrically defined fractional spirals exposed their geometry and potential application. Spiral behavior is seen in many areas of natural science. In this chapter, we study the applicability of the fractional trigonometric spirals to the morphology and growth of shelled sea animals. The spiral shape is nearly ubiquitous in shelled sea animals. The beauty and large variety of these creatures have been the basis of numerous scientific and common interest studies. These include an encyclopedic book on shells by Dance [25], a natural history study by Vermeij [120], and an excellent X-ray study by Conklin [24], which nicely expose the inner structure of sea shells. Here, a detailed study of the shell of the Nautilus pompilius as well as a simple morphological study of six additional animal shells are presented.

17.1 Nautilus pompilius This section studies the morphology and evolutionary growth of the Nautilus pompilius based on the fractional R1 -trigonometry. This study, Lorenzo [86], was performed prior to the development of the fractional meta-trigonometry. It is likely that the fractional meta-trigonometric functions would show somewhat improved results over those based on the R1 -trigonometry. Morphological models based on the fractional trigonometry are shown to be superior to those of the commonly assumed logarithmic spiral. The R1 -trigonometric functions further infer fractional differential equations (FDEs) which, together with power law parametric functions, will be used to develop a fractional growth equation modeling evolution of the Nautilus from conception to maturity. 17.1.1

Introduction

One of the earliest studies concerning the mathematical morphology of the N. pompilius shell was the classic work of Thompson [118]. Use of the logarithmic or equiangular spiral to model this shell has been forwarded by several authors. From a scientific point of view, there is the book by Thompson [118] and the later effort by McMahon and Bonner [93]. In books to popularize mathematics and its application, there are contributions by Land [61] and Hargittai [43]. Because the Nautilus is considered endangered, there has been significant scientific interest in its growth rate. Intensive studies in this area include Landman et al. [62, 63], Cochran [23], and Westermann et al. [122]. Studies on the fractal nature of the Nautilus [21] have been done by Castrejón et al. The objective here is to create a model for the Nautilus morphology based on the fractional trigonometry or more specifically on the R1 -fractional trigonometry, for comparison to the earlier modeling based on the logarithmic spiral. Beyond the issues of morphology, the fractional The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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17 Shell Morphology and Growth

Table 17.1 Shell dimensions. 𝓵r max. radius mm

Shell number

Lmax maximum dimension (diameter) mm

1

170.5

108.0

2

180.5

112.5

3

174.0

111.0

4

175.0

111.5

Number of septa

Lsp spiral length mm

c recession mm/∘

–

–

33

640.4

n.a. 1 mm/8∘

35

618.0

1 mm/23∘

33

622.5

1.5 mm/25∘

Note: c is the maximum tip recession from plane in mm/number of degrees over which it tapers to zero. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.

trigonometric approach allows modeling of the growth rate of the animal and the creation of a fractional growth rate equation. The shells of four N. pompilius were studied. Three shells were sectioned along the plane of symmetry for use in the morphology studies. A fourth tri-cut shell was used to model the locus of the siphuncles. Details on the photography and measurements used in this study may be found in Lorenzo [86]. The results of these measurements are contained in Table 17.1. The computational and fitting procedures are summarized in the following section on morphology. The following sections are adapted from Lorenzo [86] with permission of ASME. 17.1.2 Nautilus Morphology

Of the four shells studied, shell 1 was used to study the siphuncle morphology. For shells 2–4, logarithmic and R1 -fractional spirals were chosen to best fit the photographed shell spiral morphology. The fitting process was performed manually for all cases. The R1 -fractional spirals are computed based on the following forms: x = ks R 1 Cosq,v (a, btm ),

tm > 0,

y = ks R 1 Sinq,v (a, btm ) ,

tm > 0,

(17.1)

where ks and b are scaling factors determined in the fitting process along with the parameters q, v, a, and tm . Note that tm is the formal parameter of the fractional spiral, and it is a spatial parameter rather than a temporal parameter. Because the fractional trigonometric functions are defined by infinite series and are not periodic as are the circular functions, evaluation for large arguments requires high-precision computations. Thus, the unscaled functions are evaluated in the symbolic computer program Maple . In the Maple code, a minimum of 100 digits of precision was used to allow the functions to be determined for the number of rotations needed to define the shell morphologies. The output of the Maple code is linked to code written in Matlab where the results are scaled by k, and rotated to match the shell orientation in the photographs. The results are then plotted atop the shell image together with the logarithmic spiral that is computed in Matlab . The fitting process for both the R1 -fractional spirals and the logarithmic spirals was done manually. The logarithmic or equiangular spirals are defined by the polar equation

®

®

®

®

®

r = C2 ebl 𝜃 .

(17.2)

17.1 Nautilus pompilius

The results presented are based on the equivalent rectangular parametric forms: xl = C2 cos(𝜃 )ebl 𝜃

and yl = C2 sin(𝜃 )ebl 𝜃 ,

(17.3)

where 𝜃 is the angle of rotation. Thus, only two parameters are required to fit the logarithmic spirals, C2 the logarithmic scaling constant, and bl the logarithmic spiral growth rate parameter. Figures 17.1–17.3 present the results of the morphological fitting of shells 2–4, respectively. For each shell, four image panels are given: (a) with the logarithmic spiral, (b) with the R1 -trigonometric spiral, (c) with both spirals on the same plot, and lastly, (d) various enlarged views of panel (c) to show important inner details. The numerations on the axes of these figures are in pixels and a slightly different (pixel/mm) calibration applies for each figure. The calibration factors for shells 2–4 are 13.30, 13.20, and 13.16 pixels/mm, respectively. The axes of the plots are nominally −ks R1 Cosq,v (a, btm ) for the x-axis and −ks R1 Sinq,v (a, btm ) for the y-axis. The negation is required because spiral rotation of each (half ) shell is clockwise. The axis orientation of each shell is not known prior to the photography and fitting process. Thus, an angular offset is applied during the fitting process for the fractional spirals. These offsets rotate the plots in the clockwise direction. For shells 2–4, the angular offsets were found to be 85∘ , 84∘ , and 90∘ , respectively. Thus, the axes are reversed for shell 4 and nearly so for shells 2 and 3. The morphological results for the three shells showed a remarkable similarity. Curiously, the geometrical centers of the shells as inferred by both the logarithmic and the fractional spirals were displaced (∼1–2 mm) from the physical center of the shell. In all cases, the geometric center was slightly below and to the left of the physical center (i.e., the void center). This can be seen in the enlargement panels of the figures. A common center was found to best fit both the logarithmic and fractional spirals for each shell. The logarithmic spiral parameters for all three shells were found to be C 2 = 75 (in pixels) and growth rate bl = 0.171. The logarithmic spirals with these parameters fit the shell morphologies quite well after approximately a half revolution of the shell spiral (see enlargements, panel d, of Figures 17.1–17.3). With only two parameters, the fitting process for the logarithmic spirals was simple and direct. However, the fit of the logarithmic spiral to the Nautilus morphology became progressively worse approaching the origin from the one-half revolution point. And, of course, the infinite number of rotations around the origin from the half revolution point inward does not match at all. The resulting R1 -trigonometric spiral parameters required to fit the three shells are presented in Table 17.2. As mentioned earlier, the resulting parameters are remarkably similar. The following observations are made. As expected, the parameter q controls the growth rate of the R1 -spirals and the maximum difference in q (growth rate) is only 0.0010 measured against an average q = 1.1220. Furthermore, the fitting process was very sensitive to the magnitude of q. The parameter v was found to be of relatively weak influence and could easily have been selected at v = −0.2 with an attendant adjustment of the scaling factor k. Similarly, because the spiral bar was so small, the quency-related parameter a could have ranged by over an order of magnitude with little apparent change in the form of the fractional spiral; again however, the scale factor k would require adjustment. Thus, the morphological fits are not unique. The parameter b is a scale factor on the logarithmic spiral spatial growth rate and also the increment between the plotted points. Finally, the value of tm (the spatial parametric variable of the fractional spiral) increases from zero at the origin of the plots to a maximum at the tip. The last datum point did not necessarily coincide with the maximum radius point of the shell.

319

320

17 Shell Morphology and Growth

0

0

500

500

1000

1000

1500

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2500

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3000

3000

3500

0

500

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1500

2000

2500

3500

0

(a)

500

1000

1500

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2500

(b)

0

2000

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2050 2100

1000 2150 1500 2200 2000

2250

2500

2300 2350

3000 2400 3500

0

500

1000

1500 (c)

2000

2500

1400 1450 1500 1550 1600 1650 1700 (d)

Figure 17.1 Shell number 2. Morphological models: (a) Logarithmic spiral, (b) R1 -trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

17.1 Nautilus pompilius

0

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1000

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2500

3000

3000

3500

0

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0

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(b)

(a) 1800

0

1900

500

2000 1000 2100 1500 2200 2000

2300

2500

2400 2500

3000

2600 3500

0

500

1000

1500 (c)

2000

2500

1200

1300

1400

1500

1600

1700

1800

(d)

Figure 17.2 Shell number 3. Morphological models: (a) Logarithmic spiral, (b) R1 -trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

321

322

17 Shell Morphology and Growth

0

0

500

500

1000

1000

1500

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2500

2500

3000

3000

3500

3500 0

500

1000

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0

(a)

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(b)

0 2000 500

2050

1000

2100 2150

1500 2200 2000 2250 2500

2300 2350

3000

2400 3500

0

500

1000

1500 (c)

2000

2500

1350 1400 1450 1500 1500 1600 1650 (d)

Figure 17.3 Shell number 4. Morphological models: (a) Logarithmic spiral, (b) R1 -trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

17.1 Nautilus pompilius

Table 17.2 Parameters for morphological fit using R1 -trigonometric fractional spirals. Offset (∘ )

tm at max. radius

0.180 0.01353

85∘

94

2 × 10−7

0.167 0.01265

84∘

93

2 × 10−7

0.169 0.01284

90∘

91

Shell no.

Figure no.

b

q

v

a

k pix mm

2

17.1

200,000

1.1215

−0.3

2 × 10−7

3

17.2

200,000

1.1220

−0.3

4

17.3

200,000

1.1225

−0.3

Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.

17.1.2.1 Fractional Differential Equations

The morphological data presented in Table 17.2 may be used together with the Laplace transforms of Table 6.1 to determine the FDEs representing the Nautilus fractional spiral morphology. It may be readily shown that R1 Cosq,v (a, bt) = bq−1−v R1 Cosq,v (abq , t) and R1 Sinq,v (a, bt) = bq−1−v R1 Sinq,v (abq , t). Thus, the simultaneous FDEs representing the morphology (in pixels) are

and

2q 0 Dt x(tm )

+ (abq )2 x(tm ) = ks bq−1−v 0 Dt 𝛿 (tm ),

tm > 0

(17.4)

2q 0 Dt y(tm )

+ (abq )2 y(tm ) = ks a bq−1−v 0 Dvt 𝛿 (tm ),

tm > 0,

(17.5)

v+q

where 𝛿 (tm ) is the unit impulse function. The scaling and functional form of tm must be determined from additional data (see Section 17.1.5). Note that the order of the FDE is twice the trigonometric order. The solutions to equations (17.4) and (17.5) are x(tm ) = ks bq−1−v R1 Cosq,v (abq , tm ),

tm > 0,

(17.6)

R1 Sinq,v (ab , tm ),

tm > 0.

(17.7)

y(tm ) = ks b

q−1−v

q

The R1 -fractional spiral fits were viewed to be excellent over nearly the entire range of rotation, from the origin to the point of maximum radius. Of course, Nautilus, similar to all natural creatures, have differing degrees of perfection of form, and minor deviations are found. It is observed that the logarithmic spiral and the R1 -fractional spiral become (approximately) asymptotic to each other after about one-half to one revolution of the fractional spiral for all cases studied. It can be seen that near the shell origin, the logarithmic spiral continues to rotate and does not match the morphology of the Nautilus and has an infinite error relative to the actual shell morphology! The maximum deviation of the fractional spiral from the shell spiral occurred over the range from the origin to the next plotted point; see Figures 17.1d and 17.3d. The logarithmic spiral, of course, spirals forever around the mathematical origin with ever decreasing radius. Clearly, the logarithmic spiral cannot represent the Nautilus over this important range and thus cannot be used as a basis for further detailed scientific study of its evolution. Notice that the two FDEs are independent of each other except for the common synchronizing impulse that starts the dynamic process. The addition of cross-coupling terms to equations (17.4) and (17.5) may allow improved modeling near the origin but is beyond the scope of this study. It is of interest to take a detailed view of the R1 -fractional spirals used to fit the shells. The data shown in Figures 17.4 and 17.5 apply to shell 4, which is taken as typical. It is in the range

323

17 Shell Morphology and Growth

1500 Radii

Spiral components (pixels)

1000

500

0 –500 y-Components –1000 x-Components –1500

0

5

10 Θ (radians)

15

20

Figure 17.4 Shell 4, x, y, and radius components of R1 -trigonometric spiral shown in Figure 17.3b versus fractional spiral rotation Θ in radians. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure. 350 300

Spiral radius (pixels)

324

250 200 150 100 Logarithmic spiral 50 0 –10

Fractional spiral –5

0 Θ (radians)

5

10

Figure 17.5 Expanded view spiral radii (in pixels) for logarithmic and fractional spirals versus spiral angle theta (in radians). Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure.

tm < 12, where the important deviation from the logarithmic spiral occurs. This deviation from exponential behavior is also seen in the plot of radius versus tm . Figure 17.4 shows the x and y components of the fractional spiral shown in Figure 17.3b. In the figure, the components are scaled to match those of Figure 17.3. The angular rotation of the spiral required to match the image is accounted for. It is also noted that the y component is proportional to R1 Cosq,v (a, btm ) and the x component is proportional to R1 Sinq,v (a, btm ). Figure 17.5 shows the spiral radius

17.1 Nautilus pompilius

(in pixels) for the logarithmic and fractional spirals versus spiral angle theta (in radians). In Figure 17.5, exponential behavior of both the logarithmic and the R1 -spirals for theta greater than approximately 2.5 radians is seen. It is in the range tm < 12, where the important deviation of the logarithmic spiral from the actual morphology occurs. This deviation from exponential behavior is also seen in the plot of radius versus tm . Behavior in this range is important to possible understanding of the conceptional period of shell growth.

17.1.3

Spiral Length

The length of an arc expressed in parametric form is given by (equation (15.9)) tf

Ls (tm ) =

∫t i

((

dx(tm ) dtm

(

)2 +

dy(tm ) dtm

)2 )1∕2 dtm .

(17.8)

The relationships dtd R1 Cosq,v (a, t) = R1 Cosq,v+1 (a, t) and dtd R1 Sinq,v (a, t) = R1 Sinq,v+1 (a, t) from equations (6.36) and (6.40) are applied to an R1 -spiral in the form of equation (17.1) to provide the length from origin to tip as Ls (tm ) = ks bq−1−v

tf

∫0+

((R1 Cosq,v+1 (abq , tm ))2 + (R1 Sinq,v+1 (abq , tm )2 )1∕2 dtm .

(17.9)

This equation was numerically integrated for each of the shells, using the various parameters as provided in Table 17.2, to give the computed lengths for shells 2–4, respectively, as 658.1, 612.3, and 596.0 mm with a maximum error of 4.3% of the measured results indicated in Table 17.1. The singularity of the R1 Cosq,v+1 (a, t) was avoided by starting the integration at tm = 0+, the integration was stopped at the datum point corresponding to the largest radius; see, for example, Figure 17.4.

17.1.4

Morphology of the Siphuncle Spiral

The morphology associated with the locus of siphuncles of shell 1 is considered in this section. To expose most of the siphuncles, a shell sectioned with two lateral cuts created a 14 mm thick slice that contained the center portion of the shell used for this study. Figure 17.6a shows an image of the shell slice without modeling data. The calibration factor for this image is 13.03 pixels/mm. Table 17.3 contains the parametric constants used for the R1 -trigonometric spiral seen in Figure 17.6b–d. The enlargement shown in panel c indicates a high-quality match between the model and the locus. The dark triangles shown in the further enlargement (panel d) extrapolate the model inward presumably modeling the inner siphuncles that are hidden by the outer shell of the animal in this cut. Notice that the siphuncle locus has a more rapid divergence than seen for the main spirals of shells 2–4. This is validated by the fact that the order parameter q for the siphuncle is q = 1.1375 as opposed to q ≈ 1.1220 for shells 2–4.

17.1.5

Fractional Growth Rate

There has been considerable interest in the growth rate of the Nautilus. Landman et al. [63] have used radiometric methods to determine the apertural growth rates in

325

326

17 Shell Morphology and Growth

0

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0

500

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(b)

(a) 1700 1400 1800 1600 1900 1800 2000 2000 2100 2200 2200 2400 2300 2600 2400 2800 600

800

1000

1200 (c)

1400

1600

900

1000

1100

1200

1300

1400

(d)

Figure 17.6 Shell number 1. Siphuncle morphological models: (a) Image without model, (b) R1 -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. Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

17.1 Nautilus pompilius

Table 17.3 Parameters for morphological fit of shell 1 siphuncle locus using the R1 -trigonometric fractional spirals. Shell no.

Figure no.

b

q

v

a

k pix mm

1

17.6

200,000

1.13575

−0.3

2 × 10−7

0.12 0.00921

Offset (∘ )

tm at max. radius

−120∘

59

Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.

immature N. pompilius of 0.19–0.31 and 0.12 mm/day in submature animals. Cochran and Landman [23] report apertural growth rates of 0.07–0.14 mm/day for Nautilus belauensis for growth between the last two septa for the mature animal and a minimum of 10 years to reach maturity. Westermann et al. [122], using X-ray techniques to study the shell growth of the N. pompilius, indicate apertural growth of 0.11–0.18 mm/day for juvenile and early adolescents and estimates 2673 days (7–8 years) to reach maturity. Detailed analysis of these results and more are beyond the scope of this work. The motivation of these studies has been to develop an understanding that will contribute to the survival of this last existing cephalopod. This study attempts to use the results of these investigators to forward a possible growth model for the Nautilus. As a crude guideline for the analysis to follow, we arbitrarily accept growth rates of 0.20–0.40 mm/day for immature Nautilus at age of about 2–3 years and 0.10–0.14 mm/day for mature Nautilus at maturity of 8–10 years. In this section, we consider the issue of growth of the Nautilus as inferred by fractional spiral morphology. The parametric fractional trigonometric functions that morphologically match the Nautilus spirals contain the formal spatial parameter tm . It is clear that tm can be virtually any continuous function of actual time t and still match the morphology. The problem is to determine tm = f (t) to fit the constraints. The fractional morphology of the Nautilus is expressed as the pair of scaled simultaneous equations (equation (17.1)). This description differs from that of the logarithmic spiral in that its basis derives from the fundamental FDE (Hartley and Lorenzo [45]). Might the dynamic differential equations associated with the fractional spirals in fact define the evolution of the Nautilus shell and by implication the animal growth? Now, equation (17.9) tmf (( )2 )1∕2 )2 ( R1 Cosq,v+1 (abq , tm ) + R1 Sinq,v+1 (abq , tm dtm Ls (tm ) = ks bq−1−v ∫tmi = 0+ (17.10) provides an expression for the scaled length of an R1 -fractional spiral in pixels. We now let tm = f (t) ⇒ dtm = f ′ (t) dt then t = f −1 (tm ) ⇒ tf = f −1 (tmf ) and ti = f −1 (tmi ). Then, Ls (t) = ks bq−1−v

tf

∫ti = 0+

(( )2 ( )2 )1∕2 ′ R1 Cosq,v+1 (abq , f (t)) + R1 Sinq,v+1 (abq , f (t) f (t)dt. (17.11)

The growth rate of the spiral is the derivative of this expression, namely (( )2 ( )2 )1∕2 ′ Gr (t) = ks bq−1−v R1 Cosq,v+1 (abq , f (t)) + R1 Sinq,v+1 (abq , f (t) f (t) , t > 0. (17.12)

327

17 Shell Morphology and Growth

To determine f (t) to satisfy the observed growth rate constraints of the Nautilus, several linear, exponential, and logarithmic forms were considered for f (t). A power law relation for tm was found to reasonably satisfy the constraints, namely tm = km t u ⇒ dtm = km u t u−1 dt, and f ′ (t) = Then, Gr (t) = ks km u t u−1 bq−1−v

((

dtm = km ut u−1 . dt

R1 Cosq,v+1 (abq , km t u

)2

)2 )1∕2 ( + R1 Sinq,v+1 (abq , km t u ,

(17.13) (17.14)

t > 0. (17.15)

Scaling this result to millimeters, and selecting km = 16.4 and u = 0.211, the growth rate versus tm is presented in Figure 17.7. The final point on the graph corresponds to t = 3650 days or approximately 10 years. At that point, the growth rate is 0.13 mm/day. The growth rate when t = 300 days is approximately 0.2 mm/day. It is noted that t = 0 relates to the time of conception of the Nautilus and the times estimated here relate to that starting point. Thus, the beginning of life is seen to be, both literally and mathematically, a singular event. The power law form appears to be reasonable model for the Nautilus evolution. Clearly, the mapping tm = km t u from equation (17.13), together with morphological equations (17.4) and (17.5), then represent a complete inferred dynamic description of the evolution of the Nautilus. Of importance beyond this particular application is the fact that the aforementioned process provides a possible methodology for converting fractional trigonometric morphological information into a dynamic model using auxiliary data. 17.1.6 Nautilus Study Summary

The morphology and growth rate of the N. pompilius has been studied based on the R1 -fractional trigonometry. Very good fits of the Nautilus spiral were shown for the three Figure 17.7 Shell number 4, Growth rate in mm/day versus t – time (days), (tm = km tu = 16.4 t0.211 ). Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.

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shells studied. The fits did not appear to be unique in that various ranges of some of the parameters would allow equivalent quality fits of the Nautilus spiral. Beyond roughly one-half revolution of the shell’s spiral, the logarithmic and fractional spiral fits were both very good. However, the morphology fits of the shells using the fractional trigonometric spirals were superior to those of the logarithmic spiral in that they matched the much more closely near the origin and allowed the modeling of growth rate. The growth rate associated with the evolution of the Nautilus was also studied. It was found that a power law relationship between the morphological parameter tm and actual time t provided a reasonable approximation to observed growth rates. An important feature of the growth rate model is that it provides growth estimates from conception to maturity. A fractional growth rate equation (equation (17.12)) based on the R1 -fractional trigonometry was determined. A further generalization of the fractional growth equation (17.15) is easily obtained by replacing the R1 -trigonometric functions by the more general the fractional meta-trigonometric of Chapter 9. Because of the importance of growth rate of agricultural animals and because of the occurrence of spiral (and helical) morphology in other animals, plants, and microbiological life forms, the breadth of application of such an equation may be very wide. This study is considered preliminary in that superior fits of the morphology may be possible with the more general fractional meta-trigonometry of Chapter 9. Also, the inclusion of coupling terms between the x(t) and y(t) terms may further improve modeling near the origin. In addition, more detailed analysis of the tm = f (t) relationship with observed growth may yield superior growth rate relationships.

17.2 Shell 5 Super Family: Strombacea Family: Strombidae Genus: Strombus alatus Common name: Florida fighting conch In this and the following sections, the morphology of a variety of shells is studied. The identifications of the shells from this point forward have been made on the image based approach of Dance [25], in which only the characteristics of the shell are considered, ignoring the soft anatomy of the animal (see also Harasewych [41]). They are fitted with parametrically defined fractional meta-trigonometric spirals defined by x = K Cosq,v (a, 𝛼, 𝛽, k, 𝜇), y = K Sinq,v (a, 𝛼, 𝛽, k, 𝜇) .

(17.16)

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( ) ( ) where 𝜎 = (𝛼 + 𝛽q) 𝜋2 , 𝜆 = 𝛽[q − 1 − v] 𝜋2 and k = 0. Note that the right sides of equations (17.17) and (17.18) may be significantly simplified by using selected Laplace transforms from Section 12.4. Rather than repeat these FDEs for these shells, we simply list the defining parameters for each case which in turn define the constants in equations (17.17) and (17.18). The reader should also be aware that the spiral plots are x and y biased and rotated as required to align with the images. Shell 5 is identified as a Florida fighting conch based on its knobbed spire and coloration; however, the thin lip does not match that species which has a much thicker lip (see Figure 17.8f and g). The shell has been cut to expose an inner spiral; this cut, perpendicular to the axis of coiling, is shown in Figure 17.8a. The prominent spire viewed from the side is also seen here. The spire as viewed from above is shown in Figure 17.8b and c. In panel c the projected spiral, in pixels, is fitted with the parameters q = 1.055, v = −0.400, a = 1.5, 𝛼 = 0.910, 𝛽 = 0.09, k = 0, K = 34.8, 𝜇 = 0 · · · 13. The calibration factor for this image is 391.0 pixels/cm. Note in panel c, because of the many rotations of the physical spiral, the fractional spiral (center) is extended by a logarithmic spiral after approximately three rotations (see numerical issues 14.3). This is more clearly seen in the magnified views of panels (d) and (e) where the dark gray data points are the logarithmic extension. A few overlapping points are shown. Further note that at the origin (panel d), it appears that two spirals start the growth process and then merge into a single spiral form. The logarithmic spiral is described by x = C2 cos(𝜆)eb𝜆 and y = C2 sin(𝜆)eb𝜆 ,

(17.19)

with C2 = 78 and b = 0.069 and appropriate bias and rotation. Of greater interest for this shell is the body whorl (volume below the spire). The cut through this volume exposes the spiral shown in Figure 17.8f and g. This spiral is well fit with the parameters q = 1.16, v = −1.0, a = 2.5, 𝛼 = 0.850, 𝛽 = 0.14333, k = 0, K = 135, 𝜇 = 0, … , 47. The calibration factor for this image is 480.4 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth. The reason for this is unexplained. However, this behavior is also seen on the remaining shells. As mentioned earlier, the cuts through the body whorl were perpendicular to the axis of coiling, slightly different results would likely be obtained if the cut were made parallel to the inclination of the final rotation of the spire. Finally, it is interesting to note that a spiral with approximately 10 rotations exists for the spire spiral and one with approximately 2 3∕4 rotations fills nearly the same space in the body whorl leading to the conclusion that spiral growth is fundamental to these creatures.

17.3 Shell 6 Family: Conus Genus: tinianus or vexillum This shell (Figure 17.9) is characterized by its shallow spire and rounded shoulder. The coloration seen in this specimen is different from that seen in reference books (Dance pp. 214–215). The spiral of the spire is indistinctly defined and, therefore, not modeled. However, it appears to have approximately six or more rotations.

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The cut through the body whorl exposes the spiral shown in Figure 17.9c and d. This spiral is well fit with the parameters q = 1.089, v = −0.2, a = 4000, 𝛼 = 0.995, 𝛽 = 0.02381, k = 0, K = 1660, 𝜇 = 0, … , 47. The calibration factor for this image is 480.4 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth.

17.4 Shell 7 Super Family: Cypraeacea Family: Cypraea Tigris Genus: Cypraea linnaeus Common name: Tiger Cowry This is the first of two different-sized cowries to be studied, Figure 17.10. Viewed from the outside, the shape of these shells does not suggest a spiral structure. Figure 17.10a shows the shell of the larger cowry with a cut perpendicular to its length. The cut through the body exposes the spiral shown in Figure 17.10c and d. This spiral is well fit with the parameters q = 1.150, v = −0.1, a = 1, 𝛼 = 0.910, 𝛽 = 0.090, k = 0, K = 147, 𝜇 = 0, … , 12.6. The calibration factor for this image is 300 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth. Figure 17.10c shows the fractional spiral fit to the shell cross section, and Figure 17.10b provides a reference without the analytical result. Figure 17.10d gives a magnified view of the fit near the origin.

17.5 Shell 8 Super Family: Strombacea Family: Strombidae

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Genus: Tibia insulaechorab Common Name: Arabian Tibia The lip of this shell (Figure 17.11a) is normally thicker and has several blunt projections, Dance [25] pp. 76–77. However, the shell has been damaged and shows possible attack wounds. This shell has been fit with two fractional spirals: the inner spiral beginning at the origin and an outer spiral starting a common point (black circle), which ends the inner spiral and becomes the origin of the outer spiral. Panels (b) and (d) show the internal spiral structure with the common point shown on the magnified view. Panels (d) and (e) show the inner and outer spirals, respectively.

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The outer spiral is well fit with the parameters q = 1.089, v = −0.4, a = 1.2, 𝛼 = 0.630, 𝛽 = 0.380, k = 0, K = 500, 𝜇 = 0, … , 7.9. The inner spiral is well fit with the parameters q = 1.18, v = −0.4, a = 1.2, 𝛼 = 0.950, 𝛽 = 0.050, k = 0, K = 35, 𝜇 = 0, … , 4.0. The calibration factor for this image is 374 pixels/cm.

17.6 Shell 9 Super Family: Cypraeacea Family: Cypraea Tigris Genus: Cypraea linnaeus Common name: Tiger Cowry This is the second and smaller of the cowries studied. Figure 17.12a shows the shell of the smaller cowry with a cut perpendicular to its length. The cut through the body exposes the spiral shown in Figure 17.12b and c. This spiral is well fit with the parameters q = 1.185, v = −0.1, a = 1.0, 𝛼 = 0.910, 𝛽 = 0.113, k = 0, K = 170, 𝜇 = 0, … , 11.0. The calibration factor for this image is 368 pixels/cm. Observe that, again, there is an inward closure of the shell away from the fractional spiral near the end of growth. It is important to note that in spite of the size change of approximately 0.5×; the parameters are nearly identical with those of Shell 7. Figure 17.12b and c shows the fractional spiral fit and a magnified view. Evidence of some chipping, from the cut, is seen in the magnified view.

17.7 Shell 10 Super Family: Cardiacea Family: Cardiidae 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.

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Genus: Trachycardium magnum or Acrosterigma burchardi It is of some interest to know if the shells of bivalves, which differ greatly from the shells previously studied, showed any evidence of fractional geometry. Figure 17.13a–c shows the top view of the uncut and cut shell and side view of cut shell. The side view (Figure 17.13d) and the magnified view (Figure 17.13e) show an excellent fit of a fractional meta-trigonometric

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spiral defined by the parameter set q = 1.90, v = 0.0, a = 1.5, 𝛼 = 0.910, 𝛽 = 0.113, k = 0, K = 147, 𝜇 = 0, … , 43. The calibration factor for this image is 347.6 pixels/cm. The important difference between this shell and the predecessors is the relatively large value of the fractional trigonometric order parameter q, which defines the spiral divergence rate. Clearly, fractional trigonometrically defined geometry also applies to the shell of this creature.

17.8 Ammonite

17.8 Ammonite An ammonite is the fossil shell of a sea creature (cephalopod) likely dating back to the Mesozoic age. The ammonite studied here (Figure 17.14) was found in Madagascar and was estimated to be about 100,000 years old. Unfortunately, there was some grinding of the outer perimeter, possibly 1–2 mm in places, presumably to remove sharp edges of the fossil. Figure 17.14b and d shows a very good fit using the fractional meta-trigonometric functions q = 1.116, v = 0.116, a = 0.23, 𝛼 = 0.83, 𝛽 = 0.17, k = 0, K = 53.2, 𝜇 = 0, … , 98. 0

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The calibration factor for these images is 125.0 pixels/cm, and the spiral has been rotated 32∘ to align with the image. This ammonite fossil appears to be quite similar to the Nautilus, and it is interesting to note that the ammonite has 75 clearly defined septa, while the N. pompilius studied in Section 17.1 had only 33–35. However, the growth rate parameters were quite similar, with for the Nautilus and for the ammonite. The fractional meta-trigonometric functions used here allowed a very good fit over much of the spiral.

17.9 Discussion With the need to fit six parameters for each fractional spiral fit, the fidelity of the manually fit fractional spirals is far from optimal. Several hours were dedicated to each fitting. It is likely that greatly improved fidelity may be achieved with either more dedicated time or application of some automated optimization process. It seems clear that the geometric form of these creatures (shells) is well defined by the fractional trigonometric spirals. As mentioned previously, the cuts made on the shell were approximately perpendicular to the axis of coiling. It would be of significant interest to study the spirals that would occur if the cuts were made through the body whorl but parallel to the last spire whorl. Surprisingly, even the shell of the Cowry is internally of spiral form, while it does not appear to be so when viewed externally. An important common feature of all, but shell 10, is that the spiral growth rate as indicated by the value of q, is found to be limited to the narrow range of 1.089 ≤ q ≤ 1.185. Does this indicate some optimum in form for strength or efficient use of space? Or, does it possibly indicate the limits of nourishment available to support the growth rate? Finally and quite importantly, for the Nautilus, we have seen that when the temporal growth can be related to the spiral geometry, dynamic fractional growth equations may be inferred.

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 18.1 Introduction Knowledge of the galaxies and astronomical bodies and their behavior has long been a human striving. In particular, the spiral galaxies have held our attention since their discovery. It has long been our desire to understand and codify their shapes and evolution. This chapter studies the application of the fractional trigonometry to the mathematical definition of the morphology of the spiral and ring galaxies. This chapter is adapted from Lorenzo and Hartley [89] with permission of the ASME: Perhaps the most fundamental knowledge that we can have about an object or collection of objects is its morphology, that is, its size, shape, and dimension. In most instances, this is the first differentiator between objects. The more accurately or completely this information is known the better we can proceed to more defining issues of kinematics, dynamics, and evolution. This chapter studies the morphology of the spiral and ring galaxies based on the fractional meta-trigonometric spiral functions. There have been numerous classifications proposed and used since that proposed by Hubble [52] in 1936. However, to the knowledge of the authors, there are no classifications of the galaxies that provide a mathematical description of the morphology of the spiral arms. Hopefully, the work to be presented here augments and helps remove some of the subjectivity of the existing morphological classifications. Van den Bergh [12] has helpfully provided coverage of many galactic classifications including those of Hubble, de Vaucouleurs, and Elmegreen. In an earlier work, Sandage et al. p. 28 [115], references five classification schemes of galaxies that are obtained from direct photography. These are the various modifications and extensions of the Hubble classification, by Pettit [108], Holmberg [53], de Vaucouleurs [29], van den Bergh [13–16], and Morgan [100, 101]. The classification by Hubble [52] was originally based on the so-called “tuning fork diagram” (Figure 18.1). Basically, galaxy morphology is divided into elliptical and spiral shapes with the spirals further divided into normal and barred types, which are further subdivided into classes based on the openness of the spiral arms. A circular type, the S0 class, provided transition between the elliptical and the spiral types. This useful classification method was further refined by de Vaucouleurs to include r and s types, which describe the manner in which the spiral arms emanate from the end of the bar. For the r type, the arms emanate tangential to a ring at the end of the bar, and for the s type, the arms emanate more or less radially from the ends of the bar. The classifications of van den Bergh and Morgan introduce luminosity effects and along with many later classification schemes provide useful information to the understanding The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies

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Figure 18.1 The morphological classification of the galaxies as suggested by Edwin Hubble. Source: Hubble 1936 [52]. Reproduced with permission of Yale University Press.

of the nature of the galaxies. Because this study will deal exclusively with purely morphological issues, we will not delve further into these and various other galaxy classifications. In the last decade or so, computer-based studies identified as Mathematical Morphology (MM) have been forwarded by Candeas et al. [20], Moore et al. [99], Baillard et al. [9], and Aptoula et al. [8]. These studies are computer-based analyses of astronomical images for star/galaxy differentiation and for automatic galaxy classification Baillard et al. [9], and Aptoula et al. [8]. These capabilities are quite important but are also very different from the mathematical classification to be presented here. The study that follows originates from Lorenzo and Hartley [89]. In a 1981 paper, Kennicutt [57] attempted to mathematically describe the shape of galactic spiral arms based on the logarithmic spiral, which is linear in log r versus 𝜑, and the hyperbolic spiral, which is linear in r𝜙 versus r. He concluded: “It quickly became apparent that neither form precisely represents the arms, even the most regular ones.” The work described here ascribes mathematically defined fractional barred and apparently unbarred (normal) spirals, based on the fractional trigonometric functions, to the description of the galactic spiral arms. That is, we work to analytically define the galaxy spiral arm morphologies. The classification methods suggested in this study hopefully augment those currently in use or possibly be implemented to work with some of the existing computer identification schemes. A primary objective of this study is to bring to the attention of those members of the astrophysical community dealing with galactic morphology, the existence of the fractional trigonometry and its potential for describing galactic spiral and ring morphologies, and its fundamental mathematical character.

18.2 Background – Fractional Spirals for Galactic Classification In Chapters 9 and 15, a variety of fractional spirals associated with the fractional meta-trigonometric functions was presented. In this section, fractional spirals of particular interest relative to galaxy classification are presented, and some important relevant behaviors discussed. For the initial spiral, we have Sinq,v (a, 𝛼, 𝛽, k, t) versus Cosq,v (a, 𝛼, 𝛽, k, t) shown in Figure 18.2, here, the effect of the primary order variable q ranges from q = 1.0–1.8, with q = 1.0 defining the unit circle. Clearly, as the variable q increases the growth rate of the spiral increases.

18.2 Background – Fractional Spirals for Galactic Classification

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Note, in Figures 18.2–18.8, as the parameter t increases, the spiral arms diverge either from the origin or the unit circle. Furthermore, t here is the formal parameter of the spirals and does not necessarily represent time since the resulting spirals are considered to be spatial morphologies. In Figure 18.3, another phase plane plot of Sinq,v versus Cosq,v , the value of 𝛼 is varied to show its effect with q = 1.2, v = 0.2. Here, barred spirals result with the rate of divergence of the spiral arms decreasing as 𝛼 increases. These changes effectively start from the end of the bars. That these spirals are barred and are based on this fundamental mathematics should be cause for interest in astrophysical application. Importantly, the Laplace transforms of the fractional meta-trigonometric functions defining fractional differential equations in space may be easily obtained. In Figure 18.4, we use the 𝛼 = 0.8 spiral from Figure 18.3 as the basis of a study of the effect of the secondary order parameter v. In the figure, v is varied from −0.3 to 0.2 in steps of 0.1. The base spiral, v = 0.1, jumps immediately from the origin to the value of 1.0, making it a barred spiral. It is observed that as v is decreased from the value of v = 0.2, the spiral transitions from a hard barred spiral progressively toward a soft barred, or normal type of spiral. That is, the transition to the spiral arms is softened. From Figure 18.4, it can be seen that the choice of the v parameter may control the location of a spiral from being on the SB tine or the SA tine (or anywhere in between) relative to the Hubble tuning fork. The effect of the v variable also applies to the parity-based spiral functions. A family of barred spirals is shown in Figure 18.5 based on variation in the order variable q and enabled by the relationship v = q − 1. The parity functions may also be used as the basis for spirals of possible application to galaxy morphology as shown in Figures 18.6–18.8. Figure 18.6 studies Flutq,v versus Cof lq,v . Here, it may be seen that increasing 𝛽 increases the divergence rate of the spiral arms while q and v are held constant at q = 0.5, v = 0.0, respectively.

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A different set of barred spirals is shown in Figure 18.7 where the parity functions Vibq,v versus Covibq,v form the spiral basis with q, v, and 𝛽 held constant and 𝛼 varied. The divergence of these spirals increases as 𝛼 increases. The acute divergence angle from the end of the bars may remove these spirals from interest in astrophysical application. The following sections are adapted from Lorenzo and Hartley [89] with permission of ASME:

18.2 Background – Fractional Spirals for Galactic Classification

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This section has introduced some of the aspects of the fractional trigonometry as it applies to galactic morphology and has shown some of the infinite number of fractional spirals that may be applicable to the classification of spiral galactic morphologies. It is important to note that the fractional trigonometric functions provide, or are elements of, the solutions to linear commensurate, constant-coefficient, fractional

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differential equations as shown in Chapter 12. This, of course, includes the linear constant-coefficient ordinary differential equations as a subset. Thus, if a meaningful mathematical classification of the spiral (and ring) galaxies based on the fractional trigonometric functions can be created, defining fractional differential equations in space can be inferred, from which there is the potential of deeper analytical understanding of the galactic processes.

18.3 Classification Process

18.3 Classification Process Our intention here is to provide a purely mathematical description of the spiral arm morphology. That is, we exclude any attention to the luminous intensity or the spectral qualities of the galaxy nucleus or spiral arms. The possibility of morphological matching of an image in a particular spectral frequency that better displays the spiral geometry is not precluded. In fact, composite images containing a variety of wavelength information may best describe the morphological features of interest. It is noted that the morphology or arm location may change slightly at different wavelengths. In general, only galaxies with well-defined spiral arms or rings are considered in this initial study. 18.3.1

Symmetry Assumption

For two-armed spiral galaxies, the initial assumption for the mathematical fitting process will be that the spiral arms are (approximately) symmetrical. In the situation where the symmetrical model fits the arms but the center of symmetry does not align with the optical galaxy center, the offset is reported. For the most part, for this initial effort, we avoid galaxies with interacting or tidal effects. Three-armed or multiarmed spirals occasionally occur, they may be treated as n-single arms or if possible with n-symmetric arms. Out of plane or warped disk galaxies are not considered initially. In general, a symmetric form is preferred, but the arms are defined individually if necessary.

18.3.2 Galaxy Image to Spiral or Spiral to Galaxy Image

Two approaches to the morphological matching of mathematical spiral to the galaxy image are possible. First, the image of the galaxy may be manipulated to a “normal” view and then compared with the mathematical spiral. The second approach does the manipulations with the mathematical spiral. That is, the mathematical spiral is corrected for inclination by correcting either the x- and/or y-axis to achieve the proper morphology, and then rotated in-plane and scaled to match the image spiral. The approach used here is to manipulate the mathematical spiral and superimpose it on the unaltered galaxy image. 18.3.3

Inclination

The galaxies do not usually present themselves to us in face-on spirals and we must account for the inclination of the plane of the galactic disk relative to the plane that is normal to our line of sight. Two approaches are possible: first published values for the inclination may possibly be used to correct the spiral for comparison to the galaxy image; alternatively, the spiral model may be rotated by some amount (the inclination) and matched to the galaxy image and thereby infer the inclination. This later approach is used in this study. Consider the galactic plane, that is, the plane in which the disk is located. It can be brought into alignment with the plane normal to the line of sight by two rotations, one around each of the x or y axes of the normal plane. Each rotation is defined by the reciprocal of the cosine of the angle through which each axis of the plane must be rotated. Thus, it is clear that, in general, two datum are required to define or correct for the inclination. In common usage, only a single datum the inclination is quoted. Inferred is the line of nodes about which the inclination is measured or corrected.

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The following discussion describes how the issue of galactic inclination is treated in this study. Figure 18.9a shows a face-on fractional spiral defined by y0 = Sinq,v (a, 𝛼, 𝛽, k, t) and x0 = Cosq,v (a, 𝛼, 𝛽, k, t) but could also be a spiral based on the parity functions. The result of a rotation of 40∘ about y0 and 65∘ about the x0 axes is indicated by Figure 18.9b and c, respectively. The result of further rotations of 65∘ about x1 and 40∘ about the y3 axes are plotted in Figure 18.9d indicating that the order in which the rotations occur is not important. Finally, the spiral in x4 , y4 plane is rotated in-plane to yield a spiral that may be scaled and translated into position for comparison with the galactic image. 18.3.4

Data Presentation

The basis functions used for mathematical classification in this study are x0 = C • Cosq,v (a, 𝛼, 𝛽, k, 𝜂) and y0 = C•Sinq,v (a, 𝛼, 𝛽, k, 𝜂), 𝜂 > 0.

(18.1)

where 𝜂 is a spatial parameter and C is a scaling constant. Further for simplicity, we have taken a = 1 and k = 0 limiting consideration to only a subset of the principal functions. In a temporal context, the parameter a is a fractional generalization of the natural frequency; thus, its primary effect is scaling the curves with minor effect on morphology. The scaling may be compensated

18.3 Classification Process

by the choice of C. Variations in a, however, may allow some improvement in the modeled morphology. To account for the apparent inclination of the galactic plane, the following corrections are made in the fitting program: x1 = cos(𝜓) x0 , correction for x inclination,

(18.2)

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(18.3)

where 𝜓 and 𝜙 are in degrees. Then, the x1 , y1 coordinate system is rotated 𝜃 degrees in-plane to match the orientation of the galaxy. The spiral arms of galaxies usually display varying amounts of scattering of the stars defining the spirals. Thus, spirals that bound the scattering are added to main spiral fit when appropriate. In general, any of the fractional spiral parameters may be used as the basis for the bounding spirals; however, in all the cases that follow, only bounds based on variations in C are used. Furthermore, with the bounding spirals, a median or average spiral is always given. It is important to note that a significant number of different parameters or combinations of parameter may be used to determine the bounding spirals. Figure 18.10 illustrates a bounded symmetric spiral pair with an inclined nucleus outline and median spiral. The results in this figure are for illustration only. Estimates of the galactic nucleus outline may be added or not. In Section 18.4, morphology classifications are listed following the galaxy name. These subjective classifications are taken from NED (NASA/ipac Extragalactic Database) when available and represent the judgments of various investigators listed there. Differences in these judgments

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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. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Hillary Mathis/NOAO/AURA/NSF. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

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usually indicate the need for interpolation between the discrete categories. The axes numerations on the figures that follow are all in pixels as found for the original image.

18.4 Mathematical Classification of Selected Galaxies In this section, the spirals of various galaxy images are modeled using equation 18.1. The parameters are determined manually by a trial-and-error process. With some experience, a fitting of the quality discussed later may usually be done in a few hours. The galaxies were selected to give some coverage of the Hubble classifications with galaxies having well-defined spirals. With the exception of M51, the galaxies were selected to be free of tidal effects. In the figures that follow, the axes numerations are all in pixels as found in the original image.

18.4.1

NGC 4314 SB(rs)a

This galaxy (Figure 18.11) is highly symmetric and was analyzed on that basis. Furthermore, in addition to the primary arms it appears to have a weak secondary pairs of arms, which are also symmetric and seem to start from the bar. Scale 123.0 pixels/arc-min. The galaxy appears to us truly face-on; thus, common parameters for both primary and secondary arms are 𝜓 = 𝜙 = 0. Also, 𝜃offset = −75∘ , a = 1, k = 0. Primary arm parameters: q = 1.125, v = −0.1, 𝛼 = 0.10, 𝛽 = 0.90, C = 210. 𝜂 = 0 to 3.2. Secondary arm parameters: q = 1.08, v = −0.05, 𝛼 = 0, 𝛽 = 1.017, 𝜂 = 0.0 to 2.8. 18.4.2

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

NGC 1365 (Figure 18.12) is a dramatic barred spiral at a distance of 60 million light-years. In this infrared image, the bar appears to be horizontal; however, the fitting of the well-defined spiral arms seems to indicate a phantom bar (dark gray dashed lines) at approximately −45∘ . The implications of this are not clear, it may indicate a discontinuous evolutionary process. The analysis considers the arms as symmetric, and the results show good containment by close bounding fractional spirals. Common parameters: q = 1.23. v = −0.20, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 49o , 𝜃offset = −70o , C = 270 ± 35. Primary arm parameters: 𝜂 = 0.65 to 3.1. Phantom arm parameters: 𝜂 = 0.0 to 0.65. 18.4.3

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

M95 (NGC3351) (Figure 18.13) is a barred spiral with the bar sitting at about 50∘ to the x-axis in the image. The fitting parameters are q = 1.07, v = 0.0, 𝛼 = 0.450, 𝛽 = 0.541, 𝜓 = 0.0, 𝜙 = 0.0, 𝜃offset = −70o , and C = 65 ± 4, 𝜂 = 0 to 4.05. The fitting only applies to the bar and the inner part of the spiral arms. The nucleus shown has a radius rn = 22 pixels. The scale is approximately 214 light-years/pixel.

18.4 Mathematical Classification of Selected Galaxies

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Figure 18.11 Classification for NGC 4314. See text for fit parameters. Solid curve is primary arm and dashed is secondary arm. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: David W. Hogg, Michael R. Blanton, and the Sloan Digital Sky Survey Collaboration. Please see www.wiley.com/ go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

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Figure 18.12 Classification for NGC 1365. See text for fit parameters. Dashed curve is primary arm and phantom arm extends primary to origin. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO/P. Grosbol. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

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Figure 18.13 Classification for M 95. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit & Copyright: Adam Block, Mt. Lemmon SkyCenter, University of Arizona. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4 Mathematical Classification of Selected Galaxies

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

The spiral arms of NGC 2997 (Figure 18.14) were assumed to be symmetric for the fractional spirals used to fit the image. This galaxy is an excellent example of a normal spiral. The modeling parameters are q = 1.19, v = 0.19, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 30o , 𝜃offset = 0.0, C = 18.5 ± 1.5, 𝜂 = 0 to 10.7. The radius of the nucleus is rn = 17 pixels. The scale is approximately 88.2 pixels/arc-min.

18.4.5 NGC 4622 (R′ )SA(r)a pec/Sb

The galaxy NGC 4622 (Figure 18.15) appears to be rotating clockwise while the spiral arms give the inference of counterclockwise rotation. The classification of this galaxy assumed a symmetric form for its two spiral arms. While the arms could have been modeled slightly better individually, the symmetric assumption yielded good results. The galaxy appears to us face-on. The modeling parameters for Figure 18.15 are q = 1.07, v = 0.07, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 0.0, 𝜃offset = 35o , C = 174 ± 6, and 𝜂 = 0 to 7.5. The nucleus radius was arbitrarily set at 130 pixels. A size calibration was not available for the image.

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

M 66 or NGC 3627 is modeled here as a symmetric barred spiral, although significant asymmetry may be seen in the control image. The galaxy is presented to us at an inclination of about 65∘ . In this image from the European Space Organization, “North is towards upper left,

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Figure 18.14 Classification for NGC 2997. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO. Please see www.wiley.com/go/Lorenzo/Fractional_ Trigonometry for a color version of this figure.

18.4 Mathematical Classification of Selected Galaxies

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Figure 18.15 Classification for NGC4622. See text for parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image credit: NASA and The Hubble Heritage Team (STScI/AURA), Acknowledgement: Dr. Ron Buta (U. Alabama) and Tarsh Freeman (Bevill State Community College). Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

West towards upper right.” The modeling parameters for Figure 18.16 are q = 1.28, v = 0.0, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 65.0, 𝜃offset = 50o , C = 690 ± 150, and 𝜂 = 0 to 3.2. The geometric center is shifted from the optical center by 10 pixels to the left in x and 20 pixel down in the y direction. The nucleus radius was rn = 200 pixels. A size calibration was not available for the image.

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

Initially, the galaxy NGC 4535 was modeled as a symmetric barred spiral. This is shown in Figure 18.17a where the fit parameters are; q = 1.14, v = 0, a = 1.0, 𝛼 = 1.0, 𝛽 = 0, 𝜓 = 0, 𝜙 = 20o , 𝜃 = −47o , 𝜂 = 0.0008 to 3.7, rn = 32, and C = 148.5 ± 23.5. However, the arms appeared to be tangent to the nucleus, type s, so a second model attempted to capture that feature and the result is shown in Figure 18.17b where the fit parameters are; q = 1.26, v = 0.26, a = 1.0, 𝛼 = 0.8, 𝛽 = 0.2, 𝜓 = 0, 𝜙 = 40o , 𝜃 = −30o , 𝜂 = 1.0 to 9.7, rn = 32 pixels, and C = 38 ± 5. Note the considerable differences in the parameters of the two models, even the apparent inclination parameter, 𝜙, changes from 20∘ to 40∘ . 18.4.8 NGC 1300 SBc/SBb(s)/SB(rs)bc

NGC 1300 (Figure 18.18) is often described as the prototypical barred spiral galaxy. The modeling done here assumes symmetry of the spiral arms although it is clear that the left-most arm fit could be improved if modeled individually. The modeling parameters for Figure 18.18 are q = 1.145, v = −0.05, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 53.0, 𝜃offset = 10o , C = 290 ± 45, and 𝜂 = 0 to 5.2. The nucleus radius was rn = 65 pixels. The galaxy size has been estimated at 85,000

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Figure 18.16 Classification for M 66 or NGC 3627. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO/P. Barthel. Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure.

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light-years from which the calibration may be estimated at 65.4 light-years/pixel. It is interesting to note that various investigators Lindblad et al. [66] and Peterson and Huntley [107] estimated the inclination at 35∘ , 35.5∘ , 40∘ , 44∘ , 48∘ , 49.3∘ , and 53∘ . The value of 𝜙 = 53.0, found here appears to validate the largest estimate and indicates the galaxy is far from face-on. It should also be noted that the optical center of the galaxy indicated by the white axis was not used as the geometrical center but a shift of 10 pixels to the short (dark gray) y-axis provided a much better fit. This represents a distance of 654 light-years!

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Figure 18.18 Classification for NGC 1300. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Hillary Mathis/NOAO/AURA/NSF. Please see www.wiley .com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.9

Hoag’s Object

Hoag’s Object (Figure 18.19) is a ring galaxy that is about 100,000 light-years in diameter, giving an approximate calibration of 100 light-years/pixel. Initial symmetric (two arm) modeling showed satisfactory results, however, a single-arm barred spiral was found to be superior. The spiral parameters (Figure 18.19) are q = 1.015, v = −0.011, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0.0, 𝜙 = 0, 𝜃offset = −85o , C = 330, and 𝜂 = 0.1 to 10.5. The nucleus radius as shown is rn = 40 pixels, but could easily be considered to be a larger value. Because there is no evidence of a bar between the nucleus and outer ring, the initiation angle of the spiral is somewhat arbitrary. Thus, at an angle of −85∘ a phantom semi-bar with 𝜂 = 0 to 0.1 marks the assumed starting point of the spiral. The very low value of q = 1.015 indicates the closeness to a circular geometry, at q = 1.0. Thus, in this mathematical classification approach the ring galaxies are clearly viewed as extremely weak barred galaxies.

18.4.10

M 51 Sa + Sc

The image (Figure 18.20) shows M 51 (NGC 5194) and its nearby neighbor NGC 5195. By modeling the two arms as a symmetric pair, based on the right-most arm, we can see the effect of the gravitational attraction of NGC 5195 on the spiral arms of NGC 5194 as the difference between the model and the physical location of the arms in the upper portion of the image. The spiral parameters of Figure 18.20 are q = 1.19, v = −0.19, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 42o , 𝜙 = 0, 𝜃offset = 0o , C = 4.6, and 𝜂 = 9 to 20.6. The galaxy diameter is approximately 65,000 light-years giving a scale factor of approximately 65 light-years/pixel. The nucleus radius (not outlined) is approximately rn = 48 pixels. The geometric center of the galaxy is displaced downward from the optical center by 30 pixels or 1950 light-years. This is marked by the green subaxis. The in-plane (projected) deflection of the upper arm is approximately 100 pixels (6500 light-years) at the tip.

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Figure 18.19 Classification for Hoag’s Object. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: NASA, R. Lucas (STSc/AURA). Please see www.wiley .com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure. Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure. 400

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Figure 18.20 Classification for M 51. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: T.A. Rector and Monica Ramirez/NOAO/AURA/NSF. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.11

AM 0644-741 Sc/Strongly peculiar/

The galaxy AM_0644-741, a Lindsay-Shapley Ring Galaxy, is considered to be a ring collisional galaxy. Because of the regularity of the ring, a mathematical fitting was attempted, and is shown in Figure 18.21a. The ring is fit quite well as a single spiral arm with a very low growth rate, that is, q = 1.018. The geometrical center is shown by the white subaxis, which is shifted along the x-axis by 408 pixels from the optical center of the image. Since the uncropped image is 260,000 light-years wide the calibration is 65.8 light-years/pixel. Thus, the displacement appears to be approximately 27,000 light-years. The spiral parameter values for the fitting are q = 1.018, v = 0.018, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0o , 𝜙 = 57o , 𝜃offset = 8o , C = 650, xoffset = 408 pixels from galaxy

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 0

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Figure 18.21 Classification for AM 0644-741. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: NASA, ESA, and The Hubble Heritage Team (AURA/STScI) Acknowledgement: J. Higdon (Cornell U.) and I. Jordan (STScI). Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure. Please see www.wiley.com/go/Lorenzo/ Fractional_Trigonometry for a color version of this figure.

optical center, and 𝜂 = 8.58 to 16.5. The low value of q appears to indicate that this is a type S0 galaxy. There is a faint string of stars near the nucleus not modeled previously, the parameter values for this segment (Figure 18.21b) were found to be; q = 1.018, v = −0.3820. 𝛼 = 0.7, 𝛽 = 0.3, 𝜓 = 0o , 𝜙 = 57o , 𝜃offset = 20o , C = 470, and 𝜂 = 0 to 2.1. The two segments appear to match smoothly at the interface. Notice both segments have the same inclinations 𝜓 and 𝜙, and same value of q.

18.4.12

ESO 269-G57 (R’)SAB(r)ab/Sa(r)

Two models are applied for the galaxy ESO 269-G57, that is, the symmetric outer arms and the inner ring are modeled separately. The spiral arm parameter values for the fitting are q = 1.130, v = 0.130, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0o , 𝜙 = 55o , 𝜃offset = 20o , C = 99 ± 10, and 𝜂 = 4.8 to 8.0. The inner ring was well fit by a single-arm spiral with parameter values of q = 1.016, v = 0.016, 𝛼 = 1.0, 𝛽 = 0.0, 𝜓 = 0o , 𝜙 = 25o , 𝜃offset = 15o , C = 65, and 𝜂 = 2.0 to 8.4. The nucleus radius is 17 pixels. The cropped image of the galaxy (Figure 18.22) is approximately 4.06 × 3.55 arc-min,

18.4 Mathematical Classification of Selected Galaxies 300

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Figure 18.22 Classification for ESO 269-G57. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO, Data: Henri Boffin. Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure.

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies

and the galaxy width was given as 200,000 light-years yielding a calibration of approximately 250 light-years/pixel. The galaxy optical center is displaced upward from the geometrical center by about 10 pixels, 2500 light-years. Interestingly, the inclination of the spiral arm and the inner ring differ by Δ𝜙 = 30o , possibly indicating a warping of the disk were it possible to view from the side. 18.4.13 NGC 1313 SBc/SB(s)d/SB(s)d

NGC 1313 has been called the Topsy-Turvy galaxy because of its very active state. This is reflected in the large value of q in the fitting parameters. The galaxy in Figure 18.23 has been fit with symmetric spirals with the parameters; q = 1.57, v = 0.31, 𝛼 = 0.98, 𝛽 = 0.02, 𝜓 = 20o , 𝜙 = 0o , 𝜃offset = 95o , C = 137, and 𝜂 = 0 to 2.1. The apparent optical center is marked by the white axes. The geometrical center for the spiral fit was found to be 52 pixels above that and is marked by the dark gray subaxis. The size of the galaxy has been given as 50,000 light-years, giving a calibration of about 83 light-years/pixel. 18.4.14

Carbon Star 3068

This star was found to have a faint spiral structure of great regularity [20]. It was of some interest to determine the relevance of the fractional spirals to this structure. The results are presented in Section 18.7.

18.5 Analysis Continuing from Ref. [89] with permission of ASME: A summary of the galaxy morphologies based on the fittings of the fractional trigonometric spirals is given in Table 18.1. Also contained in the table are the Hubble classifications of various investigators as given in the NASA Extragalactic Database, NED. Furthermore, the table also contains the parameter (q), which is plotted against Hubble class in Figure 18.24. The Hubble class is assumed to be linear in the plot. The q parameter was chosen because of the strong effect on spiral growth rate that is seen in Figures 18.2 and 18.5. Note that NGC 1313 has not been included in the plots that follow. This galaxy does not follow our initial requirement of well-defined spiral arms. Also, Hoag’s Object, a ring galaxy, has been plotted as type S0. A roughly linear correlation of q with Hubble class is seen in Figure 18.24. Because the number of galaxies studied is small, the scatter appears to be somewhat large. In particular, the single datum at Hubble class b appears to be high and the lower datum point at Hubble class c appears to be low. There are at least two possible explanations here: first, as can be seen from the table, the Hubble classifications by various investigators are substantially not self-consistent and can easily slide horizontally as indicated by the horizontal bars. Furthermore, as can be seen in Figure 18.3, 𝛼 and possibly 𝛽, also influence the spiral growth rate and their effects are not included in the q parameter. It is believed by the authors that a larger sample of galaxies will give a stronger correlation. This, however, is beyond the scope of this initial investigation. de Vaucouleurs [28, 29] expanded the Hubble classification by adding the r–s class. The r type indicates the spiral arms exit tangent to an external ring at the termination of a bar while the s type the arms exit the end of the bar. In Figure 18.25, q − 1 is plotted as a function of v for galaxies in which 𝛼 is near unity. The galaxy number as listed in

18.5 Analysis 0

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Figure 18.23 Classification for NGC 1313. See text for fit parameters. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Henri Boffin (ESO), FORS1, 8.2-meter VLT, ESO. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

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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. 𝜶

𝜷

#

Galaxy

q

v

1

NGC 4314

1.125

−0.1

0.10

2

NGC 1365

1.23

−0.2

1

3

M 95

1.07

0.0

0.45

0.541

SB(r)b/SBb(r)/SBa/SBb

r

4

NGC 4622

1.07

0.07

1

0

(R′ )SA(r)a pec/Sb

r

5

NGC 2997

1.19

0.19

1

0

Sc/SAB(rs)c/Sc(s)

r s/s

6

M 66

1.28

0

1

0

SAB(s)b/Sb(s)/Sb

s

7

NGC 4535

1.14

0

1

0

SABc/SBc/Sc

s

Hubble Class.

r/s

0.90

SB(rs)a

rs

0

SBb/SBc/SB(s)b/SBb(s)

s

NGC 4535

1.26

0.26

0.8

0.2

–

–

8

NGC 1300

1.145

0.05

1

0

SBc/SBb(s)/SB(rs)bc

s/r s

9

Hoag’s Ob.

1.015

−0.011

1

0

–

–

10

M 51

1.19

0.19

1

0

Sa + Sc?

–

11

AM 0644-741

1.018

0.018

1

0

Sc/Strongly peculiar/

–

12

ESO 269-G57

1.130

0.130

1

0

(R′ )SAB(r)ab/Sa(r)

r

13

NGC 1313

1.57

0.31

0.98

0.02

SBc/SB(s)d/SB(s)d

s

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME.

1.3 q-Fractional order variable

364

Figure 18.24 Galaxy order parameter q versus Hubble class. Source: Lorenzo and Hartley 2011 [89]. Adapted with permission of ASME.

6 2

1.2 12 1

1.1

8

5 7

3,4

9,11

1 0.9 0.8 0.7 0.6 0.5 S0

S0a

a

ab b Hubble class

bc

c

Table 18.1, and the r–s type are given next to each data point. Importantly, all points falling on or near the v = q − 1 line are of the r type. This, of course, is expected from the results shown in Figure 18.5. Furthermore, as v moves away from that line toward v = 0 or negative v, the results become increasingly r–s to s types. Thus, we see that the value of v relative to q − 1 dictates the nature of the exit of the spiral arms and falls directly out of the fitting process. Interestingly, in this limited study, no value of v was found that was more negative than the v = 1 − q line (shown dashed). This may indicate that the v = 1 − q line gives pure s types and the space between these lines are the mixed types. The following question arises: are there theoretical limits on the values of v for spiral galaxies of the 𝛼 ≈ 1.0, 𝛽 ≈ 0.0 and k = 0 type? This was studied numerically, applying the program used for the mathematical classification study. The results are shown in Figure 18.26. The upper limit results in a hard barred spiral similar to that seen in Figure 18.4 when v = 0.1. When v is larger

18.5 Analysis

v = 1–q

(q–1) Order parameter

0.3

v = q–1 6,s 7a,r

0.25

Figure 18.25 Galaxy order parameter q − 1 versus v for galaxies with 𝛼 ≈ 1.0 and 𝛽 ≈ 0.0. Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME.

2,s 0.2

10,?

5,r

0.15

7,s 8,s/rs 12,r

0.1 4,r

0.05

11,? 0

–0.3

–0.2

–0.1

0 v

0.1

0.2

0.3

Figure 18.26 Limits for v parameter, for galaxies with 𝛼 = 1, 𝛽 = 0, and k = 0.

0.5 Hard barred spiral limit, 𝜈 = q–1

𝜈 Parameter

0

Observed data limit, 𝜈 = 1–q

–0.5

–1

–1.5 Spiral arm interference limit 𝜈 = –5.8 (q–1) –2

0

0.05

0.2 0.1 0.15 (q–1) Order parameter

0.25

0.3

than q − 1, the program yields a nonspiral result; thus, it appears that v cannot be larger than q − 1. A search was then conducted at various values of q to probe the lower limits for v. At any given value of q − 1, the value of v was reduced until the spiral arms interfered with each other. These spirals were similar to that seen in Figure 9.42. Analytically, and possibly physically, of course v could be reduced further; however, the results would not be representative of the spirals of this class. This process yielded the resulting circular markers in Figure 18.26. Surprisingly, this is closely approximated by the linear condition v = −5.8 q. Note the lines from Figure 18.25 are repeated in Figure 18.26, but not the data. The results of Figure 18.26 raise the further speculative question; if the spiral arms have a significant material content, does the interference limit possibly indicate a collapse of the spiral structure of a galaxy? On the other hand, it is more widely accepted that the spiral arms are the result of the hypothesized density waves. If this is the case, then, for galaxies of the 𝛼 ≈ 1.0, 𝛽 ≈ 0.0 type, we should expect wave interference affecting the wave amplitude.

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18.5.1

Fractional Differential Equations

Continuing from Ref. [89] with permission of ASME: The spiral fitting values for the galaxies may now be used to infer fractional-order differential equations describing the geometry. From the forms, we have used x0 = C • Cosq,v (a, 𝛼, 𝛽, k, 𝜂) and y0 = C • Sinq,v (a, 𝛼, 𝛽, k, 𝜂)

(18.4)

and the Laplace transform equations (9.66) and (9.69); the following fractional differential equations are inferred: 1 2q q v+q 2 (18.5) (cos(𝜆)0 D𝜂 − a cos(𝜆 − 𝜎))𝛿(𝜂), 0 D𝜂 x(𝜂) − 2a cos(𝜎)0 D𝜂 x(𝜂) + a x(𝜂) = C 1 v+q (18.6) (sin(𝜆)0 D𝜂 − a sin(𝜆 − 𝜎))𝛿(𝜂), C where a = 1, 𝜎 = (𝛼 + 𝛽q)(𝜋∕2), and 𝜆 = 𝛽(q − 1 − v)(𝜋∕2). We see that the left-hand sides of both of these simultaneous fractional differential equations are identical in form. The right-hand side filtering, however, changes for the x and y components. A common spatial synchronizing impulse drives both equations. In these equations, the independent variable 𝜂 is the spatial parameter defining the resulting fractional spiral. This parameter, however, does not define the physical length along the spiral which must be calculated from the x and y position variables defining the spirals. 2q 0 D𝜂 y(𝜂)

q

− 2a cos(𝜎)0 D𝜂 y(𝜂) + a2 y(𝜂) =

18.5.2 Alternate Classification Basis

In consideration of alternatives to the use of the fractional Sinq,v versus Cosq,v as the basis of galactic morphology, we note the following. In the classical trigonometry, the sin(t) versus the cos(t) provides the basis of oscillation of the harmonic oscillator. Of all the meta-trigonometric functions, we note that Covibq,v most accurately generalizes the cos(t), because of its even and real basis, that is from Chapter 9: ( ( ( ( ))) ))) ( ( = Ea Re Rq,v ai𝛼 , i𝛽 t . (9.55) Covibq,v (a, 𝛼, 𝛽, k, t) = Re Ea Rq,v ai𝛼 , i𝛽 t Similarly, the Flutq,v most accurately generalizes the sin(t), because of its odd and imaginary basis, also from Chapter 9 ( ( ( ( ))) ))) ( ( = Oa Im Rq,v ai𝛼 , i𝛽 t . (9.58) Flutq,v (a, 𝛼, 𝛽, k, t) = Im Oa Rq,v ai𝛼 , i𝛽 t Because of these features, galactic morphology modeling-based Flutq,v versus Covibq,v is worthy of future consideration. Here, we take an initial look at a possible parameter set for this phase plane combination. Figure 18.27 is phase plane of Flutq,v versus Covibq,v and −Flutq,v versus −Covibq,v , which is highlighted. The spatial t parameter ranges from 0 to 5. For this plot, we have set 𝛼 + 𝛽 = 1.14, and with q = 1.14 and v = 0.14; this results in a barred spiral with a very hard corner. We have set v = 0.1399 to show the transition from the origin to the beginning of the spiral, which would otherwise not display. The various combinations of 𝛽 and 𝛼 produce a set of spirals, which may very well match galaxy spiral morphology. In Figure 18.28, the parameter set is identical except that the v parameter is set to v = −0.400. Comparison of the two figures reveals a startling difference in the transition from the origin to spiral arms. In fact, the spiral appears to be a normal one. Furthermore, the arms of the spirals in Figure 18.28 has been rotated backward relative to those of Figure 18.27. Clearly, this combination of fractional meta-functions is of interest for this task. Other combinations of functions may also have application.

18.6 Discussion

Figure 18.27 Effect of 𝛼 and 𝛽, Hard barred spiral. Flutq,v (a, 𝛼, 𝛽, k, t) versus Covibq,v (a, 𝛼, 𝛽, k, t) and −Flutq,v (a, 𝛼, 𝛽, k, t) versus −Covibq,v (a, 𝛼, 𝛽, k, t), q = 1.14, v = 0.1399, a = 1.0, 𝛼 + 𝛽 = 1.14, 𝛽 = 0.5 to 1.3 in steps of 0.2, k = 0.

Flutq,𝜈(a,α,β,k,t)

5

β = 1.3

0

β = 0.5 –5 –6

–4

–2

0

2

4

6

Covibq,𝜈(a,α,β,k,t)

Figure 18.28 Effect of 𝛼 and 𝛽. Normal spiral. Flutq,v (a, 𝛼, 𝛽, k, t) versus Covibq,v (a, 𝛼, 𝛽, k, t) and −Flutq,v (a, 𝛼, 𝛽, k, t) versus −Covibq,v (a, 𝛼, 𝛽, k, t), q = 1.14, v = −0.4, a = 1.0, 𝛼 + 𝛽 = 1.14, 𝛽 = 0.5 to 1.3 in steps of 0.2, k = 0.

5

Flutq,𝜈(a,α,β,k,t)

β = 1.3

0 β = 0.5

–5 –6

–4

0 2 –2 Covibq,𝜈(a,α,β,k,t)

4

6

18.6 Discussion Current methods of classification of galaxy morphology are subjective comparisons to idealized galactic shapes established originally by Hubble in 1936 and later extended by de Vaucouleurs to include the effect of the manner in which the spiral arms exit barred spirals, the r–s property. This chapter defines a classification system that uses the fractional trigonometric sine and cosine functions to describe the spiral and ring galaxies on a mathematical basis. The foundation for this basis is quite fundamental, being ultimately sourced by the fractional generalization of the exponential function in the form of the R-function. The significance of this is the connection back to the solutions of fractional differential equations as seen in Chapters 12 and 2. Importantly, the mathematical basis of this classification method may allow further studies of arm evolution and kinematics.

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies

The study was based on a small selection of galaxies intended to give coverage of the Hubble classification scheme. The galaxies were also chosen based on the quality of the definition of the spiral arms. The classification basis used only the fractional trigonometric sine and cosine functions with no cross coupling and with constant growth rate, q. Furthermore, to simplify the manual fitting process the variable a, a spatial fractional generalization of the natural frequency was set to one. The matches of the analytical spirals to the physical spirals were considered to be quite good over the ranges shown thus providing mathematical descriptions over significant segments of the spiral arms. However, in some cases, the extension of the analytical model beyond that shown resulted in deviations from the spiral arms. It appears reasonable to claim that the method allows the classification of spiral galaxy morphology to be set on an analytical basis. 18.6.1

Benefits

This study allows the spiral arms to be mathematically defined, a fact that greatly refines the classification process and presents the possibility of detailed correlation studies involving the spiral parameters. The following benefits were found from this mathematical approach. 1) The process yielded mathematical descriptions of the spiral arms, which are superior to subjective descriptions. The mathematical descriptions allow mathematical correlations, projections, and inferences not possible with the subjective classification. It is important to note that the fractional trigonometric-based classifications, and the fractional differential equations based on them, serve to model both the normal and barred spirals, as well as the r–s variants of the spiral initiation and the morphology of the ring galaxies with just a few parameters. 2) Using the fractional trigonometric functions with programs to account for the inclinations of the galactic planes, good to excellent agreement was found between the actual galaxy morphology and the mathematical spiral morphology. 3) A single fractional trigonometric formulation was found to unify descriptions of both barred and normal spirals, and both r and s spiral-exits. Ring morphologies are also unified for most cases. 4) For most cases, the mathematical spiral q parameter related well to galaxy spiral growth rate, or in Hubble terms, to the openness of the spiral arms. The mathematical spiral v parameter correlated to the barred versus normal galactic spiral property. 5) Plots of v versus q − 1 were found to predict limiting bounds for the r, s behavior, thus relating the galactic spiral arm exit-behavior to the galactic spiral growth rate. 6) The method provides a spiral based method to determine inclination of the galactic plane. A total inclination comparable to contemporary quoted values may be determined by vector considerations of the 𝜓 and 𝜙 computationally inferred inclinations. 7) For NGC 1313, ESO 269-G57, AM 0644-741, M 66, and NGC 1300, major deviations of the optical center of the galaxy from the galaxy geometric center were found based on the mathematical symmetry of the spiral arms. Some of these offsets were many light-years! An indication of galaxy warping was seen for ESO 269-G57. 8) In the case of M 51, the method provides the potential for computing the tidal driven deflection of the upper areas of the spiral arms from an inferred undisturbed position, based on a symmetry assumption for the spiral arms. 9) In this mathematical classification system, the ring galaxies may be interpreted as one-armed barred galaxies (with bars not visible). 10) Finally, availability of the fractional trigonometry-based mathematical classification of the galactic spiral morphology has allowed the easy computation of the value of v for any q,

18.6 Discussion

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Figure 18.29 Classification for Carbon Star 3068. See text for fit parameters. (a) Original image (b) magnified view, (c) with analytical fit. Source: ESA/NASA & R. Sahai. Public domain.

which achieves interference of the galactic spiral arms. This limiting equation may indicate the conditions for transition of the spiral galaxies into elliptical galaxies. This limiting equation may allow validation of the density wave hypothesis relative to material arms. We note that models based on the fractional trigonometric functions identified by the classification can be considered to be the solution of the spatial part of a fractional wave-diffusion field equation. The development of such an equation in cylindrical form is an important challenge; see Chapter 20. It should be clear that other subsets and/or additions of the fractional trigonometric spirals, perhaps using the parity functions, may provide superior matches of the mapping of the functions to the galactic spiral arms.

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18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies

18.7 Appendix: Carbon Star As mentioned earlier a very regular spiral was found around Carbon Star AFGL 3068. It was of some interest to determine whether this spiral could be fit with the fractional meta-trigonometric spirals. This is done in this appendix. 18.7.1

Carbon Star AFGL 3068 (IRAS 23166 + 1655)

The pattern of mass ejection from this star has been considered to be “one of the most perfect geometrical forms created in space [102].” Thus, there is interest to know if the fractional spirals also fit the form seen here. The star spiral (Figure 18.29) has been fit with a single spiral with the parameters: q = 1.035, v = 0.035, 𝛼 = 1.0, 𝛽 = 0.0002, 𝜓 = 21o , 𝜙 = 20o , 𝜃offset = −150o , C = 74, and three segments for 𝜂 = 2.7 to 11.7, 13.4 to 17, and 20.7 to 22. These segments together with the dashed lines create a single spiral that matches the observed spiral quite well. A paper by Morris et al. [102] discusses the large mass ejection rates, outflow velocity, and other aspects associated with the spiral.

371

19 Hurricanes, Tornados, and Whirlpools In this chapter, we take a brief look at the potential applicability of the fractional trigonometry and the fractional spiral functions to vortex flows such as hurricanes, tornados, and whirlpools. According to Buckley et al. [18] p. 66, “… vortices may originate either when a weak area of low pressure forms or when the overall wind regime encounters a barrier…” At this time, we examine the morphology of these phenomena and other weather phenomena such as low-pressure cloud patterns based on available images. In the applications that follow, the computations of the spiral loci of Sections 19.1, 19.3, and 19.4 are based on the fractional meta-trigonometric Sinq,v - and Cosq,v -functions, and are identical to the mathematical definitions and nomenclature given in Section 18.3.4.

19.1 Hurricane Cloud Patterns Modern satellite technology has provided a new perspective into the morphology of hurricane flows. This allows the potential application of the fractional trigonometric spirals to the mapping of streamlines observed in cloud structures. In this section, we consider the morphology of the cloud patterns that are characteristic of the behavior of hurricane flows; see, for example, Kerry [33]. As discussed in the previous chapters, this is achieved by fitting fractional spirals to hurricane images and thus spatial modeling is possible. The parameters given for the spiral fits are the same as those used for the galaxy study in Chapter 18. 19.1.1 Hurricane Fran

Figure 19.1 is a satellite image of hurricane Fran off of the coast of Florida taken in September 1996. Several of the major flow patterns have been fit to the image. The fitting parameters are summarized in Table 19.1. With the exception of Leg 2, the legs are all fit well with order parameters q = 0.45 and v = 0.0. The fit for Leg 2 using q = 0.45 was only slightly inferior to that used in the table. 19.1.2 Hurricane Isabel

The form of hurricane Isabel, off the east coast of the United States in September 2003 (as seen in Figure 19.2) appears to be quite symmetric. This is possibly due to its distance from land. In this study, it was decided to attempt a fully symmetric fit, which is shown. In this figure, the six spirals seen are the primary spiral and five identical spirals symmetrically spaced about the origin, which is at the center of the hurricane eye. The parameters for this model were q = 0.48, v = 0, a = 1, 𝛼 = 0.20, 𝛽 = 0.8, 𝜓 = 0.0, 𝜙 = 0.0, 𝜃 = −20, C = 1500, and 𝜂 = 7.0 → 13. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

372

19 Hurricanes, Tornados, and Whirlpools 0

0

100

100

200

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300

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600

600

700

700

200

300

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600 700 (a)

800

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900 1000

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600 700 (b)

800

900 1000

Figure 19.1 Hurricane Fran off the Florida coast. (a) Reference image, (b) model. See text for spiral parameters. Source: NASA GSFC. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure. Table 19.1 Fitting parameters for Hurricane Fran. Leg

q

𝜶

v

𝜷

𝝍

𝝓

𝜽

𝜼

C

1

0.45

0

0.20

0.80

40

0

90

0 → 4.8

220

2

0.35

0

0.20

0.80

0

40

−100

0.1 → 2.5

120

3

0.45

0

0.20

0.80

40

0

−30

0 → 6.8

170

4

0.45

0

0.20

0.80

40

0

−90

1.5 → 3.6

120

100

100

150

150

200

200

250

250

300

300

350

350

400

400

450

450

500

500

550

550

600

600 0

100

200

300

(a)

400

500

0

100

200

300

400

500

(b)

Figure 19.2 Hurricane Isabel. (a) Hurricane Isabel reference image, (b) model. See text for spiral parameters. Source: NASA. Public domain. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

19.2 Tornado Classification

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.

280

300

320

340

360

380

400 220

240

260

280

300

320

In the magnified view of Figure 19.3, the spirals were extended by continuing 𝜂 to 15 to expose the morphology entering the eye. Relative to the set of parameters quoted for this spiral, it was found that changing the order parameter q to q = 0.5 resulted in coalescence of the spirals into an exact circle. However, q = 0.48 provided a better match for the streamline morphology. It is important to note for both cases studied that 𝜂, the parametric variable along each streamline, is interpreted as a spatial variable. In the fittings, this variable increases as the eye is approached. At the upper level of the hurricane, it is thought that the direction of rotation reverses into an outflow jet (Emanuel [33], p. 16). Such a reversal is not apparent in these images. The cloud structure seen appears to be that associated with the inflow of the lower level of the hurricane. The fractional trigonometric order for a hurricane is likely a variable-order process (Lorenzo and Hartley [72]), that is, q1 for the unstable outflow at the top of the vortex. The differential order of the associated fractional differential equations will be twice the trigonometric order for the fractional sine and cosine. This can be seen from the Laplace transforms for the functions.

19.2 Tornado Classification With tornadic flows, it is not currently possible to obtain images looking along the axis of rotation exposing the circular or spiral nature of the flow. However, images are often available showing tornado profiles. The current classification for tornados is the Enhanced Fujita scale, the EF scale. The Fujita scale, Bluestein [17], p. 5, is based on the nature of the damage that is observed after the event and a wind speed range is associated with each of the scale elements. For example, an EF-5 tornado is estimated to have winds in excess of 200 miles per hour. In this study, the fractional trigonometric function k index will be used as the basis for a possible classification of tornadic forms based on their profiles.

373

19 Hurricanes, Tornados, and Whirlpools

19.2.1

The k Index

The fractional trigonometric functions by their definitions, ∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 sin (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

Sinq,v (a, 𝛼, 𝛽, k, t) =

t > 0, (19.1)

∞ ( ( )) ∑ an t (n+1)q−1−v 𝜋 cos (n(𝛼 + 𝛽q) + 𝛽[q − 1 − v]) + 2𝜋k , t > 0, Γ((n + 1)q − v) 2 n=0 (19.2) are indexed by the variable k, where k = 0, 1, 2, … , D − 1 and D = Dq Dv D𝛼 D𝛽 is the product of the denominators of the rational arguments q = Nq ∕Dq , v = Nv ∕Dv , 𝛼 = N𝛼 ∕D𝛼 , and 𝛽 = N𝛽 ∕D𝛽 with D in minimal form. The source of this indexing is the fractional exponent (n + 1)q − 1 − v, which leads to the arguments for the sine and cosine in the definitions of the meta-trigonometric functions. When these functions occur as the solutions to fractional differential equations, the various values of k lead to equivalent solutions. However, as we have seen in Chapters 7–9, the functions are different when viewed individually. In the near-order effect (Section 14.7), one or more of the denominators becomes large, leading to a large value for k. For properly selected functions, this can be utilized for the classification of tornado profile morphology.

Cosq,v (a, 𝛼, 𝛽, k, t) =

19.2.2

Tornado Morphology Animation

A very-much simplified simulation of a tornado is shown in Figure 19.4. The animation is done in Matlab (version R2012a) and is based on the R2 -trigonometric functions with height h, replacing t as the main variable. To study the tornado profile, the order q was set at q = 48/49, v = 0, and 12 equally spaced streams (functions) are set into rotation. The figure shows a fixed time slice in a 3D plot. Note that the R2 -functions may be written in terms of the meta-trigonometric functions, for example, R2 Sinq,v (a, k, t) = Sinq,v (a, 0, 1, k, t). The program for the simulation is given in Appendix C.

®

Fractional dynamic tornado animation: k = 2 10 8 h-Height

374

6 4 2 0 –10 0

R2Sin(48/99,0,1,k,h)

10

10

–10 –5 0 5 R2Cos(48/49,0,1,k,h)

Figure 19.4 Simplified tornado model, 3D time slice.

19.4 Whirlpool

19.2.3

Tornado Morphology Classification

If the index k = 0, 1, 2, 3, … is allowed to vary, the profile morphology varies as shown in the sequence of panels of Figure 19.5. As can be seen in the sequence, as the index variable k increases, the change in radius with height, dr∕dh, also increases and subjectively the strength of the tornado increases. The fineness of such a classification system can be readily increased by increasing the size of the denominator Dq . In Figure 19.6, the value of Dq is increased from q = 48/49, by setting q = 98/99. With q = 98/99 and the new index k = 1, this clearly yields a morphology that interpolates between k = 0 and k = 1 in the q = 48/49 system and includes these points as k = 0 and k = 2 in the refined q = 98/99 based system. Two additional features are desired for a classification system based on these functions. First, for the morphologies shown in Figure 19.5, the footprint of the tornadoes is set by choosing the raw or unmultiplied functions. Clearly, the larger the footprint is, the more severe the tornado. This can be accommodated by multiplying the fractional trigonometric functions by an appropriate factor depending on k. For example, C(k) = k + 1. This assumes that severity increases as dr∕dh increases. Furthermore, such a factor, of course, will change with the choice of Dq . Also, a choice of Dq must be made to provide satisfactory resolution particularly for the weaker tornados. Second, the velocity at or near ground level should be keyed to the index value to calibrate the classification system. This of course will require observational input. These last steps are beyond the scope of this effort. Application of the tornado morphology classification scheme requires tornado profile images that are at least roughly size calibrated. From such images, dr/dh versus h and the tornado footprint may be estimated, and this information can be used to classify the tornado.

19.3 Low-Pressure Cloud Pattern A rather phenomenal satellite image of the cloud pattern of a low-pressure system over Iceland is of significant interest relative to the fractional trigonometric spirals. The question being: could fractional spirals be found that are representative of the flow pattern inferred by the cloud structure? Figure 19.7 shows the reference image along with two images showing a fractional spiral model and a close-up view of the center of the spiral. The fractional spiral provides a good representation over a significant number of rotations of the spiral. The two spirals shown are scaled and rotated versions of each other, and the spiral parameters are q = 0.90, v = 0.0, a = 1.0, 𝛼 = 0.95, 𝛽 = 0.05, k = 0, 𝜓 = 35∘ , 𝜙 = 20∘ , for both spirals. For the inner spiral, C = 1100, 𝜃o = 50∘ , and for the outer spiral, C = 1680, 𝜃o = 40∘ .

19.4 Whirlpool Associated with the disastrous tsunami that struck Japan in March 2011, an enormous whirlpool was formed off the east coast of Japan near the port in Oaraj, Ibaraki Prefecture. The scale of the whirlpool may be judged by the ship seen in the image (Figure 19.8) and located on the x axis at x = 200. The morphology of the whirlpool is slightly deformed by the adjacent land masses; however, it is of considerable interest to evaluate the applicability of the fractional trigonometric spirals to this phenomenon. Three streamlines were fit for this model. The common parameters for the three legs are v = 0, a = 1, 𝛼 = 0.2, 𝛽 = 0.8, k = 0, 𝜓 = 0, and 𝜙 = 70∘ . The remaining parameters are given in Table 19.2. In spite of the deformation caused by the adjacent land masses, the fractional spirals provided good fidelity to the visible streamlines.

375

19 Hurricanes, Tornados, and Whirlpools

Fractional dynamic tornado animation: k = 1

Fractional dynamic tornado animation: k = 0 10

8

8

6

6

h-Height

h-Height

10

4 2 0

4 2

–10

–5

0

5

0

10

–10

R2Cos(48/49,0,1,k,h) (a) 10

10

8

8

6

6

4

10

Fractional dynamic tornado animation: k = 3

4 2

2 0

5 0 –5 R2Cos(48/49,0,1,k,h) (b)

Fractional dynamic tornado animation: k = 2

h-Height

h-Height

–10

0

R2Cos(48/49,0,1,k,h)

–5 0 5 R2Cos(48/49,0,1,k,h)

(c)

(d)

–5

5

0

10

10

–10

10

Fractional dynamic tornado animation: k = 4

8 h-Height

376

6 4 2 0

–10

–5 0 5 R2Cos(48/49,0,1,k,h)

10

(e)

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.

19.4 Whirlpool

Fractional dynamic tornado animation: k = 1

10

Figure 19.6 Tornado morphology k = 1, for q = 98/99.

8

h-Height

6

4

2

0

–10

–5

5 0 R2Cos(98/99,0,1,k,h)

10

0

0 Low-pressure system

Low-pressure system

500

500

1000

1000

1500

1500

2000

2000

2500

2500

3000 0

500

1000

1500

2000

2500

3000

3000 3500 0

500

1000

(a)

1500

2000

2500

3000

3500

(b)

1400 1600 1800 2000 2200 2400 1200

1400

1600

1800

2000

2200

2400

(c)

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. Source: NASA. Public domain. Please see www.wiley.com/go/ Lorenzo/Fractional_Trigonometry for a color version of this figure.

377

378

19 Hurricanes, Tornados, and Whirlpools

0

100

200

300

0

100

200

300

400

500

600

400

500

600

(a) 0

100

200

300

0

100

200

300 (b)

Figure 19.8 Whirlpool off the coast of Japan, March 2011. (a) Reference image, (b) fractional spiral model. See text for model parameters. Source: Reuters Pictures. Reproduced with permission of Reuters. Please see www .wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

Table 19.2 Parameters for Whirlpool model. Leg

q

𝜽

𝜼

C

1

0.65

0 → 13.0

180

2

0.62

−5∘ −5∘

1.5 → 9.5

300

3

0.62

−10∘

4.0 → 9.0

425

19.5 Order in Physical Systems

19.5 Order in Physical Systems In this and the previous two chapters, we have studied the applicability of the fractional spirals, and by inference the fractional trigonometric functions, to various physical and biological systems. While it is premature to draw any definitive conclusions, the following observations are forwarded. All of the studies were based on the complexity functions, the fractional sine and cosine, as applied to the spatial behavior of the processes involved. Only in the case of the Nautilus, were we able to use a modicum of auxiliary data to extend our understanding to a dynamic model that inferred the growth rate of the animal from conception to maturity. Such extensions, in general, are beyond the scope of the current effort. However, in Chapter 20, we suggest possible extension into this domain. It is of interest to consider the nature of the order variable, q, on the behavior of the processes. However, we digress briefly, to note the following. First, there two types of order: there is the order of the trigonometric functions, q that has been discussed in many places in the book. Second, there is the order of the associated fractional differential equations, that we abbreviate as the differential order, and use the notation, u. Based on the Laplace transforms (Section 9.4 and Table 9.1), the associated differential order for the complexity functions, the fractional sine and cosine, is given by u = 2q, and for the parity functions by u = 4q. Table 19.3 summarizes the order associated with the fractional spirals used to fit the various morphologies. The stability listings in the table are computed based on equation (13.36) and relate to the spatial character of the morphologies. It should be noted for the whirlpool case that to four decimal places the computation yielded neutral stability. The unstable result for the sea shells is not surprising, for clearly a growth process is an unstable one. In this case, the temporal process is also unstable during the growth period since the differential order (Table 7.3) is greater than 2. For the galaxy spiral fittings, all spirals started from the origin and the spatial variable 𝜂 increased with radial distance. Having differential order greater than two, these spirals are also spatially unstable. It should be noted that if a spiral could be fit with 𝜂 increasing approaching the origin, the stability result could be changed. With the hard barred spirals (Figures 18.3–18.6), this does not seem to be a likely possibility. In the remaining three cases, the spatial variable 𝜂 increases approaching the origin and the processes are spatially stable. This is a bit surprising for the hurricane case, but as explained in Section 19.1, the process is thought to be of variable order with order at the upper level likely greater than two and unstable. At this time, we are unable to make any general statements concerning the relation between spatial and temporal stability. Table 19.3 Order and spatial stability associated with various physical processes. Process

Trig. order

Differential order

Stability

Naut.& Shells

q >1

u >2

Unstable

Galaxies

q >1

u >2

Unstable

Hurricane inflows

q ≈ 0.45 − 0.48

u ≈ 0.90 − 0.96

Stable

Low pressure

q = 0.9

u = 1.8

Stable

Whirlpool

q = 0.62–0.65

u = 1.24–1.30

Stable

379

381

20 A Look Forward In an effort such as that leading to this book, many questions and opportunities arise that may not contribute to the main theme, or were too difficult to complete in a timely manner. Some are simple questions, such as what does a half-order triangle look like? Yet, they may not have easy answers. Others might require the development of new theory, for example, fractional orthogonality. In this final chapter, we attempt to briefly outline areas where further development is needed both in regard to the theoretical aspects of the fractional trigonometric and fractional exponential functions and also relative to application of the theory to scientific problems. It is useful to start by looking backward, that is, by considering the scope of knowledge that has been uncovered relative to the classical exponential and trigonometric functions, which is shown graphically in Figure 20.1. At the core of the classic theory is the common exponential function. From this, the connection is made to the classic trigonometry via the Euler equation. The exponential function and the trigonometric functions provide solutions to ordinary linear differential equations and are often components in the solution of integer-order partial differential equations. The trigonometric functions are important because the solutions derived from them are real functions that are seen in the physical world as opposed to complex exponential functions. The development of Fourier series was enabled because of the orthogonal nature of the classical sine and cosine. From the Fourier series, important extensions to Fourier transforms and spectral analysis were possible. If we now generalize the common exponential function to the F- or R-function, as has been done for the fractional calculus, and further develop a fractional generalization of the trigonometry and hyperboletry, the focus of this book, it is appropriate to consider possible fractional generalizations of the important fruits of the classical trigonometry. As part of the development of the fractional trigonometry, important generalizations of the Euler equation were uncovered. A key result of the fractional trigonometry has been its use for the solution of linear commensurate-order fractional differential equations (Chapter 12). Relative to generalizing the classical Fourier methods, the central question becomes “can orthogonal subsets of the fractional trigonometric functions other than the classical functions be found or created?” For example, such creation may be possible by multiplication of the functions by appropriate functions to yield a constant envelope. Or, more broadly, is a fractional orthogonality possible that allows the creation of fractional versions of the Fourier series and subsequently fractional-order Fourier transforms and spectral analysis? While some initial effort has been done toward this end (Lorenzo and Hartley [79]), significant challenges and opportunities are found here. Finally, the fractional generalization of the natural logarithm, the inverse fractional exponential, if accomplished may allow generalization of the multiplication operation. This is addressed in more detail in Section 20.2. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

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20 A Look Forward

Fourier series ∞

f(t) = ∑ cneniπt/p −p < t < p −∞

Logarithmic function ln(x)

ln(z) = lnr + i(ϕ + 2πn) z = r(cosϕ + isinϕ)

Prob. and Stat., distributions

f(x) =

a π

2

e−a(x−μ)

x

loge n = L eL = n

1 2

(ex − e−x)

cosh x =

1 2

(ex + e−x)

1

Laplace and Fourier transforms

Hyperbolic functions sinh x =

∫ 1t dt

ln(x) =

x

z

e ,e

∞

∫

L{f(t)} = e−stf(t)dt 0

eiy = ex(cos y + i sin y) Euler equation Eigenfunctions dnex = ex n = 1,2,3... dxn

Circular functions sin y = cos y =

1 (eiy 2i 1 (eiy 2i

– e−iy) + e−iy)

Figure 20.1 Classical connections.

20.1 Properties of the R-Function While numerous properties for the R-function have been developed in Chapters 3 and 4, much more is needed. For example, can simple forms be found for the evaluation of (Rq,v (a, 𝛼, 𝛽, k, t))x or Rq,v (a, 𝛼, 𝛽, k, t)•Ru,v (a, 𝛼, 𝛽, k, t), and many more similar operations?

20.2 Inverse Functions It is easy to numerically obtain a plot for the inverse of the R-function; however, determining the analytical expressions for the fractional inverse turns out to be problematic. The difficulty arises because the inverse is multivalued over some domains. Figure 20.2 shows the R-function plotted as a function of t-time. The same data set is shown in Figure 20.3 but as the inverse function. In the plots, the minimum values of the R-function are approximated numerically. An approximate equation for the locus of minima as seen in Figure 20.2 (black dashed line) is Rq,0 (1, t)|Min ≈ 1 − ln(1 − 4.2373 t), 0 < t ≤ 0.236,

(20.1)

and for Figure 20.3, this is expressed as t ≈ 0.236(1 − exp(1 − Rq,0 (1, t)|Min )), Rq,0 (1, t)|Min > 1, t > 0.

(20.2)

20.2 Inverse Functions

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.

20

15 Rq,0(1,t)

q = 0.25 10

q = 2.5

5

0 0

1

2 t-Time

3

4

Figure 20.3 Inverse R-Function, Rq,0 (1, t), 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.

4

t, Inverse R-function

3.5 3

q = 2.5

2.5 2 1.5 1

q = 0.25

0.5 0

0

5

10 Rq,0(1,t)

15

20

Importantly, there are three domains in Figure 20.3 that must be separately analyzed to define the inverse function. Based on the approximate minima equation (20.1), these domains are defined by ⎫ ⎧q ≥ 1 ⎪ ⎪ ⎨q < 1 with t > 0.236(1 − exp(1 − Rq,0 (1, t)|Min )), Rq,0 (1, t) > 1⎬ . ⎪q < 1 with t < 0.236(1 − exp(1 − Rq,0 (1, t)|Min )), Rq,0 (1, t) > 1⎪ ⎭ ⎩

(20.3)

At this time, the inverse function is only analytically known for two values of the q parameter. One of these is for q = 1, of course, where the inverse function is t = ln(Rq,0 ), R > 1. The other, for q = 2, is inferred from equation (3.41), that is, R2,0 (1, t) = sinh(t). From this, we have t = sinh−1 (Rq,0 (1, t)). These values are highlighted by dashed curves in the figures. Series expansions are readily available for both functions. The end goal, of course, is the development series expressions that directly define the inverse in each domain defined. Recent progress has been reported on the determination of the inverse for the closely related Mittag-Leffler function by

383

384

20 A Look Forward

Hanneken and Achar [126]. While those results do not appear to be directly relatable to the R-function, it may be possible that the techniques used there can be applied to the determination of the inverse R-function. The character of the fractional hyperbolic functions (Chapter 5) is quite similar to that of the R-functions that compose it. Thus, the problem of determining inverse series for R1 Sinhq,v (a, t) and R1 Coshq,v (a, t) will be difficult, but important. Finding direct inverse functions for the fractional trigonometric functions may not be feasible. Importantly, most of the functions are not periodic, and their inverses are multivalued. Furthermore, the number of functions involved is large. From a practical point of view, perhaps the most useful approach is the development of software routines that can be applied to the forward function to determine its inverse. This would be the modern equivalent of the trigonometric tables.

20.3 The Generalized Fractional Trigonometries The formal definitions of the generalized fractional trigonometries were presented in Chapter 11. The development of this area presents significant challenges that place it beyond the scope of this book. Based on their Laplace transforms, these functions provide the solutions to fractional differential equations of more complexity than those discussed in Chapter 12. Because of this, it is important that the detailed properties of these functions be determined.

20.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments The meta-trigonometry introduces the possible study of imaginary and complex time as well as imaginary and complex quency; see Chapter 13 and Appendix E. It is expected that such capability will be of interest in physics research. Furthermore, because the fractional trigonometries are based on the R-function, which is defined for t > 0, it is of some interest to note that for 𝛽 = 2, we have the basis Rq,v (ai𝛼 , i2 t) = Rq,v (ai𝛼 , −t). An important question is: can a fractional trigonometry based on Rq,v (ai𝛼 , i2 t) = Rq,v (ai𝛼 , −t) extend the fractional trigonometry based on Rq,v (ai𝛼 , t) into the negative time domain? Or the more general question is: can complementary trigonometries be formed, for example, using the basis pairs Rq,v (ai𝛼 , i𝛽 t) and Rq,v (ai𝛼 , i𝛽+2 t)? Limited numerical studies have not yet conveyed a clear picture in this regard. It presently appears that, with certain choices of the q, v, 𝛼, and 𝛽 variables, this may be possible for the fractional trigonometric functions. A particular case is shown in Figure 20.4 where for various values of 𝛼 the Cov1∕2,1∕2 (1, 𝛼, 2.5, 0, t) is interpreted as running backward in negative time and the complementary Cov1∕2,1∕2 (1, 𝛼, 0.5, 0, t) is running in forward time. As may be observed from the plot, this interpretation smoothly continues the functions into negative time, that is, the functions have the same values and slopes at the origin. Note, particularly in cases where a singularity exists at t = 0, the results are inconclusive. In forward time, of course, the backward running slopes are the negative of those of the forward interpretation. Furthermore, the functions’ apparent stabilities change between the forward and backward interpretations. This explains the observations made in Figures 9.47 and 9.48 where both the Flutter and Coflutter functions each smoothly continue backward in time because of the 1–3 relationships in the 𝛽 term. A second example is shown for the Cos1,0 (1, 0.8, 𝛽, 0, t)-function in Figure 20.5. The Sin1,0 (1, 0.8, 𝛽, 0, t)-function can also be extended into negative time in a similar manner. It

20.5 Applications: Fractional Field Equations

Figure 20.4 Negative time issue. Cov1∕2,1∕2 (1, 𝛼, 𝛽, 0, t) versus t-time for a = 1.0, with q = 0.5, v = 0.5, k = 0, 𝛼 = 0.1 to 0.3 in steps of 0.1; 𝛽 = 0.5 for t > 0, and 2.5 for t < 0. For 𝛽 = 2.5, the function is interpreted as Cov1∕2,1∕2 (1, 𝛼, 𝛽, 0, −t). Source: Lorenzo 2009b [83]. Reproduced with permission of ASME. Please see www.wiley .com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

4 3

0.3

0.2

Covq,𝜈(a,α,β,k,t)

2 1 0

β = 2.5 β = 0.5 0.3

0.2 0.1

–1 –2 –3 –4 –10

α = 0.1 –5

0 t-Time

5

10

Figure 20.5 Negative time issue. Cos1,0 (1, 𝛼, 𝛽, 0, t) versus t-time for a = 1.0, with q = 1.0 and v = 0, 𝛼 = 0.8, k = 0, 𝛽 = 0.1 to 0.7 in steps of 0.3. For 𝛽 + 2, the function is interpreted as Cos1,0 (1, 𝛼, 𝛽 + 2, 0, −t). Please see www.wiley .com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

4 β = 0.1

3 β+2 β

Cosq,𝜈(a,α,β,k,t)

2

0.7 1 0.4

0.4

0

0.7

–1 –2 –3 –4 –10

0.1 –5

0 t-Time

5

10

should be noted that this method has not been found to provide extension to negative time for all of the meta-trigonometric functions nor for all selections of the function parameters. The fractional trigonometry brings meaning to the R-function with complex arguments. Clearly, a future development to the effort presented in this book is to determine the properties of the fractional trigonometric functions with complex arguments.

20.5 Applications: Fractional Field Equations In the second half of this book, considerable effort has been spent to identify potential application areas for the fractional trigonometry. The discovery and identification of the fractional spirals was very helpful in this regard. These spirals, when oscillatory in the time domain, are generalizations of the harmonic oscillator in that they allow both decay and growth. The occurrence of spiral behavior in nature is very well known (e.g., Hartmann and Mislin [50]). Historically, in most instances, these were thought to be or were at least approximated by logarithmic spirals. The logarithmic spiral, however, is infinite in both directions; that is, it can become infinitely large or infinitely small. The problem encountered in applying the logarithmic

385

386

20 A Look Forward

spirals is in the small direction, because it does not start its growth at a finite time (or position). This is most clearly apparent in the cases of shell growth and galactic morphology. This issue is alleviated by the application of the barred and apparently unbarred fractional spirals. In retrospect, it is amazing to observe the broad application of these spirals that were generated by only cross-plotting the raw fractional trigonometric functions, without any cross-coupling or variation in the order variable q. Within the scope of these studies, temporal data have not been easily available to allow determination of fractional field equations that generalize the wave or diffusion equations and yield the fractional spirals in the spatial or temporal variables. For example, it would be of great interest to acquire a fractional field equation that yields the spatial barred spirals used to match the morphology of the spiral galaxies found in Chapter 18. Such an equation may provide a path to evolutionary or trajectory information. Only for the case of growth of the Nautilus was it possible to use some auxiliary data to infer a dynamic equation for growth of the animal (equation (17.15)). However, in this and the other cases, nonpartial fractional differential equations in space have been shown; see equations (17.4), (17.5), (18.5), and (18.6). So the question becomes: how can we devise a partial fractional field equation where, for example, after an assumed separation of variables, the spatial parts of the equation yield the fractional barred spirals seen in the preceding applications? Let us consider the following partial fractional differential equation: a1

𝜕 2q w 𝜕q w 𝜕 2q w 𝜕q w 𝜕 2u w 𝜕uw + b + c w + a + b + c w = a + b + c3 w. 1 1 2 2 2 3 3 𝜕x2q 𝜕xq 𝜕y2q 𝜕yq 𝜕t 2u 𝜕t u

(20.4)

Applying the method of separation of variables, we assume a solution of the form w(x, y, t) = X(x)Y (y)T(t).

(20.5)

Introducing equation (20.5) into equation (20.4) gives 𝜕 2q X(x) 𝜕 q X(x) + b T(t)Y (y) + c1 T(t)Y (y)X(x) 1 𝜕x2q 𝜕xq 𝜕 2q Y (y) 𝜕 q Y (y) + a2 T(t)X(x) + b2 T(t)X(x) + c2 T(t)X(x)Y (y) 2q 𝜕y 𝜕yq 𝜕 2u T(t) 𝜕 u T(t) = a3 X(x)Y (y) + b3 X(x)Y (y) + c3 X(x)Y (y)T(t). 2u 𝜕t 𝜕t u

a1 T(t)Y (y)

(20.6)

Now, dividing by X(x)Y (y)T(t) yields a1 𝜕 2q X(x) b 𝜕 q X(x) + 1 + c1 2q X(x) 𝜕x X(x) 𝜕xq a3 𝜕 2u T(t) b3 𝜕 u T(t) a 𝜕 2q Y (y) b2 𝜕 q Y (y) + 2 + + c = + + c3 . 2 Y (y) 𝜕y2q Y (y) 𝜕yq T(t) 𝜕t 2u T(t) 𝜕t u Regrouping the results gives } { 𝜕 2q X(x) 𝜕 q X(x) 1 + b + c X(x) a1 1 1 X(x) 𝜕x2q 𝜕xq } { 𝜕 2q Y (y) 𝜕 q Y (y) 1 + + b + c Y (y) a2 2 2 Y (y) 𝜕y2q 𝜕yq } { 𝜕 2u T(t) 𝜕 u T(t) 1 = + b + c T(t) . a3 3 3 T(t) 𝜕t 2u 𝜕t u

(20.7)

(20.8)

20.7 Numerical Improvements for Evaluation to Larger Values of atq

In the two previous equations, each group is independent of the other two, that is, the variables have been separated. Thus, each group may be treated as nonpartial fractional differential equation and can only be equated to a constant, giving the following set: } { d2u T(t) du T(t) 1 + b3 + c3 T(t) = 𝜇, (20.9) a3 T(t) dt 2u dt u } { d2q X(x) dq X(x) 1 + b + c1 X(x) = 𝜇1 , (20.10) a1 1 X(x) dx2q dxq } { d2q Y (y) dq Y (y) 1 + b + c2 Y (y) = 𝜇2 . (20.11) a2 2 Y (y) dy2q dyq where 𝜇 = 𝜇1 + 𝜇2 . Now, each of these equations may be solved individually by applying appropriate boundary and initialization conditions. By selection of appropriate values for a1 , a2 , b1 , b2 , c1 , and c2 , equations (20.10) and (20.11) may be Laplace transformed by the methods of Chapter 12 to yield the fractional sines and cosines needed to model various spatial barred and apparently unbarred spirals in the time-sensitive context, as seen in the applications of previous chapters. Selection of a1 = a2 , b1 = b2 , and c1 = c2 will yield most of the fractional spirals used in modeling the various physical phenomena in this book. By proper selection of a3 , b3 , c3 , and u, wave or diffusion behavior or some combination may be modeled. Of course, u can also be fractional. Furthermore, as we have seen in the applications, the spatial differential-order can be greater than two. Even more generally, the time derivatives are not necessarily of commensurate order and also may not be limited in order to be less than or equal to two. Hence, we have apparently achieved our goal in rectangular coordinates. However, to achieve a reasonable statement for the boundary conditions for phenomena such as hurricanes, galactic evolution tornadoes, and the like, equation (20.4) must be converted into cylindrical or in some cases spherical coordinates. This has not yet been done.

20.6 Fractional Spiral and Nonspiral Properties The studies in Chapter 15 only begin to explore the properties of the fractional spirals and have not considered those trigonometric functions not formally defined as spirals. Further study is needed to determine whether these functions can be further characterized. Relative to the fractional spirals, for example, the cross-plot of r sin(t) vs r cos(t) becomes x2 + y2 = r2 in nonparametric, that is, rectangular coordinates. Can simple rectangular forms for some of the fractional spirals be determined? In a similar manner, since the circumference of a circle is well known, can the perimeter of a single rotation of, for example, an R1 spiral be determined as a simple algebraic function of the order, q and the number of rotations of the spiral? The logarithmic spiral is characterized by a growth rate logarithmically proportional to the angle 𝜃. The spiral of Archimedes is characterized by growth proportional to the angle 𝜃. The question becomes: can appropriate characterizations be determined for the various classes of the fractional spirals?

20.7 Numerical Improvements for Evaluation to Larger Values of atq For the applications we have studied, with values of the trigonometric order q > 1, typically 3 or 4 rotations of the fractional spirals have been adequate to model the physical behavior. This is

387

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20 A Look Forward

fortunate because as we have shown in Chapter 14 as atq grows, numerical evaluation becomes problematic. Serious and creative effort is needed to address this issue.

20.8 Epilog The integer-order trigonometry is typically thought of as a study of triangles and their properties and applications, in spite of the fact that we call the functions the circular functions. It is clear from the new perspective obtained by the development of the fractional trigonometry that the integer-order trigonometry is about circles and that triangles are an analysis tool used to study the properties of the circle. The fractional trigonometry releases the constant circular radius of the integer-order trigonometry to allow the vast array of spiral and spiral-like functions that we have seen in Chapters 9 and 15. Also, unlike the logarithmic spiral, the fractional spirals have a beginning. Furthermore, as we have seen in Chapter 12, these functions provide the solutions to a broad array of linear fractional differential equations of any real order in terms that can be related to real-world physics.

389

A Related Works A.1 Introduction In this appendix, we briefly present related efforts by other investigators. In all cases, the efforts were auxiliary to studies of the fractional calculus or the study of fractional differential equations and were not the primary focus.

A.2 Miller and Ross The development of Miller and Ross [95] pp. 314–317 is built around the incomplete gamma function, 𝛾 ∗ , and it is defined as 𝛾 ∗ (v, t) = e−t

∞ ∑ n=0

tn . Γ(v + n + 1)

(A.1)

They then take the fractional integral of the exponential as Et (v, a) = D−v eat = t v eat 𝛾 ∗ (v, at), or as infinite series Et (v, a) =

∞ ∑ n=0

an t n+v . Γ(v + n + 1)

(A.2)

(A.3)

Analogs of the classical trigonometric functions are then based on Et (v, ia) as follows: Et (v, ia) = t v

∞ ∑ n=0

(i a t)n , Γ(v + n + 1)

∞ ∞ ∑ ∑ (−1)n ( a t)2n (−1)n ( a t)2n+1 v + it , Et (v, ia) = t Γ(v + 2n + 1) Γ(v + 2n + 2) n=0 n=0 v

giving the definitions Ct (v, a) = t v St (v, a) = t v

∞ ∑ (−1)n (at)2n , Γ(v + 2n + 1) n=0

∞ ∑ (−1)n ( at)2n+1 n=0

Γ(v + 2n + 2)

.

presented here in forms comparable with the meta-trigonometric forms of Chapter 9. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

(A.4)

(A.5)

(A.6)

(A.7)

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A Related Works

Clearly, Et (v, ia) = Ct (v, a) + i St (v, a).

(A.8)

Miller and Ross provide various properties of the functions along with the following Laplace transforms: 1 L{Et (v, ia)} = v , Re v > −1, (A.9) s (s − a) s , Re v > −1, (A.10) L{Ct (v, a)} = v 2 s (s + a2 ) 1 , Re v > −2. (A.11) L{St (v, a)} = v 2 s (s + a2 )

A.3 West, Bologna, and Grigolini West, Bologna, and Grigolini [121], pp. 101–102, define their generalized fractional exponential function as the 𝜇th fractional derivative of the common exponential function, namely E𝜇t ≡

∞ ∑ n=0

t n−𝜇 . Γ(n + 1 − 𝜇)

(A.12)

Then, the WBG generalized trigonometric functions are defined as the real and imaginary parts of E𝜇i t ; thus, E𝜇i t =

∞ ∑ n=0

t n−𝜇 (cos ((n − 𝜇)𝜋∕2) + i sin((n − 𝜇)𝜋∕2)). Γ(n + 1 − 𝜇)

(A.13)

From this, cos𝜇 t =

∞ ∑ n=0 ∞

sin𝜇 t =

∑ n=0

t n−𝜇 cos ((n − 𝜇)𝜋∕2), Γ(n + 1 − 𝜇)

(A.14)

t n−𝜇 sin((n − 𝜇)𝜋∕2), Γ(n + 1 − 𝜇)

(A.15)

and E𝜇i t = cos𝜇 t + i sin𝜇 t.

(A.16)

WBG note the relationship of E𝜇a t with the Miller and Ross’ Et (v, a) as E𝜇a t = a−𝜇 Et (−𝜇, a).

(A.17)

The authors also supply a second generalized exponential function that applies for negative t; it is 𝜇th fractional derivative of the negative exponential function ∗ −t E𝜇

≡ ei𝜋𝜛 E𝜇−t = D𝜇t [e−t ].

(A.18)

A.4 Mittag-Leffler-Based Fractional Trigonometric Functions Podlubny [109], pp. 17–19, presents the two parameter Mittag-Leffler function, due to Erdelyi et al. [35] ∞ ∑ zn , 𝛼 > 0, 𝛽 > 0. (A.19) E𝛼,𝛽 (z) = Γ(𝛼 n + 𝛽) n=0

A.5 Relationship to Current Work

Podlubny further notes the use by Plotnikov and Tseytlin of fractional trigonometric functions expressible as Mittag-Leffler functions, namely Sc𝛼 (z) = Cs𝛼 (z) =

∞ ∑ (−1)n z(2−𝛼)n+1 , = zE2−𝛼,2 (−z2−𝛼 ), Γ((2 − 𝛼)n + 2) n=0 ∞ ∑ n=0

(−1)n z(2−𝛼)n , = E2−𝛼,1 (−z2−𝛼 ). Γ((2 − 𝛼)n + 1)

(A.20) (A.21)

Further referenced here by Podlubny are the functions suggested by Luchko and Srivastava; their functions are defined as ∞ ∑ (−1)k z2k+1 sin𝜆,𝜇 (z) = (A.22) , = zE2𝜇,2𝜇−𝜆+1 (−z2 ), Γ(2𝜇k + 2𝜇 − 𝜆 + 1) k=0 cos𝜆,𝜇 (z) =

∞ ∑ k=0

(−1)k z2k , = zE2𝜇,𝜇−𝜆+1 (−z2 ). Γ(2𝜇k + 𝜇 − 𝜆 + 1)

(A.23)

Bagley [10], pp. 73–77, in an appendix, bases his generalized functions on the single parameter Mittag-Leffler function, namely E𝛽 [−(at)𝛽 ] =

∞ ∑ (−(at)𝛽 )n . Γ(1 + n𝛽) n=0

(A.24)

Based on this generalized exponential, fractional-order sine and cosine functions are defined as E𝛽 [i (at)𝛽 ] − E𝛽 [−i (at)𝛽 ] , (A.25) sin𝛽 [(at)𝛽 ] = 2i and E𝛽 [i (at)𝛽 ] + E𝛽 [−i (at)𝛽 ] cos𝛽 [(at)𝛽 ] = . (A.26) 2

A.5

Relationship to Current Work

Because the preceeding definitions of the fractional trigonometric functions are all based on the real and imaginary parts of some generalized exponential, comparison of the definitions given here with those presented in this book are best done by comparing the exponential generalizations to the R-function. Then, for t>0, we have Miller and Ross Et (v, a) =

∞ ∑ n=0

an t n+v = R1,−v (a, t). Γ(v + n + 1)

(A.27)

West, Bologna, and Grigolini E𝜇t =

∞ ∑ n=0

t n−𝜇 = R1,𝜇 (1, t). Γ(n + 1 − 𝜇)

(A.28)

Mittag-Leffler Based t 𝛽−1 E𝛼,𝛽 (t 𝛼 ) =

∞ ∑ t n𝛼−1+𝛽 = R𝛼,𝛼−𝛽 (1, t), 𝛼 > 0, 𝛽 > 0. Γ(𝛼 n + 𝛽) n=0

(A.29)

391

392

A Related Works

Bagley E𝛽 [−(at)𝛽 ] =

∞ ∑ (−(at)𝛽 )n = R𝛽,𝛽−1 (−a, t). Γ(1 + n𝛽) n=0

(A.30)

Thus, it may be seen that the four generalizations of the common exponential used by these investigators are all related to special cases of the R-function. Thus, it is clear that the fractional trigonometric functions defined by these investigators will also be special cases of the fractional meta-trigonometric functions as defined in Chapter 9.

393

B Computer Code B.1 Introduction This appendix presents computer programs in MatlabⓇ version 6.1, and Maple 10Ⓡ for the R and fractional trigonometric functions (Figures B.1 and B.2).

B.2

MatlabⓇ R-Function

function [y]=Rfun(q,v,a,t) %Rfun computes the R-function % using an infinite series % % q=order of s in denominator % v=order of s in numerator % a=fractional time constant in % (s ̂ v/(s ̂ q-a)) % t=time vector % ns=number of terms in series % nt=max time value % nt=max(abs(t)); ns=round((60/q)*.1*nt*max(abs(a))); if ns130 ns=130; end % ns=130; % Alternative fixed setting for number of terms. y=t*0; for k=0:ns, y=y+(t. ̂ (q-v-1)).*(((a)*t. ̂ q). ̂ k)/gamma(q-v+k*q); end

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

B Computer Code

1.5

Figure B.1 R-Function evaluation output.

q = 0.25 to 2 in steps of 0.25

1

Rq,0 (–1,t)

0.5 0 –0.5 –1 –1.5

0

2

4

6

8

10

t-Time

Figure B.2 Evaluation program output.

1 0.8 RCosq,v(a,α,β,k,t)

394

0.6 0.4 0.2

q = 1.2 q = 0.8

0 –0.2 –0.4 0

2

4

6

8

10

t-Time

B.3 MatlabⓇ R-Function Evaluation Program clear %Evaluation program for the R-function v=0; a=-1; Tmax=10; for q=.25:.25:2.0, t=0.001:Tmax./2000:Tmax; y=Rfun(q,v,a,t); plot(t,(y),'b','linewidth',2) axis([0 Tmax -1.5 1.5]) xlabel('t-Time','fontsize',11) ylabel('R_q_,_0(-1,t)','fontsize',11) text(.2,1.4,'q=.25 to 2 in steps of .25') grid on hold on end hold off

B.5 MatlabⓇ Cosine Evaluation Program

B.4

MatlabⓇ Meta-Cosine Function

function [y]=RCos(q,v,a,alp,bet,k,t) % Done in MatlabⓇ (version R2012a) % This program computes the master fractional Cosine function % using an infinite series % Note to convert to RSin function set % [y]=RSin(q,v,a,alp,bet,k,t) and % replace cos in second last line with sin % q=characteristic order % v=differintegral order % a=characteristic frequency % alp= alpha, power of i in a*i ̂ alpha % bet= beta, power of i in (i ̂ beta)*t % t=time vector % ns=number of terms in series % nt=maximum time value % note ns and nt limits may be changed to suit application %Note for meta-sine function replace cos in second last % line with sin % nt=max(abs(t)); ns=round((60/q)*.1*nt*max(abs(a))); y=t*0; %i=sqrt(-1); nt=max(abs(t)); ns=round((60/q)*.2*nt*max(abs(a))); if ns110 ns=110; end t=t; % ns=130; % Alternative fixed setting for number of terms. y=t*0; for n=0:1.5*ns, xi=n.*(alp+bet.*q)+bet.*(q-1-v); y=y+(t. ̂ (q-v-1)).*(((a)*t. ̂ q). ̂ n).*cos(xi.*(pi.*(0.5+2.*k))) ./gamma(q-n.*q); end

B.5

MatlabⓇ Cosine Evaluation Program

%Rcos_vs_time.m %Evaluation Program %Done in MatlabⓇ (version R2012a)

395

396

B Computer Code

clear v=0; a=1.0; k=0; alp=1; % Note to simulate R2Cos let alp=0, k=0, and bet=1 bet=.5; for q=.8:.20:1.2; t1=.1:.1:10; t2=.001:.001:.099; t3=.000001:.000001:.00099; t4=.00000001:.00000001:.00000099; t=[t4 t3 t2 t1]; y=RCos(q,v,a,alp,bet,k,t); plot(t,y,'k','linewidth',2) axis([0 10 -.4 1]) hold on xlabel('t - Time','fontsize',11) ylabel('RCos_q_,_v(a,\alpha,\beta,k,t)','fontsize',11) text(.1,.3,'q=0.8') text(1.5,.3,'q=1.2') grid on end hold off

B.6 MapleⓇ 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine > This Maple 10 program calculates phase plane plot for fractional sine vs cosine. RSin ∶= proc(q, v, a, alp, bet, t, k); local y, nmax; nmax ∶= 200; ( ( 1 (a ⋅ t q )n ⋅ sin((n(alp + bet ⋅ q) y ∶= add ((n + 1) ⋅ q(− v − 1) ! ))) ) 𝜋 + bet ⋅ (q − 1 − v)) ⋅ +2⋅𝜋⋅k , n = 0 ..nmax ; 2 y ∶= (t (q−1−v) ) ⋅ y; end proc; RSin ∶= proc(q, v, a, alp, bet, t, k) local y, nmax; nmax ∶= 200; y ∶= add((a∗ t ∧ q)∧ n∗ sin((n(alp + bet∗ q) + bet∗ (q − 1 − v))∗ (1∕2∗ 𝜋 + 2∗ 𝜋 ∗ k) )∕factorial((n + 1) ∗ q − v − 1), n = 0 ..nmax); y ∶= t ∧ (q − 1 − v) ∗ y end proc

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

> RCos ∶= proc(q, v, a, alp, bet, t, k); local y, nmax; nmax ∶= 200; ( ( 1 (a ⋅ t q )n ⋅ cos((n(alp + bet ⋅ q) y ∶= add ((n + 1) ⋅ q(− v − 1) ! ))) ) 𝜋 + bet ⋅ (q − 1 − v)) ⋅ +2⋅𝜋⋅k , n = 0 ..nmax ; 2 y ∶= (t (q−1−v) ) ⋅ y; end proc; RCos ∶= proc(q, v, a, alp, bet, t, k) local y, nmax; nmax ∶= 200; y ∶= add((a∗ t ∧ q)∧ n∗ cos((n(alp + bet∗ q) + bet∗ (q − 1 − v))∗ (1∕2∗ 𝜋 + 2∗ 𝜋 ∗ k ))∕factorial((n + 1)∗ q − v − 1), n = 0 ..nmax); y ∶= t ∧ (q − 1 − v)∗ y end proc > > with(plots): > tmax ∶= 340000; tmax ∶= 340000 1 ; > tinc ∶= 5000 1 tinc ∶= 5000 > tfinal ∶= tinc ⋅ tmax tfinal ∶= 68 > q ∶= 1.15; q ∶= 1.15 > > v ∶= −0.6; v ∶= −.6 1 ; > a ∶= 100000 1 a ∶= 100000 > alp ∶= 1.4; alp ∶= 1.4 > bet ∶= 1; bet [∶= 1( )) ( ( t ,0 , > R1cos ∶= seq evalf[10] RCos q, v, a, alp, bet, tinc t = tinc ..tmax ⋅ tinc)]: )) [ ( ( ( t ,0 , R1sin ∶= seq evalf[10] RSin q, v, a, alp, bet, tinc t = tinc ..tmax ⋅ tinc)]: Rcos1 ∶= − R1cos: Rsin1 ∶= − R1sin: > with(linalg):

397

398

B Computer Code

x > f ∶= x → : tinc > t ∶= vector(tmax ⋅ tinc, f ): > t ∶= convert(t, list): > dat ∶= convert(t R1cos R1sin, list): > dat ∶= convert(dat, array): > dat ∶= convert(dat, Matrix): > pair1 ∶= (R1cos, R1sin) → [R1cos, R1sin]; pair2 ∶= (Rcos1, Rsin1) → [Rcos1, Rsin1]; P1 ∶= zip(pair1, R1cos, R1sin): P2 ∶= zip(pair2, Rcos1, Rsin1) ∶ #plot([Rcos(1), Rsin(1)]) # this gives both functions vs time plot([P1, P2]); pair1 ∶= (R1cos, R1sin) → [R1cos, R1sin] pair2 ∶= (Rcos1, Rsin1) → [Rcos1, Rsin1]

ExportMatrix(“C∶∕ MATLAB6p1 ∕work∕ dat, ” dat, target = Matlab, format = rectangular)# export to Matlab

399

C Tornado Simulation See Figure C.1. %Tornado Simulation %Done in MatlabⓇ (version R2012a) %Program provides very simplified tornado simulation based on %fractional meta-trigonometric functions with alpha=0, beta=1 %Program requires computer functions RSin(q,v,a,alpha,beta,k,t) %and RCos(q,v,a,alpha,beta,k,t) clear q=98/99; v=0; a=1; k=4; % Tornado strength t1=.1:.02:10; %t2=.001:.001:.099; %t3=.000001:.000001:.00099; t4=.000000000001:.00000001:.00000099; %t=[t4 t3 t2 t1]; t=[t4 t1]; Nt=80; %Nt=number of frames/rev Nf=8; %Nf=number of filaments per frame %t=[t4 t3 t2]; %t=[t4 t3]; %t=[t4]; %t=[t2 t1]; %t=logspace(-10,1,500); for n=Nt+1:-1:1; axis equal yc=2.*RSin(q,v,a,0,1,k,t); xc=2.*RCos(q,v,a,0,1,k,t); thetac=atan2(yc,xc); yc1=RSin(q,v,a,0,1,k,n.*max(t)./Nt); xc1=RCos(q,v,a,0,1,k,n.*max(t)./Nt); thetac1=atan2(yc1,xc1); thetainc=2*pi./(Nt); yc=RSin(q,v,a,0,1,k,t); xc=RCos(q,v,a,0,1,k,t); The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

C Tornado Simulation

Fractional dynamic tornado animation: k = 4 10 8 6 h-Height

400

4 2 0 –10 0 Rsin(q,0,a,0,1,k,t)

10

10

5

0

–5

–10

Rcos(q,0,a,0,1,k,t)

Figure C.1 Simulation final frame.

for fila=0:1:Nf-1 inc2=2.*pi.*fila./Nf; yc1=RSin(q,v,a,0,1,k,n.*max(t)./Nt); xc1=RCos(q,v,a,0,1,k,n.*max(t)./Nt); y=yc.*sin(thetac+n.*thetainc+inc2)./sin(thetac); x=xc.*cos(thetac+n.*thetainc+inc2)./cos(thetac); y1=yc1.*sin(thetac1+n.*thetainc)./sin(thetac1); x1=xc1.*cos(thetac1+n.*thetainc)./cos(thetac1); plot3(y,x,t,’r:’,y1,x1,n.*max(t)./Nt,’bO’,y1,x1,n.*max(t). /Nt,’b*’,y1,x1,n.*max(t)./Nt,’b.’)%debris on %plot3(y,x,t,’k:’) %debris off view(150,20) axis([-12 12 -12 12 0 10]) grid on hold on M(n)= getframe; end hold off end xlabel(’RCos(q,0,a,0,1,k,t)’) ylabel(’RSin(q,0,a,0,1,k,t)’) title(’Fractional Dynamic Tornado Animation: k=4’) zlabel(’h-Height’) movie(M,12,60) movie2avi(M,’Tor4K3.avi’) % Note, after generation movie may be run from command window % by entering movie(M,12,60)

401

D Special Topics in Fractional Differintegration D.1 Introduction In Chapter 9 and previous chapters, fractional differintegrals of the fractional trigonometric functions have been derived. In all cases, the fractional differintegration has started from t = 0, and because the functions are zero for t < 0, initialization or history effects have not been an issue. However, when a function is segmented and preceded by another different function, fractional differintegrations must consider initialization effects caused by the preceding function (see Section 1.2.2 and [68, 71]). Clearly, there will be times when it is desired to start a fractional differintegration at times other than t = 0. This appendix addresses this topic.

D.2 Fractional Integration of the Segmented tp -Function We start our discussion seeking the fractional differintegral of a segmented form of the function t p , namely the composite function { fi (t) b = 0 < t < c (D.1) f (t) = t p t > c, where we assume that f (t) and all of its derivatives are zero for all t −1, q [d(t − a)] Γ(−q) Γ(p − q + 1)

(D.2)

where B is the complete beta-function. From Miller and Ross [95] (equation 6.3, p. 36) −q p 0 Dt t

=

Γ(p + 1) p+q t , Re q > 0, p > −1, t > 0. Γ(p + q + 1)

(D.3)

In both derivations, the Riemann–Liouville fractional integral is the basis for a fractional integration from a to t − a, or 0 to t. That is, the result ONLY can be started from those points and carries the further implication that the function f = t p and all of its derivatives are zero for all t prior to the start of the integration. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

402

D Special Topics in Fractional Differintegration

We now use the theory of initialization [68, 71] to generalize the classical results just presented and thereby substantially increase their utility. The initialized fractional-order integration operator (see Section 1.2.2) is defined as −q c Dt x(t)

or

−q

≡ c dt x(t) + 𝜓(xi , −q, b, c, t) , q > 0, t ≥ c,

(D.4)

t

1 (t − 𝜏)q−1 x(𝜏)d𝜏 + 𝜓(xi , −q, b, c, t) , q > 0, t ≥ c, Γ(q) ∫c

−q

c Dt x(t) =

where

c

𝜓(xi , −q, b, c, t) ≡

∫b

(t − 𝜏)q−1 xi (𝜏) d𝜏 , t ≥ c Γ(q)

(D.5)

(D.6)

and c ≥ b with x(t) = 0 for ∀t < b. Note, in equations (D.4) and (D.5) and from this point on, D represents the initialized derivative and d is the uninitialized operator (i.e., the operator we would usually find in the literature). Furthermore, 𝜓(xi , −q, b, c, t) is called the initialization function and is generally a time-varying function that must be added to the fractional-order operator to account for the effects of the past. Note that xi (t) is not required to be the same as x(t). Then, for the initialized fractional integration of t p t

1 (t − 𝜏)q−1 𝜏 p d𝜏 + 𝜓(fi , −q, b, c, t), q > 0, p > −1, t > c, Γ(q) ∫c

−q

p c Dt t =

(D.7)

or t

−q

p c Dt t =

c

1 1 (t − 𝜏)q−1 𝜏 p d𝜏 + (t − 𝜏)q−1 fi (𝜏)d𝜏, q > 0, p > −1, t > c, Γ(q) ∫c Γ(q) ∫b=0 −q

(D.8)

−q

where t > c > b. Note that when fi (t) = t p , c Dt t p = 0 Dt t p . The forms of equations (D.7) and (D.8) now properly account for the history (initialization) effects the fractional integration beginning at t = c, with an initialization history starting at t = b = 0. In the general case, b may be less than zero; however, here b > 0 is selected so as to apply to the fractional trigonometric functions. The 𝜓 term can be thought of as a generalization of the constant of integration in the integer-order calculus; both bring into consideration the effects of the past. Now, consider the first integral in equation (D.8): t

t

c

1 1 1 (t − 𝜏)q−1 𝜏 p d𝜏 = (t − 𝜏)q−1 𝜏 p d𝜏 − (t − 𝜏)q−1 𝜏 p d𝜏, t > c, q > 0. Γ(q) ∫c Γ(q) ∫0 Γ(q) ∫0 (D.9) For the middle integral in equation (D.9), we let 𝜏 = tu ⇒ d𝜏 = tdu 1

t

1 1 (t − 𝜏)q−1 𝜏 p d𝜏 = (t − tu)q−1 (tu)p tdu, q > 0, t > c, Γ(q) ∫0 Γ(q) ∫0 1

t

1 t q+p (t − 𝜏)q−1 𝜏 p d𝜏 = (1 − u)q−1 (u)p du, q > 0, t > c. Γ(q) ∫0 Γ(q) ∫0 Temporarily, let p = r − 1; then, 1

t

1 t q+r−1 (t − 𝜏)q−1 𝜏 p d𝜏 = (1 − u)q−1 (u)r−1 du, r > 0, q > 0, t > c. Γ(q) ∫0 Γ(q) ∫0 The integral is seen to be a beta-function [116] p. 419: t

1 t q+r−1 (t − 𝜏)q−1 𝜏 p d𝜏 = B(r, q), Γ(q) ∫0 Γ(q)

r > 0, q > 0, t > c.

D.2 Fractional Integration of the Segmented tp -Function

Replacing r = p + 1, we get for the middle integral of equation (D.9) t Γ(p + 1) 1 t q+p (t − 𝜏)q−1 𝜏 p d𝜏 = B(p + 1, q) = t q+p , p > −1, q > 0, t > c, (D.10) Γ(q) ∫0 Γ(q) Γ(p + 1 + q)

in agreement with equation (D.3). For the last integral in equation (D.9), let 𝜏 = tu ⇒ d𝜏 = tdu c

−

1 1 (t − 𝜏)q−1 𝜏 p d𝜏 = − Γ(q) ∫0 Γ(q) ∫0

c∕t

c

−

1 t q+p (t − 𝜏)q−1 𝜏 p d𝜏 = − Γ(q) ∫0 Γ(q) ∫0

(t − tu)q−1 (tu)p tdu ,

q > 0, t > c,

c∕t

(1 − u)q−1 (u)p du ,

q > 0, t > c.

Temporarily, let p = r − 1, we have c

−

1 t q+r−1 (t − 𝜏)q−1 𝜏 p d𝜏 = − Γ(q) ∫0 Γ(q) ∫0

c∕t

(1 − u)q−1 (u)r−1 du ,

q > 0, t > c.

The integral is recognized as an incomplete beta-function and in the notation of Spanier and Oldham [116] c ) 1 t q+r−1 ( − (t − 𝜏)q−1 𝜏 p d𝜏 = − B r, q, ct , q > 0, 0 ≤ ct < 1, r > 0, t > c. Γ(q) ∫0 Γ(q) Replacing r = p + 1, we get ) t q+p ( 1 (t − 𝜏)q−1 𝜏 p d𝜏 = − B p + 1, q, ct , q > 0, 0 ≤ ct < 1, p > −1, t > c. Γ(q) ∫0 Γ(q) c

−

− B(v, 𝜇, 1 − x), giving From [116], we have the relationship, B(𝜇, v, x) = Γ(𝜇)Γ(v) Γ(𝜇+v) { } c ) ( Γ(p + 1) 1 1 (t − 𝜏)q−1 𝜏 p d𝜏 = −t q+p − − B q, p + 1, 1 − ct , Γ(q) ∫0 Γ(p + q + 1) Γ(q) (D.11) q > 0, 0 ≤ ct < 1, p > −1, t > c, for the last integral of equation (D.9). Combining this result with equation (D.10), we have for the integral (D.9) { } t ) ( Γ(p + 1) Γ(p + 1) 1 1 q−1 p q+p c (t − 𝜏) 𝜏 d𝜏 = t − + B q, p + 1, 1 − t , Γ(q) ∫c Γ(p + q + 1) Γ(p + q + 1) Γ(q) (D.12) or t ) t q+p ( 1 (t − 𝜏)q−1 𝜏 p d𝜏 = B q, p + 1, 1 − ct , q > 0, 0 ≤ ct < 1, p > −1, t > c. (D.13) Γ(q) ∫c Γ(q) This result into equation (D.7) yields ) t q+p ( −q p B q, p + 1, 1 − ct + 𝜓(fi , −q, b, c, t), q > 0, p > −1, t > c, c Dt t = Γ(q)

(D.14)

c

where 𝜓(fi , −q, b, c, t) =

1 Γ(q) ∫

(t − 𝜏)q−1 fi (𝜏)d𝜏, t > c, and fi (t) is any desired function that 0

can be integrated. Then, the fractional integral of the segmented function equation (D.1) is −q 0 Dt f (t)

t ⎧ ⎪ 1 (t − 𝜏)q−1 fi (𝜏)d𝜏 b = 0 < t ≤ c = ⎨ Γ(q) ∫0 ⎪equation (D.14) t > c. ⎩

(D.15)

403

404

D Special Topics in Fractional Differintegration

D.3 Fractional Differentiation of the Segmented tp -Function We now consider fractional differentiation of equation (D.1), that is, of t p from time t = c > 0, preceded by a function fi (t). From the literature, equation (D.2) still holds [104] p. 67, but now with q>0; however, the initialization for the fractional differentiation is slightly different from that for the fractional integral. Thus, the initialization will affect the result differently. The initialization for the fractional derivative is given by q c Dt

−u m −u f (t) = c Dm t Dt f (t) = Dt (c dt f (t) + 𝜓( fi , −u, b, c, t)),

(D.16)

where f (t) is defined by equation (D.1). Furthermore, we take m as the least integer greater than q, with q = (m − u)>0 and t > c ≥ b = 0. Furthermore, with f (t) = 0 for all t < 0, then ) ( t dm 1 q u−1 p (t − 𝜏) 𝜏 d𝜏 + 𝜓(fi , −u, b, c, t) . c Dt f (t) = dt m Γ(u) ∫c Leaving fi general, and for simplicity we consider only 0 ≤ q < 1, ⇒ m = 1 ) ( t d 1 q p u−1 p D t = (t − 𝜏) 𝜏 d𝜏 + 𝜓(f , −u, b, c, t) , t > c, c t i dt Γ(u) ∫c q p c Dt t

t

=

c

d 1 d 1 d (t − 𝜏)u−1 𝜏 p d𝜏 − (t − 𝜏)u−1 𝜏 p d𝜏 + 𝜓(fi , −u, b, c, t). (D.17) dt Γ(u) ∫0 dt Γ(u) ∫0 dt

We now consider the first integral of (D.17) and apply Leibnitz’ rule t

t

d 1 1 𝜕 (t − 𝜏)u−1 𝜏 p d𝜏 = (t − 𝜏)u−1 𝜏 p d𝜏+(t − t)u−1 𝜏 p − 0, ∫ ∫ dt Γ(u) 0 Γ(u) 0 𝜕t t (u − 1) t d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = (t − 𝜏)u−2 𝜏 p d𝜏, t > c. dt Γ(u) ∫0 Γ(u) ∫0 Now, let 𝜏 = tv ⇒ d𝜏 = t dv; then, t (u − 1) 1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = (t − tv)u−2 (tv)p t dv, dt Γ(u) ∫0 Γ(u) ∫0 t (u − 1)t u+p−1 1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = (1 − v)u−2 (v)p dv, t > c. ∫0 dt Γ(u) ∫0 Γ(u)

Integrating by parts gives t (u − 1)t u+p−1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = dt Γ(u) ∫0 Γ(u)

] 1 1 (1 − v)u−1 p−1 (1 − v)u−1 (v)p || − p v dv , | +∫ u−1 u−1 |0 0 ] [ 1 p (1 − v)u−1 vp−1 dv , 0+ u − 1 ∫0 [

t (u − 1)t u+p−1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = dt Γ(u) ∫0 Γ(u) t 1 u+p−1 pt d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = (1 − v)u−1 vp−1 dv. dt Γ(u) ∫0 Γ(u) ∫0

The integral is recognized as the complete beta-function giving the first integral as t p t u+p−1 p t u+p−1 Γ(p)Γ(u) t u+p−1 Γ(p + 1) d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = B(p, u) = = . dt Γ(u) ∫0 Γ(u) Γ(u)Γ(p + u) Γ(p + u)

(D.18)

Replacing u, for our case u = 1 − q t p t p−q Γ(p + 1) p−q d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = B(p, 1 − q) = t , t > c, ∫ dt Γ(u) 0 Γ(1 − q) Γ(p − q + 1)

(D.19)

D.3 Fractional Differentiation of the Segmented tp -Function

in agreement with equation (D.2) but now constrained to 0 ≤ q < 1. The second integral in equation (D.17) is derived in similar manner. Application of Leibnitz’ rule gives c

−

c

u−1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = − (t − 𝜏)u−2 𝜏 p d𝜏, t > c. ∫ dt Γ(u) 0 Γ(u) ∫0

(D.20)

Letting 𝜏 = tv ⇒ d𝜏 = t dv gives c

−

(u − 1)t u+p−1 d 1 (t − 𝜏)u−1 𝜏 p d𝜏 = − ∫0 dt Γ(u) ∫0 Γ(u)

c∕t

(1 − v)u−2 (v)p dv.

Integrating by parts ] [ c∕t c∕t (1 − v)u−1 (u − 1)t u+p−1 (1 − v)u−1 (v)p || p−1 =− − p(v) dv , | +∫ Γ(u) u−1 u−1 |0 0 ] [ ( )u−1 ( c )p c∕t 1 − ct p (u − 1)t u+p−1 t u−1 p−1 (1 − v) (v) dv , − +0+ =− Γ(u) u−1 u − 1 ∫0 =

)u−1 ( c )p pt u+p−1 c∕t t u+p−1 ( 1 − ct − (1 − v)u−1 (v)p−1 dv, t > c. t Γ(u) Γ(u) ∫0

The integral on the right is seen to be an incomplete beta-function, giving the second integral of equation (D.17) as −

c )u−1 ( c )p p t u+p−1 ( ) t u+p−1 ( d 1 1 − ct (t − 𝜏)u−1 𝜏 p d𝜏 = − B p, u, ct , t ∫ dt Γ(u) 0 Γ(u) Γ(u) t > c, 0 ≤ ct < 1, p > 0, u > 0, 0 < q < 1.

(D.21)

Substituting the results in equations (D.19) and (D.21) into equation (D.17) yields q p c Dt t

=

)u−1 ( c )p p t u+p−1 ( ) Γ(p + 1) p−q t u+p−1 ( 1 − ct − t + B p, u, ct t Γ(p − q + 1) Γ(u) Γ(u) d + 𝜓(fi , −u, b, c, t), t > c, 0 ≤ ct < 1, p > 0, u > 0, 0 < q < 1. dt

(D.22)

Replacing u by 1 − q q

p c Dt t =

)−q ( c )p ) ( Γ(p + 1) p−q p t p−q t p−q ( 1 − ct − t + B p, 1 − q, ct t Γ(p − q + 1) Γ(1 − q) Γ(1 − q) d + 𝜓(fi , −u, b, c, t), t > c, 0 ≤ ct < 1, p > 0, u > 0, 0 < q < 1. dt

Spanier and Oldham [116] 58:5:3, p. 576, provide the following useful relation: B(v, 𝜇 + 1, x) =

𝜇 xv (1 − x)𝜇 B(v + 1, 𝜇, x) + . v v

(D.23)

405

406

D Special Topics in Fractional Differintegration

Applied to equation (D.23), this yields q p c Dt t

=

)−q ( c )p Γ(p + 1) p−q t p−q ( 1 − ct t + t Γ(p − q + 1) Γ(1 − q) [ ( c )p ( )−q ] 1 − ct ) p t p−q −q ( d t c − B p + 1, −q, t + + 𝜓(fi , −u, b, c, t), Γ(1 − q) p p dt

t > c, 0 ≤ ct < 1, p > 0, u > 0, 0 < q < 1, ) d ( Γ(p + 1) p−q q t p−q q p t + B p + 1, −q, ct + 𝜓(fi , −u, b, c, t), c Dt t = Γ(p − q + 1) Γ(1 − q) dt c t > c, 0 ≤ t < 1, p > 0, u > 0, 0 < q < 1,

(D.24)

where the choice of the function fi (t) that occurs over the historical or initialization period b ≤ t < c influences the result over the period of consideration, that is, t > c. Explicitly, the initialization term for this differentiation is given by c

d 1 (t − 𝜏)u−1 fi (𝜏)d𝜏, t > c. dt Γ(u) ∫b=0

(D.25)

Thus, the qth-order fractional derivative of f (t) as defined in equation (D.1) is given by

q 0 Dt f (t)

t ⎧ ⎪ 1 d (t − 𝜏)q−1 fi (𝜏)d𝜏, b = 0 < t < c, = ⎨ Γ(q) dt ∫0 ⎪equation (D.24) t > c. ⎩

(D.26)

D.4 Fractional Integration of Segmented Fractional Trigonometric Functions In this and previous chapters, we have derived expressions for the differintegrals of the fractional trigonometric functions. A common feature of these results is that all of the functions and their differintegrals start at t = 0+ . Our interest here is functions of the form { fi (t), b = 0 < t ≤ c, (D.27) f (t) = Cosq,v (a, 𝛼, 𝛽, t), c < t, where the Cosq,v -function is considered as a typical fractional trigonometric function and is segmented in the sense that it does not start from t = 0+ . In general, the effect of the past must be initialized with a time-varying function rather than a constant, as is the case for integer-order integration. Therefore, we consider the initialized fractional integration of Cosq,v (a, 𝛼, 𝛽, k, t) starting at t = c>0. We assume that fi and all fi(n) = 0∀t < 0. Then, for the principal function, we have −u c Dt Cosq,v (a, 𝛼, 𝛽, t)

= c dt−u Cosq,v (a, 𝛼, 𝛽, t) + 𝜓(fi , −u, b, t), u > 0, t > c, t

−u c Dt Cosq,v (a, 𝛼, 𝛽, t)

=

1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 + 𝜓(fi , u, b, t), Γ(u) ∫c u > 0, t > c,

D.4 Fractional Integration of Segmented Fractional Trigonometric Functions t

=

c

1 1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 − (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 ∫ Γ(u) 0 Γ(u) ∫0 +𝜓(fi , −u, b, t), t > c, u > 0. (D.28)

Consider now the first integral t

t

1 1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 = (t − 𝜏)u−1 Γ(u) ∫0 Γ(u) ∫0 ∞ ∑ an 𝜏 (n+1)q−1−v × cos(𝜆n )d𝜏, t > c, u > 0, Γ((n + 1)q − v) n=0 where 𝜆n = (𝛼n + 𝛽[(n + 1)q − 1 − v]) 𝜋2 for the principal function. Integrating term-by-term =

∞ ∑ n=0

t an cos(𝜆n ) 1 (t − 𝜏)u−1 𝜏 (n+1q−1−v d𝜏, u > 0. Γ((n + 1)q − v) Γ(u) ∫0

At this point, an important question arises: relative to the term-by-term integration of each t ( ) should the integration be initialized in the same manner as was done for tp in Section D.2? The answer depends on the intent of the analysis. If t ( ) is presumed to have a history during 0 < 𝜏 < c that affects t ( ) during c < 𝜏 ≤ t, then each t ( ) should be initialized as done in Section D.2. A more likely scenario would be that the intent is to have the segment of the Cosq,v -function that starts at t = c>0 to effectively be an overlay of the function that starts at t = 0, but to allow the possibility that the cos segment has a history function that we choose. That is, we wish to fractionally integrate f (t) such that equation (D.27) applies. Here, our intention is the latter case. Now, let 𝜏 = t𝜂 ⇒ d𝜏 = t d𝜂 1 n 1 ∑ a cos(𝜆n ) (t − t𝜂)u−1 (t𝜂)(n+1)q−1−v dt𝜂, u > 0, = Γ(u) n=0 Γ((n + 1)q − v) ∫0 ∞

1 n 1 ∑ a cos(𝜆n ) = (1 − 𝜂)u−1 (𝜂)(n+1)q−1−v d𝜂, u > 0. t u−1+(n+1)q−v ∫0 Γ(u) n=0 Γ((n + 1)q − v) ∞

Thus, the first integral term of equation (D.28) is given by t

1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 Γ(u) ∫0 ∞ n 1 ∑ a cos(𝜆n ) = t u−1+(n+1)q−v B((n + 1)q − v, u), Γ(u) n=0 Γ((n + 1)q − v) (n + 1)q − v > 0 ⇒ q > v, u > 0, t > c,

(D.29)

where 𝜆n = (𝛼n + 𝛽[(n + 1)q − 1 − v]) 𝜋2 for the principal function. Considering the second integral of equation (D.28) c

1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 Γ(u) ∫0 ∞ c ∑ an 𝜏 (n+1)q−1−v 1 (t − 𝜏)u−1 cos(𝜆n ) d𝜏, u > 0, t > c, =− Γ(u) ∫0 Γ((n + 1)q − v) n=0

−

where 𝜆n = (𝛼n + 𝛽[(n + 1)q − 1 − v]) 𝜋2 . Integrating term-by-term c n 1 ∑ a cos(𝜆n ) (t − 𝜏)u−1 𝜏 (n+1)q−1−v d𝜏, u > 0, t > c. Γ(u) n=0 Γ((n + 1)q − v) ∫0 ∞

=−

407

408

D Special Topics in Fractional Differintegration

Under the integral, let 𝜏 = t𝜂 ⇒ d𝜏 = t d𝜂 ∞ c∕t n 1 ∑ a cos(𝜆n ) =− (t − t𝜂)u−1 (t𝜂)(n+1)q−1−v t d𝜂, u > 0, t > c, Γ(u) n=0 Γ((n + 1)q − v) ∫0 c∕t n 1 ∑ a cos(𝜆n ) (1 − 𝜂)u−1 (𝜂)(n+1)q−1−v d𝜂, u > 0, t > c. t u−1+(n+1)q−v ∫0 Γ(u) n=0 Γ((n + 1)q − v) ∞

=−

Giving for the second integral of equation (D.28) c

1 (t − 𝜏)u−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 Γ(u) ∫0 ∞ n ) ( 1 ∑ a cos(𝜆n ) =− t u−1+(n+1)q−v B (n + 1)q − v, u, ct , t > c, q > v, u > 0. Γ(u) n=0 Γ((n + 1)q − v)

−

(D.30)

Substituting the results of equations (D.29) and (D.30) back into equation (D.28) then gives ∞ n 1 ∑ a cos(𝜆n ) −u D Cos (a, 𝛼, 𝛽, t) = t u−1+(n+1)q−v B((n + 1)q − v, u) c t q,v Γ(u) n=0 Γ((n + 1)q − v) n ) ( 1 ∑ a cos(𝜆n ) t u−1+(n+1)q−v B (n + 1)q − v, u, ct + 𝜓(fi , −u, b, t), Γ(u) n=0 Γ((n + 1)q − v) ∞

−

q > v, u > 0, t > c.

(D.31)

Writing equation 58:5:1 of [116] as B(𝜇; 𝜔; x) = B(𝜇 ∶ 𝜔) − B(𝜔; 𝜇; 1 − x) allows the simplification −u c Dt Cosq,v (a, 𝛼, 𝛽, t) =

∞ n ) ( 1 ∑ a cos(𝜆n ) t u−1+(n+1)q−v B u; (n + 1)q − v; 1 − ct Γ(u) n=0 Γ((n + 1)q − v)

+𝜓(fi , −u, b, t), q > v, u > 0, t > c,

(D.32)

where fi is only defined for b = 0+ < t < c and c

1 (t − 𝜏)u−1 fi (𝜏)d𝜏, q > v, u > 0, t > c. Γ(u) ∫0 The fractional integral for f (t) as defined by equation (D.27) now is given by 𝜓(fi , −u, b, t) =

t ⎧ 1 ⎪ (t − 𝜏)u−1 fi (𝜏)d𝜏, b = 0 < t ≤ c, f (−u) (t) = ⎨ Γ(u) ∫0 ⎪equation (D.32), t > c. ⎩

(D.33)

D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions Here, we are concerned with the fractional differentiation of f (t) as defined in equation (D.27). Therefore, we consider the initialized fractional differentiation of Cosq,v (a, 𝛼, 𝛽, k, t) starting at t = c. Again, we assume that fi and all fi(n) = 0∀t < 0. Then, for the principal function, we have u c Dt Cosq,v (a, 𝛼, 𝛽, t)

−w = Dm t {c dt Cosq,v (a, 𝛼, 𝛽, t) + 𝜓(fi , −w, b, c, t)},

u = (m − w) > 0, t > c,

(D.34)

D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions

where m is the least integer greater than u. Then, } { t dm 1 u w−1 (t − 𝜏) Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 + 𝜓(fi , −w, b, c, t) , c Dt Cosq,v (a, 𝛼, 𝛽, t) = dt m Γ(w) ∫c u = (m − w) > 0, t > c, u c Dt Cosq,v (a, 𝛼, 𝛽, t)

t

m

=

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt m Γ(w) ∫0 c dm 1 − m (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 dm + m 𝜓(fi , −w, b, c, t), u = (m − w) > 0, t > c. dt

(D.35)

We consider the first integral and, for simplicity, we will confine 0 < u < 1 ⇒ m = 1 t

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 ∞ t ∑ an 𝜏 (n+1)q−1−v d 1 w−1 (t − 𝜏) = cos(𝜆n )d𝜏, u = (1 − w) > 0, t > c, dt Γ(w) ∫0 Γ((n + 1)q − v) n=0 where 𝜆n = (𝛼n + 𝛽[(n + 1)q − 1 − v]) 𝜋2 . Integrating term-by-term t n 1 ∑ a cos(𝜆n ) d (t − 𝜏)w−1 𝜏 (n+1)q−1−v d𝜏, u = (1 − w) > 0, t > c. Γ(w) n=0 Γ((n + 1)q − v) dt ∫0 ∞

=

Applying Leibnitz’ rule t n 1 ∑ a cos(𝜆n )(w − 1) (t − 𝜏)w−2 𝜏 (n+1)q−1−v d𝜏, u = (1 − w) > 0, t > c. Γ(w) n=0 Γ((n + 1)q − v) ∫0 ∞

=

Now, let 𝜏 = t𝜇 ⇒ d𝜏 = t d𝜇; thus, 1 n 1 ∑ a cos(𝜆n )(w − 1) w−2+(n+1)q−v (1 − 𝜇)w−2 𝜇(n+1)q−1−v d𝜇, u = (1 − w) > 0, t > c, t ∫0 Γ(w) n=0 Γ((n + 1)q − v) (D.36) integrating by parts ∞

=

1

∫0

(1 − 𝜇)w−2 𝜇(n+1)q−1−v d𝜇 [(n + 1)q − 1 − v] 1 𝜇 (n+1)q−1−v (1 − 𝜇)w−1 || (1 − 𝜇)w−1 𝜇(n+1)q−2−v d𝜇, | + | ∫0 w−1 w − 1 |0 1

=− or 1

[(n + 1)q − 1 − v] 1 (1 − 𝜇)w−1 𝜇(n+1)q−2−v d𝜇, ∫0 ∫0 w−1 1 [(n + 1)q − 1 − v] (1 − 𝜇)w−2 𝜇(n+1)q−1−v d𝜇 = + B((n + 1)q − 1 − v, w). ∫0 w−1 (1 − 𝜇)w−2 𝜇(n+1)q−1−v d𝜇 = +

409

410

D Special Topics in Fractional Differintegration

This result substituted back into equation (D.36) gives t

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 ∞ n 1 ∑ a cos(𝜆n )(w − 1) w−2+(n+1)q−v [(n + 1)q − 1 − v] = t B((n + 1)q − 1 − v, w), Γ(w) n=0 Γ((n + 1)q − v) w−1 u = (1 − w) > 0, t > c, where w>0. This result is extended for real arguments by the substitution B(x, y) = Γ(x)Γ(y)∕Γ(x + y). Then, we have n Γ((n + 1)q − 1 − v)Γ(w) 1 ∑ a cos(𝜆n ) t w−2+(n+1)q−v [(n + 1)q − 1 − v] , Γ(w) n=0 Γ((n + 1)q − v) Γ((n + 1)q − 1 − v + w) ∞

=

u = (1 − w) > 0, t > c, ∞ ∑ an cos(𝜆n ) t w−2+(n+1)q−v , u = (1 − w) > 0, t > c. = Γ((n + 1)q − 1 − v + w) n=0 This gives for the first integral of equation (D.35) with m = 1: t ∑ an cos(𝜆n ) d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 = t (n+1)q−1−v−u , dt Γ(w) ∫0 Γ((n + 1)q − v − u) n=0 ∞

t > c.

(D.37)

We now consider the second integral of equation (D.35): c

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 ∞ c ∑ an 𝜏 (n+1)q−1−v d 1 (t − 𝜏)w−1 =− cos(𝜆n )d𝜏, t > c, dt Γ(w) ∫0 Γ((n + 1)q − v) n=0

−

c n 1 ∑ a cos(𝜆n ) d (t − 𝜏)w−1 𝜏 (n+1)q−1−v d𝜏, t > c. Γ(w) n=0 Γ((n + 1)q − v) dt ∫0 ∞

=−

Applying Leibnitz’ rule c

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 ∞ c n 1 ∑ a cos(𝜆n )(w − 1) (t − 𝜏)w−2 𝜏 (n+1)q−1−v d𝜏, t > c. =− Γ(w) n=0 Γ((n + 1)q − v) ∫0

−

Applying integration by parts to last integral gives c

∫0

c(n+1)q−1−v (t − c)w−1 w−1 (n + 1)q − 1 − v c + (t − 𝜏)w−1 𝜏 (n+1)q−2−v d𝜏, t > c. ∫0 w−1

(t − 𝜏)w−2 𝜏 (n+1)q−1−v d𝜏 = −

(D.38)

D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions

This result into equation (D.38) yields c

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 [ (n+1)q−1−v ∞ n (t − c)w−1 c 1 ∑ a cos(𝜆n )(w − 1) =− − Γ(w) n=0 Γ((n + 1)q − v) w−1 ] c (n + 1)q − 1 − v w−1 (n+1)q−2−v + (t − 𝜏) 𝜏 d𝜏 , t > c. ∫0 w−1

−

Let 𝜏 = t𝜇 ⇒ d𝜏 = t d𝜇; thus, c

d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(w) ∫0 [ (n+1)q−1−v ∞ n (t − c)w−1 c 1 ∑ a cos(𝜆n )(w − 1) =− − Γ(w) n=0 Γ((n + 1)q − v) w−1

−

] (n + 1)q − 1 − v (n+1)q−2+w−v c∕t w−1 (n+1)q−2−v + (1 − 𝜇) 𝜇 d𝜇 , t > c, t ∫0 w−1 c d 1 (t − 𝜏)w−1 Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 = − dt Γ(w) ∫0 ∞ n 1 ∑ a cos(𝜆n ) = [c(n+1)q−1−v (t − c)w−1 Γ(w) n=0 Γ((n + 1)q − v) )] ( − (n + 1)q − 1 − v) t (n+1)q−2+w−v B((n + 1)q − 1 − v, w, ct , t > c. Replacing w with w = 1 − u gives c

d 1 (t − 𝜏)−u Cosq,v (a, 𝛼, 𝛽, 𝜏)d𝜏 dt Γ(1 − u) ∫0 ∞ ∑ an cos(𝜆n ) 1 [c(n+1)q−1−v (t − c)−u = Γ(1 − u) n=0 Γ((n + 1)q − v) )] ( − (n + 1)q − 1 − v) t (n+1)q−1−u−v B((n + 1)q − 1 − v, 1 − u, ct , t > c, −

(D.39)

for the second integral. This result together with equation (D.37) is used to evaluate equation (D.35) (with m = 1) as u c Dt Cosq,v (a, 𝛼, 𝛽, t)

=

∞ ∑ n=0

an cos(𝜆n ) t (n+1)q−1−v−u Γ((n + 1)q − v − u)

∑ a cos(𝜆n ) 1 [c(n+1)q−1−v (t − c)−u Γ(1 − u) n=0 Γ((n + 1)q − v) )] d ( − (n + 1)q − 1 − v) t (n+1)q−1−u−v B((n + 1)q − 1 − v, 1 − u, ct + 𝜓(fi , −w, b, c, t), dt u = (1 − w) > 0, t > c, q > v, 0 ≤ ct < 1, and w − 1 must be real, (D.40) ∞

n

+

with 0 < u < 1 ⇒ m = 1, 𝜆n = (𝛼n + 𝛽[(n + 1)q − 1 − v]) 𝜋2 for the principal function, and c

1 d d (t − 𝜏)w−1 fi (𝜏)d𝜏, t > c. 𝜓(fi , −w, b, c, t) = dt Γ(w) dt ∫0

(D.41)

411

412

D Special Topics in Fractional Differintegration

Then, the solution for the fractional derivative of the segmented function defined by equation (D.27) is given by t ⎧ 1 d ⎪ (t − 𝜏)w−1 fi (𝜏)d𝜏, b = 0 < t ≤ c, f (u) (t) = ⎨ Γ(w) dt ∫0 ⎪ equation (D.40), t > c, ⎩

for any integrable function fi and where w = 1 − u, with 0 < u < 1.

(D.42)

413

E Alternate Forms E.1 Introduction In Chapter 9 and previous chapters, definitions of the fractional trigonometric functions have been developed. It should be noted that the form of equation (9.1) was chosen for the convenience that it allowed either the “a” and/or the “t” variables to be made complex. The cost of this convenience, however, was to introduce two new variables, 𝛼 and 𝛽, into the defining summation. Complexly proportional forms with one less variable in the summation follow. Here, useful alternative, generally nonequivalent, definitions to the previous forms are presented. The following is adapted from Lorenzo [82] with permission of ASME: Equation (9.1) is rewritten as Rq,v ( ai𝛼 , i𝛽 t) = i𝛽(q−1−v)

∞ ∑ an (t)(n+1)q−1−v in(𝛼+𝛽q) n=0

Γ((n + 1)q − v)

,

t > 0.

(E.1)

For q, 𝛼, and 𝛽 rational, equation (3.124) is applied to in(𝛼+𝛽q) , giving 𝛼

𝛽

Rq,v ( ai , i t) = i

𝛽(q−1−v)

∞ ( ( )) ∑ an (t)(n+1)q−1−v 𝜋 cos n(𝛼 + 𝛽 q) + 2𝜋k Γ((n + 1)q − v) 2 n=0

+ i i𝛽(q−1−v)

∞ ( ( )) ∑ an (t)(n+1)q−1−v 𝜋 sin n(𝛼 + 𝛽 q) + 2𝜋k , Γ((n + 1)q − v) 2 n=0

t > 0, k = 0, 1, 2, … , D1 − 1,

(E.2)

where D1 is the product of the denominators of Dq , D𝛼 , D𝛽 in minimal form. For v, q, 𝛼, and 𝛽 rational, equation (3.124) is again applied to i𝛽(q−1−v) , the equation is expanded, and real and imaginary terms are collected to give Rq,v ( ai𝛼 , i𝛽 t) =

∞ ( ( )) ( ( )) ∑ an (t)(n+1)q−1−v [ 𝜋 𝜋 cos n(𝛼 + 𝛽 q) cos 𝛽(q − 1 − v) + 2𝜋k1 + 2𝜋k2 Γ((n + 1)q − v) 2 2 n=0 )) ( ( ))] ( ( 𝜋 𝜋 + 2𝜋k1 sin 𝛽(q − 1 − v) + 2𝜋k2 − sin n(𝛼 + 𝛽 q) 2 2 ∞ ( ( )) ( ( )) ∑ an (t)(n+1)q−1−v [ 𝜋 𝜋 cos n(𝛼 + 𝛽 q) sin 𝛽(q − 1 − v) +i + 2𝜋k1 + 2𝜋k2 Γ((n + 1)q − v) 2 2 n=0

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

414

E Alternate Forms

)) ( ( ))] ( ( 𝜋 𝜋 , + sin n(𝛼 + 𝛽 q) + 2𝜋k1 cos 𝛽(q − 1 − v) + 2𝜋k2 2 2 t > 0, k1 = 0, 1, 2, … , D1 − 1, k2 = 0, 1, 2, … , D2 − 1,

(E.3)

where D2 is the product of the denominators of Dq , Dv , D𝛽 with repeated multipliers removed. From this equation, we have the following alternate definitions, where B is used to discriminate from our basic definitions: BCosq,v (a, 𝛼, 𝛽, k1 , k2 , t) ∞ ( ( )) ( ( )) ∑ an (t)(n+1)q−1−v [ 𝜋 𝜋 cos n(𝛼 + 𝛽 q) = cos 𝛽(q − 1 − v) + 2𝜋k1 + 2𝜋k2 Γ((n + 1)q − v) 2 2 n=0 )) ( ( ))] ( ( 𝜋 𝜋 , − sin n(𝛼 + 𝛽 q) + 2𝜋k1 sin 𝛽(q − 1 − v) + 2𝜋k2 2 2 t > 0, k1 = 0, 1, 2, … , D1 − 1, k2 = 0, 1, 2, … , D2 − 1, (E.4)

BSinq,v (a, 𝛼, 𝛽, k1 , k2 , t) =

∞ ( ( )) ( ( )) ∑ an (t)(n+1)q−1−v [ 𝜋 𝜋 cos n(𝛼 + 𝛽 q) sin 𝛽(q − 1 − v) + 2𝜋k1 + 2𝜋k2 Γ((n + 1)q − v) 2 2 n=0 )) ( ( ))] ( ( 𝜋 𝜋 , + sin n(𝛼 + 𝛽 q) + 2𝜋k1 cos 𝛽(q − 1 − v) + 2𝜋k2 2 2 t > 0, k1 = 0, 1, 2 · · · D1 − 1, k2 = 0, 1, 2 · · · D2 − 1. (E.5)

Complexly proportional forms with one less variable in the summation follow in the next section.

E.2 Reduced Variable Summation Forms The basis form for the meta-trigonometry is Rq,v ( ai𝛼 , i𝛽 t) =

∞ ∑ (ai𝛼 )n (i𝛽 t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

,

t > 0.

(E.6)

The following forms are derived with one less variable in the summation. Equation (E.1) is rewritten as ∞ ∑ (ai𝛼 )n (i𝛽q t q )n 𝛼 𝛽 𝛽 q−1−v , t > 0, (E.7) Rq,v ( ai , i t) = (i t) Γ((n + 1)q − v) n=0 𝛼

𝛽

Rq,v ( ai , i t) = i

𝛽(q−1−v)

∞ ∑ (ai(𝛼+𝛽q) )n t (n+1)q−1−v n=0

Γ((n + 1)q − v)

,

t > 0.

(E.8)

t > 0.

(E.9)

Letting 𝜎 = 𝛼 + 𝛽 q ⇒ 𝛽 = (𝜎 − 𝛼)∕q, Rq,v ( ai𝛼 , i𝛽 t) = i(𝜎−𝛼)(q−1−v)∕q

∞ ∑ (ai𝜎 )n t (n+1)q−1−v n=0

Γ((n + 1)q − v)

,

effectively eliminating one variable in the summation of equation (E.1). The exponent of i, of course, could then be reduced by one variable in a new form. Written in terms of the R-function, we have Rq,v ( ai𝛼 , i𝛽 t) = i(𝜎−𝛼)(q−1−v)∕q Rq,v (ai𝜎 , t), t > 0. (E.10)

E.3 Natural Quency Simplification

Here, we see a potential alternate basis for a meta-trigonometry, namely Rq,v (ai𝜎 , t). The leading complex constant merely serves to reorient or rotate the functions in the complex plane. Equation (E.10) essentially tells us that the generalized definition (E.1) with 𝛽 = 0 is sufficient to create a complete set of fractional trigonometric functions. In other words, setting 𝛽 = 0 in equation (9.1) will generate a trigonometry equivalent to that generated by equation (E.10). Conversely, the 𝛼 variable may be eliminated in the defining equation. To see this, rewrite equation (E.7) as ∞ ∑ (a)n (i𝛼+𝛽q t q )n , t > 0. (E.11) Rq,v ( ai𝛼 , i𝛽 t) = (i𝛽 t)q−1−v Γ((n + 1)q − v) n=0 and let 𝜉 = (𝛼 + 𝛽 q) ∕q, Rq,v ( ai𝛼 , i𝛽 t) = (i𝛽 t)q−1−v Rq,v ( ai𝛼 , i𝛽 t) = i

∞ ∑

an (i𝜉 t)nq , Γ((n + 1)q − v)

n=0 ∞ ∑ (𝛽−𝜉)(q−1−v) n=0

t > 0.

an (i𝜉 t)(n+1)q−1−v , Γ((n + 1)q − v)

(E.12)

t > 0,

(E.13)

which is complexly proportional to equation (E.1) with 𝛼 = 0. Again, the exponent of i could then be reduced by one variable in a new form. In terms of the equivalent R-function, Rq,v ( ai𝛼 , i𝛽 t) = i(𝛽−𝜉)(q−1−v) Rq,v (a, i𝜉 t),

t > 0.

(E.14)

E.3 Natural Quency Simplification Consider Rq,v (a, t) ≡

∞ ∑ (a)n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

,

t > 0.

(E.15)

q

Now, let a = i𝛿 Ωn , where Ωn is called the natural quency (“que-ency,” k¯u-en-s𝚤), and i𝛿 allows to become complex; this gives the simplification

q Ωn

𝛿

Rq,v (i Ωn , t) =

∞ q ∑ (i𝛿 Ωn )n (t)(n+1)q−1−v n=0

Γ((n + 1)q − v)

Then, Rq,v (i𝛿 Ωn , t) =

=t

q−1−v

∞ ∑ n=0

(Ωn t)nq i𝛿 n , Γ((n + 1)q − v)

∞ nq ∑ Ωn (t)(n+1)q−1−v 𝛿n i , Γ((n + 1)q − v) n=0

t > 0.

t > 0.

Assuming 𝛿 rational and applying equation (3.124) gives ∞ nq ( ( )) ∑ Ωn (t)(n+1)q−1−v 𝜋 Rq,v (i𝛿 Ωn , k, t) = cos 𝛿 n + 2𝜋k Γ((n + 1)q − v) 2 n=0 ∞ nq ( ( )) (n+1)q−1−v ∑ Ωn (t) 𝜋 sin 𝛿 n + 2𝜋k , +i Γ((n + 1)q − v) 2 n=0

(E.16)

(E.17)

t > 0.

This allows the following pair of simplified definitions: ∞ nq ( ( )) ∑ Ωn (t)(n+1)q−1−v 𝜋 B1 Cosq,v (Ωn , 𝛿, k, t) ≡ cos 𝛿 n + 2𝜋k , t > 0, Γ((n + 1)q − v) 2 n=0 ∞ nq ( ( )) (n+1)q−1−v ∑ Ωn (t) 𝜋 B1 Sinq,v (Ωn , 𝛿, k, t) ≡ sin 𝛿 n + 2𝜋k , t > 0, Γ((n + 1)q − v) 2 n=0

(E.18)

(E.19) (E.20)

415

416

E Alternate Forms

where the B1 notation indicates alternative trigonometric functions, which are not pursued here. These definitions may be further simplified by constraining attention to only the principal, k = 0, functions ∞ nq ∑ Ωn (t)(n+1)q−1−v cos(𝛿 n𝜋∕2), t > 0, (E.21) B1 Cosq,v (Ωn , 𝛿, t) ≡ Γ((n + 1)q − v) n=0 B1 Sinq,v (Ωn , 𝛿, t) ≡

∞ nq ∑ Ωn (t)(n+1)q−1−v sin(𝛿 n𝜋∕2), Γ((n + 1)q − v) n=0

t > 0.

(E.22)

These forms are equivalent to taking k = 𝛽 = 0, a = Ωq , and 𝛼 = 𝛿 in the original definitions, that is, equations (9.6) and (9.7). The Laplace transforms associated with this simplification are [ ] q { } sq − Ωn cos(𝛿𝜋∕2) v L B1 Cosq,v (Ωn , 𝛿, t) = s , q > v, (E.23) q 2q s2q − 2Ωn cos(𝛿𝜋∕2)sq + Ωn [ ] q } { sin(𝛿𝜋∕2) Ω n , q > v. (E.24) L B1 Sinq,v (Ωn , 𝛿, t) = sv q 2q s2q − 2Ωn cos(𝛿𝜋∕2)sq + Ωn The related R-function equations are B1 Cosq,v (Ωn , 𝛿, t) =

Rq,v (Ωq i𝛿 , t) + Rq,v (Ωq i−𝛿 , t) 2 Rq,v (Ω i , t) − Rq,v (Ωq i−𝛿 , t)

,

t > 0,

(E.25)

,

t > 0.

(E.26)

q 𝛿

B1 Sinq,v (Ωn , 𝛿, t) =

2i

417

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Index a Achar, B.N. 309, 314, 384 Agarwal, R.P. 20, 22 (Table 3.1), 31 Apostol, T.M. 27, 42 application of F-and R-function to cooling manifold 32–34 super-capacitor circuit 16–18 applications of the fractional trigonometry and/or hyperbolics to foam reactor 76–77 fractional differential equations see fractional differential equations fractional growth 325–328 fractional oscillators 309 galaxy classification 341–368 hurricanes morphology 371–373 indexed fractional differential equations 255–256 low pressure system cloud formation 375, 377 Nautilus 317–329 non-commensurate, fractional differential equations 254–255 sea shells 329–339 tornadoes 373–377 whirlpools 375 approximations, R-function 276–279 Aptoula, E. 342 asymptotic behavior, R2 -trigonometric functions 275–276 attraction of loci to circles 100, 186 origin 81, 186

Baillard, A. 342 Bergh, van den, S. 341 Bluestein, H.B. 373 Bologna, M. 390, 391 Bonner, J.T. 317 Buckley, B. 371

c Calico, R.A. 12 Candeas, A.J. 342 Carbon Star, morphology of 369–370 Castrejon Peta, A.A. 317 Churchill, R.V. 40, 273 Cochran, K.J. 317 commutative properties of, complexity-parity operations 179 compatibility, backward, of meta-trigonometry to fractional trigonometries 179–181 complex numbers, roots of 40 computer codes meta-trig functions 395–398 R-function 393–394 conjugate, sum and product for meta-trigonometry 207 R1 -trig 87–88 Conklin, W.A. 317

d Dance, P.S. 317 Davis, H.T. 243 Davis, P.J. 283 Drodzdov, A.D. 309

e b Badii, L. 10 Bagley, R.L. 12, 391, 392

Eab, C.H. 309 Elmegreen, D.M. 341 Emanual, K. 373

The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, First Edition. Carl F. Lorenzo and Tom T. Hartley. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. Companion Website: www.wiley.com/go/Lorenzo/Fractional_Trigonometry

426

Index

Erdelyi, A. 22 (Table 3.1) Euler equation 6–7 see also fractional Euler, and fractional semi-Euler equation exponentially damped sinusoids 162–165

f F-function 10 effect of a variation 13 and stability of 14–16 in complex plane 13 properties 16 fractal mathematical process 214–215 fractional calculus operations on meta-trigonometric functions 195 R1 -hyperbolic functions 72–73 R1 -trigonometry functions 89–90 R2 -trigonometry functions 119–127 R3 -trigonometry functions 146–157 fractional derivatives and integrals Grünwald definition 3 Laplace transform of 5–6 Riemann–Liouville definition 4 initialization of 4–5 fractional differential equations commensurate order 243 fundamental 9–11 of first kind 245 indexed, multiple solutions 255 of second kind 246 -higher order unrepeated roots 250–252 repeated complex roots 253 repeated real roots 252 Laplace transforms for 246–250, 254 non-commensurate order 75, 254–255 solution of linear 254–255 fractional differintegration t p -function 404 segmented 206 segmented fractional trig function 406 fractional Euler equations generalized 161 meta-trigonometric 206, 208, 209 R1 -trigonometry 86 R2 -trigonometry 114 R3 -trigonometry 142 fractional field equations 385–387 fractional hyperbolic differential equation 74–75 fractional oscillators 309–315

block diagram for 310, 315 coupled 314 intrinsic damping of 312 parity function based 311 space of 309 stability 310, 311 fractional poles 214 fractional semi-Euler equations generalized (meta-trigonometric) R2 -trigonometry 117, 118 R3 -trigonometry 144, 145 fractional trigonometries meta-trigonometry 159–216 R1 -trigonometry 79–95 R2 -trigonometry 97–127 R3 -trigonometry 129–157 related works 8, 389–392 Fujita scale 373 enhanced Fujita scale 373 functions defined F-function 10 fractional hyperbolic 63–64 for the fractional calculus Agarwal’s 31 Erdelyi’s 20 Gorenflo and Mainardi’s 32 Hartley’s 22 Matignon’s 22 Miller and Ross’ 21 Mittag-Leffler’s 31 Oldham and Spanier’s 31 Podlubny’s 32 R-function 22 Robotnov’s 32 summary 22 G-function 34–35 H-function 36–38 R-function 22 meta-trigonometric Coflutter 178 Cosine 161 Covibration 177 Flutter 178 Sine 161 Vibration 178 ratio and reciprocal 217–227 complexity 217 naming convention 227 parity 219, 221

208, 209

Index

relationships 226 R-function represent 225 fundamental fractional differential equation 9–16 complex plane behavior 13 equation 1, 9–16 impulse response of 10 transform of 10

g galaxy(ies), galaxy morphology 341 Am 0644-741 359, 360 barred, classification 347 process 347–350 of selected galaxies 350 ESO 269-G57 360, 361 fractional differential equat. 366 fractional spirals for 342–346, 366–367 Hubble classification 342, 364 Hoag’s Object 358, 359 inclination 347 interference of arms 365, 369 M51 358, 359 M66 (NGC3627) 353, 356 M95 350, 352 morphology of analysis 362–365 compared to Hubble 362–364 NGC 1300 349, 355, 358 NGC 1313 362, 363 NGC 1365 350, 352 NGC 2997 353, 354 NGC 4314 350, 351 NGC 4535 355, 357 NGC 4622 353, 355 shift of geometrical center from optical center for 355, 357, 359 Am 0644-741, 359 ESO 269-G57 362 M66 355 M51 358 NGC 1300 357 NGC 1313 362 r-s types 362, 364, 367, 368 symmetry assumption 347, 368 Gazale, M.J. 283 generalized trigonometric, expectations of 8 generalized fractional trigonometries 229–240

1,

G-function based 229–233 G-trigonometric function 229 G-trigonometric function, Laplace transform 231 H-function based 234–240 H-trigonometric function, Laplace transform 235 G-function 34–36 Glockle, W.G. 12 Gorenflo, R. 21, 22 (Table 3.1), 32, 309 Grigolini, P. 390, 391

h Hanneken, J.W. 384 Hargittai, I. 283, 317 Hartley, T.T. 4, 5, 7, 9, 14, 16, 19, 20, 22 (Table 3.1), 24, 26, 31, 38, 46, 47, 61, 63, 65, 66, 76, 79, 81, 89, 93, 97, 99, 107, 109–110, 125–126, 243, 266, 277–281, 284–285, 327, 341–342, 344, 348, 349, 351–352, 354–365, 373, 381 Hartmann, H. 283 Heaviside, O. 243 H-function 36–38, 234 high precision software 281 Holmberg, E. 341 Hubble, E. 341, 342 hurricanes 371–373 cloud patterns 371 Fran 371–372 Isabel 371–373 hyperboletry, fractional 63–77 differential equations 74–75 identities 64, 70 fractional calculus operations on 72–73 functions 64 inter-relationships 72 Laplace transforms 73 relation to R1 -trig. functions 89 R-function equivalents 69–71 hyperdamping 15 Hyslop, J.M. 45

i identities meta trig. 206–214 R-function 29–31, 47–61, 70, 71, 86, 87, 103, 116, 135, 143, 176, 177 impulse response function generalized 10

427

428

Index

inverse fractional-trigonometric functions 384 Mittag-Leffler 384 R-function, problem defined 382–283 three domains 383

Land, F. 317, 421 Landman 317 Laplace transform(s) fractional operators 5–6 meta-trig.functions 188–192 R1− hyperbolic functions 73 R1− trigonometric functions 90–92 R2 -trigonometric functions 111–113 R3 -trigonometric functions 140–141 simplified trigonometric 246–250 LePage, W.R. 14 Lim, S.C. 309 Lorenzo, C.F. 4, 5, 7, 9, 14, 15, 19, 20, 22 (Table 3.1), 24, 26, 38, 46, 47, 61, 63, 65, 66, 76, 79, 81, 89, 93, 97, 99, 107, 109–110, 125–126, 159, 160, 162, 164, 169, 171, 178, 186–188, 190, 193, 205, 243, 246, 266, 277–281, 284–285, 309, 310, 317, 318, 320–324, 326–328, 341, 342, 344, 348–349, 351–352, 354–361, 363–365, 373, 381, 385, 413 low pressure cloud pattern 375, 377

tornado animation 373 McMahon, T.A. 317 meta-trigonometry, defined 160–161, 166, 176–178 alternate forms 161, 413–416 backward compatibility 179, 215 closed form summation 165 commutative prop. 179 differintegration 195, 401–412 extension to negative time 384 fractional calculus operations 195–206 fractional poles 214 inter-relationships 206–211 Laplace transforms 188–192 meta-identities 211–214 R-function representation of 192–195 taxonomy of 190, 215 summary tables 193, 205 Miller, K.S. 12, 243, 389, 391 Mislin, H. 283 Mittag-Leffler function 8, 11, 12, 20, 22 (Table 3.1), 31, 46, 244, 245, 256, 383, 390, 422 defined 11 relation to F-function 11 Moore, J.A. 342 Morgan, W.W. 341 morphology carbon star 370 galaxy 341–362 hurricane cloud pattern Fran 371 Isabel 371 low pressure cloud 375 sea shell 317 tornado 375 whirlpool 375

m

n

Mainardi, F. 21, 22 (Table 3.1), 32, 309 Malti, R. 266–268 MapleⓇ , computer code for phase plane plot 318 Matignon, D. 22 (Table 3.1) MatlabⓇ 318, computer code for R-function 393 R-function evaluation 394 meta-cosine evaluation 395 meta-cosine function 395

Nautilus pompilius 421–435 growth rate 426, 433 growth rate equation 325–328 morphology 422 siphuncle 325 spiral length 325 near order effect 279–281, 287, 374 negative time, extensions to 384–385 Nonnenmacher, T.F. 12 numerical issues 274

k Kelly, J.M. 309 Kennicutt, R.C. Jr. Kerry, E. 371 k-index 374 Knopp, K. 45 Koh, C.G. 309

342

l

Index

o Oberhettinger, F. 10 Oldham, K.B. 2–4, 9, 12, 20, 22 (Table 3.1), 23, 30–35, 38, 45, 72, 211, 243, 401, 403, 405 order differential and trigonometric 378 in physical systems 378–379 Oustaloup, A. 3, 243

p Petit, E. 341 Podlubny, I. 4, 21, 22 (Table 3.1), 32, 35, 243–244, 252, 264, 390–391

r Radwan, A. 309 ratio and reciprocal trigonometric functions 217–227 naming convention 227 R-function representation of 225 relationships for 226 related trigonometries 8 Miller and Ross 389 Mittag-Leffler based Bagley 391, 392 Podlubny/Erdelyi 390 Podlubny/Luchko/Srivastava 391 Podlubny/Plotnikov/Tseylin 391 West et al 390 R1 -fractional trigonometry 79–95 defined 80 fractional calculus operations 89–90 fractional differential equations 94–95 inter-relationships 88 Lapace transforms 90–92 properties relation to R1 -hyperbolic functions 89 summary (table) 93 R2 -fractional trigonometry 97–127 defined, complexity 97–98 parity 102–108 fractional calculus operations 119–124 fractional Euler-like equations 116–119 fractional differential equations 127 inter-relationships 113–119 Laplace transforms 111–113 properties 113 summary (table) 125, 126 R3 -fractional trigonometry 129–157

defined, complexity 129–130 parity 134–137 fractional calculus operations 146–157 fractional Euler-like equations 142, 144, 145 inter-relationships 141–146 Lapace transforms 140–141 summary (table) 156, 157 R-function approximations to 276–279 convergence of 271–272 uniform convergence 273 differintegration of 25 eigen character of 38 fractional impulse function 27 identities 29–31 indexed forms of 41 inverse 382–384 Laplace transform 22–23 as a linear system 259 frequency response 260 time response 260 numerical issues 274 relationship to elementary functions 27–28 exponential functions 54–55 fractional calculus functions 31–32 Riemann–Liouville 3–5 Riemann sheets 13–14 Robotnov, Y.N. 9, 12, 21, 22 (Table 3.1), 32 Ross, B. 12, 243

s Samko, S.G. 3, 243 Sandage, A. 341 series approximations, R-function 275–279 asymptotic behavior of 275 convergence 44–45 high precision software for 281 numerical issues 274 near order effect 279 R−function 271 meta-trigonometric functions 272 uniform 273 term-by term operations 44 shells, shell morphology 317–340 ammonite 339 Arabian Tibia 333 Cardiidae 336

429

430

Index

shells, shell morphology (contd.) Conus tinianus 330 Florida conch 329 Nautilus pompilius 317–329 growth rate equation 325–328 Tiger cowry 332 Spanier, J. 2–4, 9, 12, 20, 22 (Table 3.1), 23, 30–35, 38, 45, 72, 211, 243, 401, 403, 405 spirals analysis of 288 barred 165, 187, 188, 283, 287, 306, 341–343, 350, 353, 355, 358, 365, 368 Belousov–Zhabotinsky 283 carbon star 362, 369 classical Archimedean 303, 304 Fermat’s 303, 304 hyperbolic 303, 304 logarithmic 303, 304, Theodorus’ 307 fractional trigonometric 283, 301 see also applications galactic 341 general 294 length and growth rate 294–297 referenced 301 relation to classical 303 retardation 301–303 scaling of 296–299 simple 294 velocities 299–301 stability 15, 17, 250, 259–261, 264, 266–269, 310–311, 379 Stanislavsky, A.A. 309 summary tables meta-trigonometry 193, 205 R1 -trigonometry 93 R2 -trigonometry 125, 126 R3 -trigonometry 156, 157 supercapacitor 16 modeling and response 17–18 symmetry(ies) 65, 100, 134, 162, 165, 184, 255–256, 260, 287, 318, 347, 349, 350, 353, 355, 358, 360, 362, 368, 371 systems meta-trig function based, frequency responses 261 R-function based 259

frequency responses 260 time responses 260 stability of 266–268 diagram 268

t term-by term operations 44–45 Thompson, D. 317 tornado animation 399 Fujita scale 373 k index 374 morphology classification 374–377 trigonometries, fractional complementary 384 generalized trigonometries G-function based 229 Laplace transform of 230 H-function based 234 Laplace transform of 235 meta-trigonometry, functions alternate forms 161 commutative properties of 179 complexity based functions 160 convergence of 272 fractional calculus operations on 195 identities 206, 211 identities based on integer trig. 211 Laplace transforms complexity based 188 parity based functions 191 as linear systems 261 numerical issues parity based functions 166 quency simplification 415 ratio and reciprocal functions 217 reduced variable forms 414 taxonomy of meta-trig. functions 190, 215 R-function representation 192 R1 -trigonometry differential equations 94 fractional calculus operations on 89 functions defined 79–80 identities 87 interrelationships 88 Laplace transforms 90 relation to, R1 –hyperbolic, functions 89 R-function equivalents 86, 87, 93

Index

R2 -trigonometry asymptotic behavior 100 fractional calculus operations on 119–127 functions defined complexity based 97–98 parity based 102–108 interrelationships 113–119 Laplace transforms 111–113 R-function equivalents 126 R3 -trigonometry fractional calculus operations on functions defined complexity based 129–130 parity based 102–108 identities 135, 143 interrelationships 142–146 Laplace transforms 140–141 R-function equivalents 157 trigonometry, traditional 6 transfer functions fractional 259–260

elementary, first kind 244, 245 elementary, second kind 24, 244 stability of 13–16, 266–268

v Vaucouleurs, de, G. Vermeij, G.J. 317

341

w 146

w-plane 14–16 responses in 15–16 West, B.J. 243, 390, 391 Westerman, B. 317 Whirlpool 375 Widder, D.J. 45 Winfree, A.T. 283 Wylie, C.R. 45

x X-ray study

317

431