Fractional Trigonometry - With Applications to Fractional Differential Equations and Science

Informationen zum Autor Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry.Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional-order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann's complementary-function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University. Klappentext Addresses the rapidly growing ­field of fractional calculus and provides simpli­fied solutions for linear commensurate-order fractional differential equations­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is the result of the authors' work in fractional calculus, and more particularly, in functions for the solutions of fractional di­fferential equations, which is fostered in the behavior of generalized exponential functions. The authors discuss how fractional trigonometry plays a role analogous to the classical trigonometry for the fractional calculus by providing solutions to linear fractional di­fferential equations. The book begins with an introductory chapter that o­ffers insight into the fundamentals of fractional calculus, and topical coverage is then organized in two main parts. Part One develops the definitions and theories of fractional exponentials and fractional trigonometry. Part Two provides insight into various areas of potential application within the sciences. The fractional exponential function via the fundamental fractional differential equation, the generalized exponential function, and R-function relationships are discussed in addition to the fractional hyperboletry, the R1-fractional trigonometry, the R2-fractional trigonometry, and the R3-trigonometric functions. ­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science also:* Presents fractional trigonometry as a tool for scientists and engineers and discusses how to apply fractional-order methods to the current toolbox of mathematical modelers* Employs a mathematically clear presentation in an e­ ort to make the topic broadly accessible* Includes solutions to linear fractional di­fferential equations and generously features graphical forms of functions to help readers visualize the presented concepts* Provides e­ffective and efficient methods to describe complex structures­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is an ideal reference for academic researchers, research engineers, research scientists, mathematicians, physicists, biologists, and chemists who need to apply new fractional calculus methods to a variety of disciplines. The book is also appropriate as a textbook for graduate- and PhD-level courses in fractional calculus.Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry.Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional-order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann's complemen...

List of contents

Preface xvAcknowledgments xixAbout the Companion Website xxi1 Introduction 11.1 Background 21.2 The Fractional Integral and Derivative 31.3 The Traditional Trigonometry 61.4 Previous Efforts 81.5 Expectations of a Generalized Trigonometry and Hyperboletry 82 The Fractional Exponential Function via the Fundamental Fractional Differential Equation 92.1 The Fundamental Fractional Differential Equation 92.2 The Generalized Impulse Response Function 102.3 Relationship of the F-function to the Mittag-Leffler Function 112.4 Properties of the F-Function 122.5 Behavior of the F-Function as the Parameter a Varies 132.6 Example 163 The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus 193.1 Introduction 193.2 Functions for the Fractional Calculus 193.3 The R-Function: A Generalized Function 223.4 Properties of the Rq,v(a, t)-Function 233.5 Relationship of the R-Function to the Elementary Functions 273.6 R-Function Identities 293.7 Relationship of the R-Function to the Fractional Calculus Functions 313.8 Example: Cooling Manifold 323.9 Further Generalized Functions: The G-Function and the H-Function 343.10 Preliminaries to the Fractional Trigonometry Development 383.11 Eigen Character of the R-Function 383.12 Fractional Differintegral of the TimeScaled R-Function 393.13 R-Function Relationships 393.14 Roots of Complex Numbers 403.15 Indexed Forms of the R-Function 413.16 Term-by-Term Operations 443.17 Discussion 464 R-Function Relationships 474.1 R-Function Basics 474.2 Relationships for Rm,0 in Terms of R1,0 484.3 Relationships for R1¨Mm,0 in Terms of R1,0 504.4 Relationships for the Rational Form Rm¨Mp,0 in Terms of R1¨Mp,0 514.5 Relationships for R1¨Mp,0 in Terms of Rm¨Mp,0 534.6 Relating Rm¨Mp,0 to the Exponential Function R1,0(b, t) = ebt 544.7 Inverse Relationships-Relationships for R1,0 in Terms of Rm,k 564.8 Inverse Relationships-Relationships for R1,0 in Terms of R1¨Mm,0 574.9 Inverse Relationships-Relationships for eat = R1,0(a, t) in Terms of Rm¨Mp,0 594.10 Discussion 615 The Fractional Hyperboletry 635.1 The Fractional R1-Hyperbolic Functions 635.2 R1-Hyperbolic Function Relationship 725.3 Fractional Calculus Operations on the R1-Hyperbolic Functions 725.4 Laplace Transforms of the R1-Hyperbolic Functions 735.5 Complexity-Based Hyperbolic Functions 735.6 Fractional Hyperbolic Differential Equations 745.7 Example 765.8 Discussions 776 The R1-Fractional Trigonometry 796.1 R1-Trigonometric Functions 796.2 R1-Trigonometric Function Interrelationship 886.3 Relationships to R1-Hyperbolic Functions 896.4 Fractional Calculus Operations on the R1-Trigonometric Functions 896.5 Laplace Transforms of the R1-Trigonometric Functions 906.6 Complexity-Based R1-Trigonometric Functions 926.7 Fractional Differential Equations 947 The R2-Fractional Trigonometry 977.1 R2-Trigonometric Functions: Based on Real and Imaginary Parts 977.2 R2-Trigonometric Functions: Based on Parity 1027.3 Laplace Transforms of the R2-Trigonometric Functions 1117.4 R2-Trigonometric Function Relationships 1137.5 Fractional Calculus Operations on the R2-Trigonometric Functions 1197.6 Inferred Fractional Differential Equations 1278 The R3-Trigonometric Functions 1298.1 The R3-Trigonometric Functions: Based on Complexity 1298.2 The R3-Trigonometric Functions: Based on Parity 1348.3 Laplace Transforms of the R3-Trigonometric Functions 1408.4 R3-Trigonometric Function Relationships 1418.5 Fractional Calculus Operations on the R3-Trigonometric Functions 1469 The Fractional Meta-Trigonometry 1599.1 The FractionalMeta-Trigonometric Functions: Based on Complexity 1609.2 The Meta-Fractional Trigonometric Functions: Based on Parity 1669.3 Commutative Properties of the Complexity and Parity Operations 1799.4 Laplace Transforms of the FractionalMeta-Trigonometric Functions 1889.5 R-Function Representation of the FractionalMeta-Trigonometric Functions 1929.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions 1959.7 Special Topics in Fractional Differintegration 2069.8 Meta-Trigonometric Function Relationships 2069.9 Fractional Poles: Structure of the Laplace Transforms 2149.10 Comments and Issues Relative to the Meta-Trigonometric Functions 2149.11 Backward Compatibility to Earlier Fractional Trigonometries 2159.12 Discussion 21510 The Ratio and Reciprocal Functions 21710.1 Fractional Complexity Functions 21710.2 The Parity Reciprocal Functions 21910.3 The Parity Ratio Functions 22110.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions 22510.5 Relationships 22610.6 Discussion 22711 Further Generalized Fractional Trigonometries 22911.1 The G-Function-Based Trigonometry 22911.2 Laplace Transforms for the G-Trigonometric Functions 23011.3 The H-Function-Based Trigonometry 23411.4 Laplace Transforms for the H-Trigonometric Functions 23512 The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry 24312.1 Fractional Differential Equations 24312.2 Fundamental Fractional Differential Equations of the First Kind 24512.3 Fundamental Fractional Differential Equations of the Second Kind 24612.4 Preliminaries-Laplace Transforms 24612.5 Fractional Differential Equations of Higher Order: Unrepeated Roots 25012.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots 25212.7 Fractional Differential Equations Containing Repeated Roots 25312.8 Fractional Differential Equations of Non-Commensurate Order 25412.9 Indexed Fractional Differential Equations: Multiple Solutions 25512.10 Discussion 25613 Fractional Trigonometric Systems 25913.1 The R-Function as a Linear System 25913.2 R-System Time Responses 26013.3 R-Function-Based Frequency Responses 26013.4 Meta-Trigonometric Function-Based Frequency Responses 26113.5 FractionalMeta-Trigonometry 26413.6 Elementary Fractional Transfer Functions 26613.7 Stability Theorem 26613.8 Stability of Elementary Fractional Transfer Functions 26713.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions 26813.10 Discussion 27014 Numerical Issues and Approximations in the Fractional Trigonometry 27114.1 R-Function Convergence 27114.2 The Meta-Trigonometric Function Convergence 27214.3 Uniform Convergence 27314.4 Numerical Issues in the Fractional Trigonometry 27414.5 The R2Cos- and R2Sin-Function Asymptotic Behavior 27514.6 R-Function Approximations 27614.7 The Near-Order Effect 27914.8 High-Precision Software 28115 The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry 28315.1 The Fractional Spiral Functions 28315.2 Analysis of Spirals 28815.3 Relation to the Classical Spirals 30315.4 Discussion 30716 Fractional Oscillators 30916.1 The Space of Linear Fractional Oscillators 30916.2 Coupled Fractional Oscillators 31417 Shell Morphology and Growth 31717.1 Nautilus pompilius 31717.2 Shell 5 32917.3 Shell 6 33017.4 Shell 7 33217.5 Shell 8 33217.6 Shell 9 33617.7 Shell 10 33617.8 Ammonite 33917.9 Discussion 34018 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 34118.1 Introduction 34118.2 Background-Fractional Spirals for Galactic Classification 34218.3 Classification Process 34718.4 Mathematical Classification of Selected Galaxies 35018.5 Analysis 36218.6 Discussion 36718.7 Appendix: Carbon Star 37019 Hurricanes, Tornados, and Whirlpools 37119.1 Hurricane Cloud Patterns 37119.2 Tornado Classification 37319.3 Low-Pressure Cloud Pattern 37519.4 Whirlpool 37519.5 Order in Physical Systems 37920 A Look Forward 38120.1 Properties of the R-Function 38220.2 Inverse Functions 38220.3 The Generalized Fractional Trigonometries 38420.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments 38420.5 Applications: Fractional Field Equations 38520.6 Fractional Spiral and Nonspiral Properties 38720.7 Numerical Improvements for Evaluation to Larger Values of atq 38720.8 Epilog 388A Related Works 389A.1 Introduction 389A.2 Miller and Ross 389A.3 West, Bologna, and Grigolini 390A.4 Mittag-Leffler-Based Fractional Trigonometric Functions 390A.5 Relationship to CurrentWork 391B Computer Code 393B.1 Introduction 393B.2 Matlab® R-Function 393B.3 Matlab® R-Function Evaluation Program 394B.4 Matlab® Meta-Cosine Function 395B.5 Matlab® Cosine Evaluation Program 395B.6 Maple® 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine 396C Tornado Simulation 399D Special Topics in Fractional Differintegration 401D.1 Introduction 401D.2 Fractional Integration of the Segmented tp-Function 401D.3 Fractional Differentiation of the Segmented tp-Function 404D.4 Fractional Integration of Segmented Fractional Trigonometric Functions 406D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions 408E Alternate Forms 413E.1 Introduction 413E.2 Reduced Variable Summation Forms 414E.3 Natural Quency Simplification 415References 417Index 425

Product details

• Authors: Tom T Hartley, Tom T. Hartley, Tom T. Lorenzo Hartley, Carl Lorenzo, Carl F Lorenzo, Carl F. Lorenzo, Carl F. Hartley Lorenzo, CF Lorenzo, Lorenzo Carl F.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 30.12.2016
• EAN: 9781119139409
• ISBN: 978-1-119-13940-9
• No. of pages: 464
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Miscellaneous; Chemie, Mathematik, Trigonometrie, Mathematics, chemistry, Mathematical analysis, Mathematik u. Statistik i. d. Chemie, Mathematik in den Ingenieurwissenschaften, Applied Mathematics in Engineering, Mathematische Analyse, Mathematics & Statistics for Chemistry, Fraktionale Differentialgleichung