Fractional Trigonometry: With Applications to Fractional Differential Equations and Science

Addresses the rapidly growing ­field of fractional calculus and provides simpli­fied solutions for linear commensurate–order fractional differential equations ­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is the result of the authors work in fractional calculus, and more particularly, in functions for the solutions of fractional di­fferential equations, which is fostered in the behavior of generalized exponential functions. The authors discuss how fractional trigonometry plays a role analogous to the classical trigonometry for the fractional calculus by providing solutions to linear fractional di­fferential equations. The book begins with an introductory chapter that o­ffers insight into the fundamentals of fractional calculus, and topical coverage is then organized in two main parts. Part One develops the definitions and theories of fractional exponentials and fractional trigonometry. Part Two provides insight into various areas of potential application within the sciences. The fractional exponential function via the fundamental fractional differential equation, the generalized exponential function, and R–function relationships are discussed in addition to the fractional hyperboletry, the R1–fractional trigonometry, the R2–fractional trigonometry, and the R3–trigonometric functions. ­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science also: Presents fractional trigonometry as a tool for scientists and engineers and discusses how to apply fractional–order methods to the current toolbox of mathematical modelers Employs a mathematically clear presentation in an e­ ort to make the topic broadly accessible  Includes solutions to linear fractional di­fferential equations and generously features graphical forms of functions to help readers visualize the presented concepts Provides e­ffective and efficient methods to describe complex structures ­The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science is an ideal reference for academic researchers, research engineers, research scientists, mathematicians, physicists, biologists, and chemists who need to apply new fractional calculus methods to a variety of disciplines. The book is also appropriate as a textbook for graduate– and PhD–level courses in fractional calculus. Carl F. Lorenzo is Distinguished Research Associate at the NASA Glenn Research Center in Cleveland, Ohio. His past positions include chief engineer of the Instrumentation and Controls Division and chief of the Advanced Controls Technology and Systems Dynamics branches at NASA. He is internationally recognized for his work in the development and application of the fractional calculus and fractional trigonometry. Tom T. Hartley, PhD, is Emeritus Professor in the Department of Electrical and Computer Engineering at The University of Akron. Dr Hartley is a recognized expert in fractional–order systems, and together with Carl Lorenzo, has solved fundamental problems in the area including Riemann s complementary–function initialization function problem. He received his PhD in Electrical Engineering from Vanderbilt University. INDICE: Preface .Acknowledgements .1. Introduction .1.1 Background .1.2 The Fractional Integral and Derivative .1.3 The Traditional Trigonometry .1.4 Previous Efforts .1.5 Expectations of a Generalized Trigonometry and Hyperboletry .2. The Fractional Exponential Function via the Fundamental Fractional Differential Equation .2.1 The Fundamental Fractional Differential Equation .2.2 The Generalized Impulse Response Function .2.3 Relationship of the F–function to the Mittag–Leffler Function .2.4 Properties of the F–function .2.5 Behavior of the F–function as the Parameter a Varies .2.6 Example .3. The Generalized Fractional Exponential Function .3.1 Introduction .3.2 Functions for the Fractional Calculus .3. 3 The R–Function: A Generalized Function .3.4 Properties of the Function .3.5 Relationship of the R–function to the Elementary Functions .3.6 R–Function Identities .3.7 Relationship of the R–Function to the Fractional Calculus Functions .3.8 Example: Cooling Manifold .3.9 A Further Generalized Function: The G–Function and H–function .3.10 Preliminaries to the Fractional Trigonometry Develpoment .3.11 Eigen Character of the R–Function .3.12 Fractional Differintegral of the Time–Scaled R–Function .3.13 R–Function Relationships .3.14. Roots of Complex Numbers .3.15 Indexed Forms of the R–Function .3.16 Term–by–Term Operations .3.17 Discussion .4. R–Function Relationships .4.1 R–Function Basics .4.2 Relationships for  in Terms of .4.3 Relationships for  in Terms of .4.4 Relationships for the Rational Form in Terms of .4.5 Relationships for  in Terms of .4.6 Relating  to the Exponential Function .4.7 Inverse Relationships Relationships for  in Terms of .4.8 Inverse Relationships Relationships for  in Terms of .4.9 Inverse Relationships Relationships for  in Terms of .4.10 Discussion .5. The Fractional Hyperboletry .5.1 The Fractional R1–Hyperbolic Functions .5.2 R1–Hyperbolic Function Relationship .5.3 Fractional Calculus Operations on the R1–Hyperbolic Functions .5.4 Laplace Transforms of the R1–Hyperbolic Functions .5.5 Complexity Based Hyperbolic Functions .5.6 Fractional Hyperbolic Differential Equations .5.7 Example .5.8 Discussion .6. The R1 Fractional Trigonometry .6.1 R1–Trigonometric Functions .6.2 R1–Trigonometric Function Interrelationship .6.3 Relationships to R1–Hyperbolic Functions .6.4 Fractional Calculus Operations on the R1–Trigonometric Functions .6.5 Laplace Transforms of the R1–Trigonometric Functions .6.6 Complexity Based R1–Trigonometric Functions .6.7 Fractional Differential Equations .7. The R2 Fractional Trigonometry .7.1 R2–Trigonometric Functions: Based on Real and Imaginary Parts .7.2 R2–Trigonometric Functions: Based on Parity .7.3 Laplace Transforms of the R2–Trigonometric Functions .7.4 R2 Trigonometric Function Relationships .7.5 Fractional Calculus Operations on the R2 Trigonometric Functions .7.5.9 Summary of Fractional Calculus Operations on the R2 Trigonometric Functions .7.6 Inferred Fractional Differential Equations .8. The R3–Trigonometric Functions .8.1 The R3–Trigonometric Functions: Based on Complexity .8.2 The R3–Trigonometric Functions: Based on Parity .8.3 Laplace Transforms of the R3–Trigonometric Functions .8.4 R3 Trigonometric Function Relationships .8.5 Fractional Calculus Operations on the R3 Trigonometric Functions .9. The Fractional Meta–Trigonometry .9.1 The Fractional Meta–Trigonometric Functions: Based on Complexity .9.2 The Meta–Fractional Trigonometric Functions: Based on Parity .9.3 Commutative Properties of the Complexity and Parity Operations .9.4 Laplace Transforms of the Fractional Meta–Trigonometric Functions .9.5 R–Function Representation of the Fractional Meta–Trigonometric Functions .9.6 Fractional Calculus Operations on the Meta–Trigonometric Functions .9.7 Special Topics in Fractional Differintegration .9.8 Meta–Trigonometric Function Relationships .9.9 Fractional Poles: Structure of the Laplace Transforms .9.10 Comments and Issues Relative to the Meta–Trigonometric Functions .9.11 Backward Compatibility to Earlier Fractional Trigonometries .9.12 Discussion .10. The Ratio and Reciprocal Functions .10.1 Fractional Complexity Functions .10.2 The Parity Reciprocal Functions .10.3 The Parity Ratio Functions .10.4 R–Function Representation of the Fractional Ratio and Reciprocal Functions .10.5 Relationships .10.6 Discussion .11. Further Generalized Fractional Trigonometries .11.1 The G–Function Based Trigonometry .11.2 Laplace Transforms for the G–Trigonometric Functions .11.3 The H–Function Based Trigonometry .11.4 Laplace Transforms for the H–Trigonometric Functions .12. The Solution of Linear Fractional Differential Equations based on the Fractional Trigonometry .12.1 Fractional Differential Equations .12.2 Fundamental Fractional Differential Equations of the First Kind .12.3 Fundamental Fractional Differential Equation of the Second Kind .12.4 Preliminaries–Laplace Transforms .12.5 Fractional Differential Equations of Higher Order: Unrepeated .Roots .12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots .12.7 Fractional Differential Equations Containing Repeated Roots .12.8 Fractional Differential Equations of Non–Commensurate Order .12.9 Indexed Fractional Differential Equations: Multiple Solutions .12.10 Discussion .13. Fractional Trigonometric Systems .13.1 The R–function as a Linear System .13.2 R–System Time Responses .13.3 R–Function Based Frequency Responses .13.4 Meta–Trigonometric Functions Based Frequency Responses .13.5 Fractional Meta–Trigonometry .13.6 Elementary Fractional Transfer Functions .13.7 Stability Theorem (Matignon) .13.8 Stability of Elementary Fractional Transfer Functions .13.9 Insights into the Behavior of the Fractional Meta–Trigonometric Functions .13.10 Discussion .14. Numerical Issues and Approximations in the Fractional Trigonometry .14.1 R–Function Convergence .14.2 The Meta–Trigonometric Function Convergence .14.3 Uniform Convergence .14.4 Numerical Issues in the Fractional Trigonometry .14.5 The R2Cos and R2Sin Function Asymptotic Behavior .14.6 R–function Approximations .14.7 The Near Order Effect .14.8 High Precision Software .15. The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry .15.1 The Fractional Spiral Functions .15.2 Analysis of Spirals .15.3 Relation to Classical Spirals .15.4 Discussion .16. Fractional Oscillators .16.1 The Space of Linear Fractional Oscillators .16.2 Coupled Fractional Oscillators .17. Shell Morphology and Growth .17.1 Nautilus Pompilius .17.2 –17.7 Various Shells .17.8 Ammonite .17.9 Discussion .18. Mathematical Classification of the Spiral and Ring Galaxy Morphologies .18.1 Introduction .18.2 Background Fractional Spirals for Galactic Classification .18.3 Classification Process .18.4 Mathematical Classification of Selected Galaxies .18.5 Analysis .18.6 Discussion .18.7 Appendix Carbon Star .19. Hurricanes, Tornados and Whirlpools .19.1 Hurricane Cloud Patterns .19.2 Tornado Classification .19.3 Low Pressure Cloud Pattern .19.4 Whirlpool .19.5 Order in Physical Systems .20. A Look Forward .20.1 Properties of the R–Function .20.2 Inverse Functions .20.3 The Generalized Fractional Trigonometries .20.4 Extensions to Negative Time, Complementary Trigonometries and Complex Arguments .20.5 Applications: Fractional Field Equations .20.6 Fractional Spiral Properties and Non–Spiral Properties .20.7 Numerical Improvements for Evaluation to Larger Values of atq .20.8 Epilog .Appendix A Related Works .Appendix B Computer Code .Appendix C Tornado Simulation .Appendix D Special Topics in Fractional Differintegration .D.1 Fractional Integration of the Segmented Function .D.2 Fractional Differentiation of the Segmented Function .D.3 Fractional Integration of Segmented Fractional Trigonometric Functions .D.4 Fractional Differentiation of Segmented Fractional Trigonometric Functions .Appendix E Alternate Forms .E.1 Reduced Variable Summation Forms .E.2 Natural Quency Simplification

• ISBN: 978-1-119-13940-9
• Editorial: Wiley–Blackwell
• Encuadernacion: Cartoné
• Páginas: 496
• Fecha Publicación: 23/12/2016
• Nº Volúmenes: 1
• Idioma: Inglés