# The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science

John Wiley & Sons, Nov 14, 2016 - Mathematics - 464 pages

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.

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### Contents

 The Fractional Exponential Function via the Fundamental Fractional 9 The RFunction and Other 19 3 20 1 29 4 47 Relating Rmp0 b Inverse RelationshipsRelationships to the Exponential Function for R10 R in 10 Terms t of ebt Rmk 54 Inverse Inverse RelationshipsRelationships RelationshipsRelationships for R10 in Terms of R1m0 61 The RsTrigonometric Functions 129
 The RFunction R2 Cos and Approximations R2SinFunction 275 Further Characterization of the Fractional 283 Fractional Oscillators 309 Shell Morphology and Growth 317 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341 Hurricanes Tornados and Whirlpools 371 A Look Forward 381 A Related Works 389

 The Fractional MetaTrigonometry 159 The Ratio and Reciprocal Functions 217 Further Generalized Fractional Trigonometries 229 Introduction to Applications 241
 Matlab MetaCosine Function 395 Special Topics in Fractional Differintegration 401 Copyright