The Fractional Trigonometry

With Applications to Fractional Differential Equations and Science

Author: Carl F. Lorenzo,Tom T. Hartley

Publisher: John Wiley & Sons

ISBN: 1119139406

Category: Mathematics

Page: 496

View: 3176

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.

Elements of plane trigonometry

In which is introduced, a dissertation on the nature and use of logarithms

Author: Francis Maseres

Publisher: N.A

ISBN: N.A

Category: Logarithms

Page: 471

View: 3566

Elements of Trigonometry, Plane and Spherical

Adapted to the Present State of Analysis : to which is Added, Their Application to the Principles of Navigation and Nautical Astronomy : with Logarithmic, Trigonometrical, and Nautical Tables, for Use of Colleges and Academies

Author: Charles William Hackley

Publisher: N.A

ISBN: N.A

Category: Trigonometry

Page: 307

View: 4887

Algebra and Trigonometry

Author: Ron Larson

Publisher: Cengage Learning

ISBN: 1111788979

Category: Mathematics

Page: 1040

View: 4916

This market-leading text continues to provide students and instructors with sound, consistently structured explanations of the mathematical concepts. Designed for a two-term course, the new Eighth Edition retains the features that have made Algebra and Trigonometry a complete solution for both students and instructors: interesting applications, cutting-edge design, and innovative technology combined with an abundance of carefully written exercises. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

The Rudiments of Mathematics

Designed for the Use of Students at the Universities : Containing an Introduction to Algebra, Remarks on the First Six Books of Euclid, the Elements of Plane Trigonometry

Author: William Ludlam

Publisher: N.A

ISBN: N.A

Category: Algebra

Page: 352

View: 9979

The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order

Author: N.A

Publisher: Elsevier

ISBN: 9780080956206

Category: Mathematics

Page: 322

View: 1294

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering

Trigonometric Sums in Number Theory and Analysis

Author: Gennady I. Arkhipov,Vladimir N. Chubarikov,Anatoly A. Karatsuba

Publisher: Walter de Gruyter

ISBN: 3110197987

Category: Mathematics

Page: 564

View: 3786

The book presents the theory of multiple trigonometric sums constructed by the authors. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. M. Vinogradov ́s estimates and use them to solve several problems in analytic number theory. They investigate trigonometric integrals, which are often encountered in physics, mathematical statistics, and analysis, and in addition they present purely arithmetic results concerning the solvability of equations in integers.

Trigonometry For Dummies

Author: Mary Jane Sterling

Publisher: John Wiley & Sons

ISBN: 1118827570

Category: Mathematics

Page: 384

View: 4698

A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers. Tracks to a typical Trigonometry course at the high school or college level Packed with example trig problems From the author of Trigonometry Workbook For Dummies Trigonometry For Dummies is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.

Logarithmic and Trigonometric Tables

To Seven Places of Decimals, in a Pocket Form ; in which the Errors of Former Tables are Corrected

Author: Ferdinand Rudolph Hassler

Publisher: N.A

ISBN: N.A

Category: Logarithms

Page: 10

View: 1224

Intelligent Mathematics: Computational Analysis

Author: George A. Anastassiou

Publisher: Springer Science & Business Media

ISBN: 9783642170980

Category: Computers

Page: 802

View: 2245

Knowledge can be modeled and computed using computational mathematical methods, then lead to real world conclusions. The strongly related to that Computational Analysis is a very large area with lots of applications. This monograph includes a great variety of topics of Computational Analysis. We present: probabilistic wavelet approximations, constrained abstract approximation theory, shape preserving weighted approximation, non positive approximations to definite integrals, discrete best approximation, approximation theory of general Picard singular operators including global smoothness preservation property, fractional singular operators. We also deal with non-isotropic general Picard singular multivariate operators and q-Gauss-Weierstrass singular q-integral operators. We talk about quantitative approximations by shift-invariant univariate and multivariate integral operators, nonlinear neural networks approximation, convergence with rates of positive linear operators, quantitative approximation by bounded linear operators, univariate and multivariate quantitative approximation by stochastic positive linear operators on univariate and multivariate stochastic processes. We further present right fractional calculus and give quantitative fractional Korovkin theory of positive linear operators. We also give analytical inequalities, fractional Opial inequalities, fractional identities and inequalities regarding fractional integrals. We further deal with semi group operator approximation, simultaneous Feller probabilistic approximation. We also present Fuzzy singular operator approximations. We give transfers from real to fuzzy approximation and talk about fuzzy wavelet and fuzzy neural networks approximations, fuzzy fractional calculus and fuzzy Ostrowski inequality. We talk about discrete fractional calculus, nabla discrete fractional calculus and inequalities. We study the q-inequalities, and q-fractional inequalities. We further study time scales: delta and nabla approaches, duality principle and inequalities. We introduce delta and nabla time scales fractional calculus and inequalities. We finally study convergence with rates of approximate solutions to exact solution of multivariate Dirichlet problem and multivariate heat equation, and discuss the uniqueness of solution of general evolution partial differential equation \ in multivariate time. The exposed results are expected to find applications to: applied and computational mathematics, stochastics, engineering, artificial intelligence, vision, complexity and machine learning. This monograph is suitable for graduate students and researchers.

Trigonometry with calculators

Author: George S. Donovan,Beverly Beyreuther Gimmestad

Publisher: Brooks/Cole

ISBN: 9780871502841

Category: Mathematics

Page: 330

View: 6189

Thesaurium Mathematicae: Or, The Treasury of the Mathematicks

Containing Variety of Useful Practices in Arithmetick, Geometry, Trigonometry, Astronomy, Geography, Navigation and Surveying. ... To which is Annex'd a Table of L0000 Logarithms, Log-sines and Log-tangents. Illustrated with Several Mathematical Sculptures on Copper-plates

Author: John Taylor

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 364

View: 3038

Physics of Fractal Operators

Author: Bruce West,Mauro Bologna,Paolo Grigolini

Publisher: Springer Science & Business Media

ISBN: 9780387955544

Category: Mathematics

Page: 354

View: 4485

This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; "Physics of Fractal Operators" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Physics of Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.

q-Fractional Calculus and Equations

Author: Mahmoud H. Annaby,Zeinab S. Mansour

Publisher: Springer

ISBN: 3642308988

Category: Mathematics

Page: 318

View: 7634

This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working towards the investigation of q-fractional difference equations; families of q-Mittag-Leffler functions are defined and their properties are investigated, especially the q-Mellin–Barnes integral and Hankel contour integral representation of the q-Mittag-Leffler functions under consideration, the distribution, asymptotic and reality of their zeros, establishing q-counterparts of Wiman’s results. Fractional q-difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of q-Mittag-Leffler functions. Among many q-analogs of classical results and concepts, q-Laplace, q-Mellin and q2-Fourier transforms are studied and their applications are investigated.

Schaum's Outline of Introductory Surveying

Author: James Wirshing,Roy Wirshing

Publisher: McGraw Hill Professional

ISBN: 9780070711242

Category: Science

Page: 360

View: 2786

Collects problems and detailed solutions related to aspects of surveying such as leveling, transits, angle measurement, topographic surveys, and slope staking

Trigonometrie für Dummies

Author: Mary Jane Sterling

Publisher: John Wiley & Sons

ISBN: 3527638563

Category: Mathematics

Page: 377

View: 7968

Trigonometrie besch?ftigt sich mit Winkeln und Dreiecken. Das h?rt sich ja ganz einfach an, aber jeder, der sich schon mit Trigonometrie besch?ftigen durfte, wei? wie verdammt kniffelig sie sein kann. ?Trigonometrie f?r Dummies? f?hrt die Leser in diese sonderbare Welt ein und versucht dabei auch zu zeigen wo, wann und warum es sinnvoll ist, sich mit diesem Thema zu besch?ftigen. Am Ende sind dann Sinus, Cosinus, Tangens und Cotangens keine Fremden mehr sondern gute alte Bekannte.