*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:** 9396

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.

This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to find high-precision solutions to fractional-order differentiations and differential equations. Numerical algorithms and their implementations are proposed to analyze multivariable fractional-order control systems. Through high-quality MATLAB programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems. Contents Introduction to fractional calculus and fractional-order control Mathematical prerequisites Definitions and computation algorithms of fractional-order derivatives and Integrals Solutions of linear fractional-order differential equations Approximation of fractional-order operators Modelling and analysis of multivariable fractional-order transfer function Matrices State space modelling and analysis of linear fractional-order Systems Numerical solutions of nonlinear fractional-order differential Equations Design of fractional-order PID controllers Frequency domain controller design for multivariable fractional-order Systems Inverse Laplace transforms involving fractional and irrational Operations FOTF Toolbox functions and models Benchmark problems for the assessment of fractional-order differential equation algorithms

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

The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. The Fractional Laplacian explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered. Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain Covers physical and mathematical concepts as well as detailed mathematical derivations Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions Discusses viscous flow and physical examples from scientific and engineering disciplines Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.

*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:** 4312

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.

' This reference book presents unique and traditional analytic calculations, and features more than a hundred universal formulas where one can calculate by hand enormous numbers of definite integrals, fractional derivatives and inverse operators. Despite the great success of numerical calculations due to computer technology, analytical calculations still play a vital role in the study of new, as yet unexplored, areas of mathematics, physics and other branches of sciences. Readers, including non-specialists, can obtain themselves universal formulas and define new special functions in integral and series representations by using the methods expounded in this book. This applies to anyone utilizing analytical calculations in their studies. Contents:Mathematical PreparationCalculation of Integrals Containing Trigonometric and Power FunctionsIntegrals Involving xγ, (p + txp)–λ Sine and Cosine FunctionsDerivation of General Formulas for Integrals Involving Powers of x, (a + bx)-Type Binomials and Trigonometric FunctionsIntegrals Involving xγ, 1/(p + txp)–λ, exp[–axγ] and Trigonometric FunctionsIntegrals Containing Bessel FunctionsIntegrals Involving the Neumann Function Nσ(x)Integrals Containing Other Cylindrical and Special FunctionsIntegrals Involving Two Trigonometric FunctionsDerivation of Universal Formulas for Calculation of Fractional Derivatives and Inverse Operators Readership: Undergraduate and graduate students interested in analytic calculations in integral calculus. Researchers from the fields of modern mathematical analysis, theoretical physics and engineering. Non-experts interested in integrals, fractional derivatives and inverse operators. Key Features:Content in this book is easy to readHas simple tables of integration and taking fractional derivatives and calculating complicated inverse operatorsMost valuable tip from the book is a simple and nice way to perform integration showing it as a design culture but not as tedious workKeywords:Mellin Representation;Contour Integration;Analytic Continuation;Taking Residue;L''Hôpital''s Rule;Gamma Function;Psi-Function;Exponential Function;Neumann;Trigonometric;Cylindrical;Struve;Modified Struve;Bessel and Modified Bessel Functions;Probability Integrals;Integral of Frenal;General Formulas for Taking Integrals;Fractional Derivatives;Inverse Operators;Euler Number'

This text investigates Waring's problem, approximation by fractional parts of the values of a polynomial, estimates for Weyl sums, distribution of fractional parts of polynomial values, Goldbach's problem, more. 1954 edition.

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.

This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the subject. Most chapters may be covered or omitted, depending upon the background of the student. For example, the text may be used as a primary reference in an introductory course for difference equations which also includes discrete fractional calculus. Chapters 1—2 provide a basic introduction to the delta calculus including fractional calculus on the set of integers. For courses where students already have background in elementary real analysis, Chapters 1—2 may be covered quickly and readers may then skip to Chapters 6—7 which present some basic results in fractional boundary value problems (FBVPs). Chapters 6—7 in conjunction with some of the current literature listed in the Bibliography can provide a basis for a seminar in the current theory of FBVPs. For a two-semester course, Chapters 1—5 may be covered in depth, providing a very thorough introduction to both the discrete fractional calculus as well as the integer-order calculus.

Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms. Provides applications of local fractional Fourier Series Discusses definitions for local fractional Laplace transforms Explains local fractional Laplace transforms coupled with analytical methods

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.

Intermediate Algebra with Trigonometry focuses on principles, operations, and approaches employed in intermediate algebra with trigonometry. The publication first elaborates on basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on polynomials, sums, and differences, multiplication of polynomials, greatest common factor and factoring by grouping, inequalities involving absolute value, equations with absolute value, and multiplication, division, and order of operation for real numbers. The manuscript then ponders on rational expressions, quadratic equations, and rational expressions and roots. Topics include equations quadratic in form, quadratic formula, completing the square, multiplication and division of complex numbers, equations with radicals, simplified form for radicals, multiplication and division of rational expressions, and addition and subtraction of rational expressions. The text takes a look at triangles, trigonometric identities and equations, introduction to trigonometry, and sequence and series, including arithmetic progressions, trigonometric functions, tables and calculators, sum and difference formulas, and the law of sines and cosines. The publication is a valuable reference for students and researchers interested in intermediate algebra with trigonometry.