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2024 | Buch

Introduction to Fractional Calculus Kapitel lesen Erstes Kapitel lesen

Fractional Calculus

High-Precision Algorithms and Numerical Implementations

verfasst von: Dingyü Xue, Lu Bai

Verlag: Springer Nature Singapore

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

Fractional calculus and its applications are fascinating research areas in many engineering disciplines. This book is a comprehensive collection of research from the author's group, which is one of the most active in the fractional calculus community worldwide and is the birthplace of one of the four MATLAB toolboxes in fractional calculus, the FOTF Toolbox. The book presents high-precision solution algorithms for a variety of fractional-order differential equations, including nonlinear, delay, and boundary value equations. Currently, there are no other universal solvers available for the latter two types of equations.

Through this book, readers can systematically study the mathematics and solution methods in the field of fractional calculus and apply these concepts to different engineering fields, particularly control systems engineering.

This book is a translation of an original German edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

Inhaltsverzeichnis

Frontmatter
1. Introduction to Fractional Calculus
Abstract
At the beginning of the development of the theory of classical calculus (called integer-order calculus in this book), the British scientist Isaac Newton and the German mathematician Gottfried Wilhelm Leibniz used different symbols for different orders of derivatives. For example, Newton used the notation \(\dot{y}(x)\), \(\ddot{y}(x)\) and \(\dddot{y}(x)\), while Leibniz used the notation \(\textrm{d}^n y(x)/\textrm{d}x^n\), where n is a positive integer. A natural question is how to extend n into fractions or even complex numbers. In a letter written by the French mathematician Marquis de l’Hôpital to Leibniz in 1695, he asked question “what would be the meaning if \(n = 1/2\) in the \(\textrm{d}^n y(x)/\textrm{d}x^n\) notation”. In a letter dated 30 September 1695, Leibniz replied, “Thus it follows that \(\textrm{d}^{1/2} x\) will be equal to \(x\sqrt{\textrm{d}x:x}\). This is an apparent paradox from which, one day, useful consequences will be drawn” [1]. In this chapter, a brief historic view of fractional calculus is presented. The tools for fractional calculus are summarized.
Xue Dingyü, Bai Lu
2. Commonly Used Special Functions: Definitions and Computing
Abstract
Special functions are dedicated mathematical functions invented by mathematicians. For instance, if the integrand is \(f(x)=\textrm{e}^{-x^2}\), the analytical solution to its indefinite integrals does not exist. Therefore, mathematicians invented a special function \(\textrm{erf}(x)\) to express the integral expression. This special function is regarded as the analytical solution of the integral problem. In different applications, many other such special functions are invented. For instance, Gamma function and Beta function and so on. Many special functions are invented for indefinite integral and differential equation problems.
Xue Dingyü, Bai Lu
3. Definitions and Numerical Evaluations of Fractional Calculus
Abstract
As previously noted, fractional calculus dates back to the days when Newton and Leibniz first invented traditional calculus. In the absence of a unified and widely accepted definitions, fractional calculus did not progress well in its early development. In this chapter, various definitions of fractional derivatives and integrals are presented. The numerical solutions of derivatives and integrals under Grünwald–Letnikov, Riemann–Liouville and Caputo fractional-order definitions are presented. The properties and geometric interpretations are explored.
Xue Dingyü, Bai Lu
4. High-Precision Numerical Algorithms and Implementation in Fractional Calculus
Abstract
The accuracy of the algorithm described earlier is at the O(h) level, also known as the first-order algorithm. The computational error is closely related to the step size h. If h is large, the computational error is also large. For example, imprecisely, if \(h = 0.01\), the computational error is almost 0.01. If there is an algorithm with \(O(h^2)\), called a second-order algorithm, it is possible to obtain a computational error of 0.0001, while a fourth-order algorithm \(O(h^4)\) may bring the error down to \(0.01^4 = 10^{-8}\). It follows that if we want to obtain a numerical solution with high accuracy, we need to increase the order of the algorithm.
Xue Dingyü, Bai Lu
5. Approximations of Fractional-Order Operators and Systems
Abstract
Dynamical systems are the basis of mathematical models for describing many physical phenomena. From the point of view of system analysis and description, systems can usually be classified into linear and nonlinear systems. Starting from this chapter, the concept of systems will be introduced.
Xue Dingyü, Bai Lu
6. Analytical and Numerical Solutions of Linear Fractional-Order Differential Equations
Abstract
In the past, the familiar dynamical systems are described by integer-order differential equations. Accordingly, the fractional-order dynamical systems are described by fractional-order differential equations (FODEs). In this chapter, linear FODEs are discussed. Analytical and numerical solutions of linear FODEs are presented. Also, some attempts on solution and stability of irrational FODEs are discussed.
Xue Dingyü, Bai Lu
7. Numerical Solutions of Nonlinear FODEs
Abstract
Analytical solution methods for linear FODEs were given in Chap. 6. High-precision numerical algorithms for Riemann–Liouville and Caputo differential equations were given. In control system research and other fields of applications, nonlinear behaviors are unavoidable. In this chapter, some numerical solution methods on nonlinear FODEs are studied, and the high-precision algorithm is presented and recommended.
Xue Dingyü, Bai Lu
8. Block Diagram-Based Solutions of FODEs
Abstract
Some numerical algorithms for nonlinear FODEs are presented in Chapter 7. In this chapter, a FOTF blockset for Simulink is designed and presented. A block diagram-based FODE solution framework is presented. The Simulation modeling and solution framework of various nonlinear FODEs is established.
Xue Dingyü, Bai Lu
9. Numerical Solutions of Special Fractional-Order Differential Equations
Abstract
Command-driven methods for nonlinear Caputo equations are introduced in Chapter 7, and the methods are mainly used in solving explicit equations and FOSS equations. In Chapter 8, block diagram-based methods are introduced for these FODEs, with wider application fields. In this chapter, various complicated FODEs such as implicit FODEs, delay FODEs and boundary value problems of FODEs are explored in Simulink. A brief introduction is made to time-fractional FODE is introduced.
Xue Dingyü, Bai Lu
Backmatter
Metadaten
Titel
Fractional Calculus
verfasst von
Dingyü Xue
Lu Bai
Copyright-Jahr
2024
Verlag
Springer Nature Singapore
Electronic ISBN
978-981-9920-70-9
Print ISBN
978-981-9920-69-3
DOI
https://doi.org/10.1007/978-981-99-2070-9

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