Fractional Calculus High-Precision Algorithms and Numerical Implementations

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.

Introduction to Fractional Calculus.- Commonly Used Special Functions: Definitions and Computing.- Definitions and Numerical Evaluations of Fractional Calculus.- High-precision Numerical Algorithms and Implementation in Fractional Calculus.- Approximations of Fractional-order Operators and Systems.- Analytical and Numerical Solutions of Linear Fractional-order Differential Equations.- Numerical Solutions of Nonlinear FODEs.- Block Diagram based Solutions of FODEs.- Numerical Solutions of Special Fractional-order Differential Equations.

Xue Dingyu, received his doctorate from Sussex University in 1992, and took his post at Northeastern University in China in 1993, where he was promoted to full professorship in 1997. He started his teaching work on MATLAB since 1990, where he designed a Lab Demonstration GUI for Classroom in Sussex University, while he we doing his D Phil research. In 1996, he published his first MATLAB book in Chinese, entitled “Computer aided Control System Design with MATLAB Applications” in Tsinghua University Press in China, which was regarded the first in China, and one of the first few earliest ones in the world. In 30+ years of time, he has published 12 books in English plus numerous books in Chinese. The recent ones are the six-volume works on scientific computing and simulation, published in De Gruyter, Berlin, in 2020-2022.

He received many teaching honors and awards in China, for National-level Elite Courses, Textbooks, and National First-class Undergraduate courses, and national-level Educational Awards. He is the co-founder and vice-chair of the Fractional-order Systems and Control Committee, China Automation Association. He received two China Natural Science Foundation grants on fractional systems. Co-author (4th) of the 2010 Springer monograph on fractional calculus, Author of the 2017 De Gruyter monograph on Fractional calculus and fractional-order control. He is the author of the FOTF MATLAB Toolbox, one of the four most widely used MATLAB tools in the fractional calculus community worldwide. 

Bai Lu, was a PhD candidate under Xue Dingyu’s supervision. He received his doctorate from Northeastern University in 2018, with many contributions in high-precision algorithms. He is now a lecturer in Shenyang University. He is also the co-author of the Chinese version of the proposed monograph.