This book presents a comprehensive investigation into the exponential and Mittag-Leffler Euler differences for semi-linear fractional-order differential equations, a subject falling within the purview of computational mathematics. The field of exponential and Mittag-Leffler Euler differences has witnessed a period of rapid development in recent times. This has led to the emergence of new techniques such as exponential Euler integrator, exponential Runge-Kutta methods, multistep exponential integrator, exponential Rosenbrock-type methods, and more. This book puts forth several difference approaches to fractional-order differential equations and offers insights into their practical applications in the study of neural networks. Adopting a holistic approach, the book presents a foundational framework for this topic, underscoring the significance of exponential and Mittag-Leffler Euler differences in the numerical theory of fractional-order differential equations.

The book is designed for graduate students with an interest in numerical solutions of fractional-order differential equations, as well as for researchers engaged in the qualitative theory of difference equations.

Autorentext

Tianwei Zhang received the B.S. degree from the School of Mathematics, Yunnan Normal University, Kunming, Yunnan, China, in 2008, and his M.S. degree in Applied Mathematics from the Yunnan University of China (2008.9-2011.7). In 2020-2023, he received his PhD of Applied Mathematics in Yunnan University. He is an active researcher in Applied Mathematics (Artificial neural networks, Stochastic theory, Differential Equations and Dynamic Systems, etc.). He has published more than 80 papers and is an active reviewer of many international journals. Since Nov. 2023, Dr. Zhang has been in School of Mathematics and Statistics, Yunnan University as a teacher.

Yongkun Li received the Ph.D. degree from Sichuan University, Chengdu, China. He is currently a Professor and doctoral supervisor in Yunnan University, Kunming, China. one of the first young and middle-aged academic and technical leaders in Yunnan Province, member of the Mathematical Discipline Evaluation Group of Yunnan Provincial Academic Degree Committee, commentator of "American Mathematical Review" and "German Mathematical Digest", director of the Chinese Mathematical Society, chairman of the Yunnan Society of Industrial and Applied Mathematics. He is the author or coauthor of more than 200 journal articles. His research interests include nonlinear dynamic systems, neural networks, and applied mathematics.

Jianwen Zhou received the B.S. degree from the School of Mathematics, Yunnan Normal University, Kunmimg, Yunnan, China, in 2004, the M.S. degree from the School of Mathematics, Yunnan Normal University, Kunmimg, Yunnan, China, in 2007, and the Ph.D. degree from the Department of Mathematics, Yunnan University, Yunnan, China, in 2010. He is currently a Professor and doctoral supervisor with the Yunnan University. He is also deputy secretary general of Industrial and Applied Mathematics Society of Yunnan Province and an Active Researcher in Nonlinear differential equations and applications. He has published more than 30 papers and is an active reviewer of many international journals.