Patnaik Sansit, Hollkamp John P, Semperlotti Fabio
School of Mechanical Engineering, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA.
Proc Math Phys Eng Sci. 2020 Feb;476(2234):20190498. doi: 10.1098/rspa.2019.0498. Epub 2020 Feb 12.
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.
变阶分数阶算子直到最近几年才被提出并在数学上进行形式化。能够制定演化控制方程使得这些算子成功应用于从力学、传输过程、控制理论到生物学等复杂现实世界问题的建模。变阶分数阶微积分（VO-FC）是微积分中一个相对鲜为人知的分支，它为模拟跨学科过程提供了显著的机会。认识到这一未开发的潜力，科学界一直在深入探索VO-FC在工程和物理系统建模中的应用。这篇综述旨在为对这个迷人领域感兴趣的读者提供一个起点。我们对VO-FC分析和计算方法的发展以及在复杂物理系统模拟中的应用所取得的进展进行了简洁而全面的总结。更具体地说，在简短介绍基本数学概念之后，我们从科学建模背景下的实际应用角度介绍VO-FC这一主题。
