Ding Wei, Patnaik Sansit, Sidhardh Sai, Semperlotti Fabio
Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.
Entropy (Basel). 2021 Jan 15;23(1):110. doi: 10.3390/e23010110.
Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.
分布阶分数阶微积分（DOFC）是分数阶微积分这一更广泛领域中迅速兴起的一个分支，在复杂系统建模方面有着重要且深远的应用。DOFC 概括了常数阶和变阶分数阶算子固有的多尺度性质，为对其行为源于在多个尺度上发生的非局部和记忆效应的复杂相互作用与叠加的系统进行建模提供了重要契机。近年来，已经产生了大量聚焦于 DOFC 的数学方面及实际应用的研究。然而，仍缺乏对现有文献以及 DOFC 最新技术水平（具体涉及实际应用方面）的系统综述。这篇综述文章旨在为读者提供一个路线图，以理解 DOFC 的早期发展以及其在对复杂实际问题建模中的逐步演变和应用。综述首先简要介绍了 DOFC 的数学知识，包括解析方法和数值方法，接着继续广泛概述了 DOFC 在诸如粘弹性、输运过程和控制理论等领域的应用，这些领域是迄今为止研究活动最为集中的领域。
