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.

中文翻译：

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分布阶分数算子的应用：回顾


分布阶分数阶微积分 (DOFC) 是分数阶微积分更广泛领域中一个快速新兴的分支，在复杂系统的建模方面具有重要而深远的应用。 DOFC 概括了常量和变阶分数算子的内在多尺度性质，为建模系统提供了重要的机会，这些系统的行为源于多个尺度上发生的非局域效应和记忆效应的复杂相互作用和叠加。近年来，大量研究集中在 DOFC 的数学方面和实际应用。然而，仍然缺乏对现有文献和 DOFC 最新技术（特别是与实际应用相关）的系统回顾。这篇综述文章旨在为读者提供一个路线图，以了解 DOFC 的早期发展以及复杂现实问题建模的逐步演变和应用。这篇综述首先简要介绍了 DOFC 的数学原理，包括分析方法和数值方法，随后对 DOFC 在粘弹性、输运过程和控制理论等领域的应用进行了广泛的概述，这些领域已成为大多数研究的焦点迄今为止的活动。