An introduction to fractional calculus and its Applications in Electric Circuits

Abstract A natural extension of differential calculus, initially proposed by l'Hôpital in a letter to Leibniz, leaded to the concept of fractional order derivatives. The application of fractional derivatives allows one to correctly describe the dynamics of many real systems, from biosystems to financial markets, where memory effects, dissipation and fractal dimensionality are present. This paper aims to present an overview of fractional derivatives and its representations, both in the form of Grünwald-Letnikov finite differences as well as in the form of Riemann-Liouville integrals, and to apply it in describing RC and RL electrical circuits of fractional order.

摘要：微积分的自然延伸最初由洛必达在致莱布尼茨的一封信中提出，由此催生了分数阶导数（fractional order derivatives）的概念。分数阶导数的应用能够精准刻画诸多实际系统的动力学行为，其覆盖范畴从生物系统延伸至金融市场，这类系统普遍存在记忆效应、耗散现象与分形维度特性。本文旨在对分数阶导数及其多种表示形式进行综述，既涵盖格伦沃尔德-列特尼科夫（Grünwald-Letnikov）有限差分形式，也包含黎曼-刘维尔（Riemann-Liouville）积分形式，并将其应用于分数阶RC与RL电路的描述中。