The commonly defined fractional derivatives, like Riemann-Liouville and Caputo ones, are non-local operators which have the long-term memory characteristic, since they are in connection with all historical data. Because of this special property, they may be invalid for modeling some processes and materials with short-term memory phenomena. Motivated by this observation and in order to enlarge the applicability of fractional calculus theories, a fractional derivative with the short-term memory property is defined in this paper. It can be viewed as an extension of the Caputo fractional derivative. Several properties of this short memory fractional derivative are given and proved. Meanwhile, the stability problem for fractional differential equations with such a derivative is studied. By applying fractional Lyapunov direct methods, the stability conditions applicable to the local case and the global case are established respectively. Finally, three numerical examples are provided to demonstrate the correctness and effectiveness of the theoretical results.

通常定义的分数导数，如黎曼-刘维尔和卡普托导数，是具有长期记忆特征的非局部算子，因为它们与所有历史数据相关。由于这种特殊性质，它们可能对一些具有短期记忆现象的过程和材料的建模无效。受此观察的启发，为了扩大分数阶微积分理论的适用性，本文定义了具有短期记忆特性的分数阶导数。它可以看作是 Caputo 分数导数的扩展。给出并证明了这种短记忆分数导数的几个性质。同时研究了具有这种导数的分数阶微分方程的稳定性问题。通过应用分数 Lyapunov 直接方法，分别建立了适用于局部案例和全局案例的稳定性条件。最后通过三个数值算例证明了理论结果的正确性和有效性。