标题： 分数阶导数：傅里叶，大象，记忆效应，粘弹性材料和异常扩散 显示英文标题

标题： Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials and anomalous diffusions

摘要： 本文将发表在《美国数学学会通知》上，从分数微积分的起源以及莱布尼茨和傅里叶所起的关键作用的简要历史记载开始。 通过称为半群方法的现代技术，介绍了傅里叶的分数阶导数定义并进行了剖析。 介绍了分数阶导数理论的最新进展。 此外，我们还讨论了一些科学界提出的问题。 最后，我们展示了三种不同的应用：具有记忆效应的人口增长、粘弹性材料和异常扩散。 显示英文摘要

摘要： This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials and anomalous diffusions.