As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end.

翻译：众所周知，已有多种方法被提出用于数值逼近Caputo分数阶导数。这些方法面临的一个共同挑战是Caputo分数阶导数的非局部特性，这导致了计算速度慢且内存消耗大的缺陷。分数阶导数的扩散表示是克服上述挑战的有效工具。本文提出了两种新的扩散表示方法来逼近阶数为$0<\alpha<1$的Caputo分数阶导数。最后给出了新方法的误差分析及若干数值算例。