中文摘要：

在那里存在，这被知道二个大多数之间的 obivious 差别通常使用了部分 derivatives-Riemann-Liouville (R-L ) 的定义定义和 Caputo 定义。在部分演算的部分衍生物的多重定义在流变学妨碍了部分演算的申请。在这份报纸，我们澄清 R-L 定义和 Caputo 定义在部分元素模型(Scott-Blair 模型) 的机械类似物的帮助下两个都是 rheologically 有瑕疵的。我们也澄清那 rheologically 使他们完美，两个定义的更低的终端应该被放到。我们进一步证明有更低的终端的 R-L 定义一 and 有更低的终端的 Caputo 定义一在足够光滑的功能和有单个点的有限数字的功能的区别是相等的。因此，我们能与更低的终端在流变学把部分衍生物定义为 R-L 衍生物一(或，相等地，有更低的终端的 Caputo 衍生物一) 不仅为不变的过程，而且为短暂过程。基于上述定义，为部分微分方程的部分操作员和起始的条件的作文规则的问题分别地被讨论。作为一个例子，我们与 Scott-Blair 模型一起学习一个部分振荡器并且在给定的起始的条件下面给这个方程的一个准确答案。

英文摘要：

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville （R-L） definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model （Scott-Blair model）. We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R-L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal a →∞ （or, equivalently, the Caputo derivatives with lower terminal a →∞） not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott-Blair model and give an exact solution of this equation under given initial conditions.