论文标题
分数衍生物和分数微积分的基本定理
Fractional derivatives and the fundamental theorem of Fractional Calculus
论文作者
论文摘要
在本文中，我们介绍了按有限间隔定义的分数积分和衍生物的单参数家族。首先，我们提醒读者，即已知的事实，即在某些合理的条件下，存在一个独特的分数积分家族，即著名的Riemann-Liouville分数积分。至于分数衍生物，它们的自然定义遵循分数演算的基本定理，即，将它们作为Riemann-Liouville分数积分作为左内算子引入。到目前为止，文献中提出了三个此类衍生物的家族：Riemann-Liouville分数衍生物，Caputo分数衍生物和Hilfer分数衍生物。我们阐明了这些衍生物在不同函数空间上的互连，并提供了其某些属性，包括投影仪的公式和拉普拉斯变换。但是，事实证明，存在无限的其他分数衍生物家族，它们是Riemann-Liouville分数积分的左内运算符。在本文中，我们专注于重要类别的这些分数衍生物，并讨论它们的某些特性。
In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.
