Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

中文翻译：

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Riemann-Liouville分数导数关于函数的运算及其应用

Mikusiński的运算演算是形式化的形式，用于理解积分和导数算子并求解微分方程，Y。Luchko和合作者已将其应用于几种类型的分数阶演算算子，例如[26]等。在本文中，我们考虑了一个函数相对于另一个函数的Riemann-Liouville分数阶微分的算符，并发现在这种更一般的设置下，也可以对Luchko的方法进行少量修改。Mikusiński的运算演算方法用于获得具有恒定系数和这类分数导数的分数阶微分方程的精确解。这些解决方案可以用Mittag-Leffler类型的函数表示。