In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

中文翻译：

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通用分数阶导数的算术运算及其应用

在本文中，我们首先使用在零点处具有幂函数类型的可积奇异性的Sonine核来解决一般分数积分和导数。讨论了这些运算符的特殊情况和组成。然后，我们为带有Sonine核的一般分数导数构建Mikusiński类型的运算演算。该操作演算被用于分析处理具有一般分数导数的分数阶微分方程的一些初值问题。解以卷积级数的形式表示，该卷积级数泛化了指数函数和Mittag-Leffler函数的幂级数。