Abstract In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.

中文翻译：

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带有Caputo分数阶导数的线性微分方程解的表示公式

摘要 本文考虑了一个具有变系数和Caputo分数阶导数的线性微分方程。对于这个方程，研究了柯西问题，当在中间点给出初始条件时，该中间点不一定与分数阶微分算子的初始点重合。对基本解矩阵的基本性质进行了详细分析。特别地，证明了该矩阵关于两个变量的 Hölder 连续性，并给出了它的对偶定义。在此基础上，提出并证明了求解柯西问题的两个表示公式。