An equivalent definition of the fractional Caputo derivative $D^\nu_t g$, for $\nu \in (0, 1)$, is found, within suitable assumptions on $g$. Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion equation subject to nonhomogeneous boundary conditions. This approach also allows to construct numerical solutions to the initial-boundary value problem for the subdiffusion equation with memory.

中文翻译：

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Caputo导数的等效定义及其在次扩散方程中的应用

在$ g $的适当假设内，找到了对于$ \ nu \ in（0，1）$的分数Caputo导数$ D ^ \ nu_t g $的等效定义。然后讨论了分数微积分和分数阶偏微分方程理论的一些应用。特别是，此替代定义用于证明服从非均匀边界条件的线性扩散方程经典解的最大原理。这种方法还允许构造具有记忆的子扩散方程的初边值问题的数值解。