Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain $m\times m$ matrices (where $m$ is the order of the collocation scheme), are verified both analytically, for all $m\ge 1$ and all sets of collocation points, and computationally, for all $ m\le 20$. The semilinear case is also addressed.

翻译：时间分数阶抛物型方程具有阶数为$\alpha\in(0,1)$的Caputo时间导数，我们采用连续配置方法对其进行时间离散化。针对此类离散格式，给出了其解存在唯一性的充分条件。本文探讨了两种方法：Lax-Milgram定理和特征函数展开。所得的充分条件涉及某些$m\times m$矩阵（其中$m$为配置格式的阶数），并通过解析方法（对所有$m\ge 1$及所有配置点集）和计算方法（对所有$m\le 20$）进行了验证。此外，还讨论了半线性情况。