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：本文针对具有$\alpha\in(0,1)$阶Caputo时间导数的时间分数阶抛物型方程，采用连续配置方法进行时间离散。针对此类离散格式，我们给出了其解存在且唯一的充分条件。研究探索了两种分析路径：Lax-Milgram定理与特征函数展开法。所得充分条件涉及特定的$m\times m$矩阵（其中$m$为配置格式的阶数），我们通过解析方法对所有$m\ge 1$及任意配置点集进行了验证，同时通过数值计算对所有$m\le 20$的情形进行了验证。文中亦讨论了半线性情形下的相应问题。