We consider the semidiscrete approximation of the Cauchy problem $$\begin{aligned} ({\mathbf {D}}_{t}^{\alpha }u)(t) = A u(t) +f\big (t,u(t)\big ), \quad t \in (0,T],\ u(0) = u^0,\ 0< \alpha <1, \end{aligned}$$ on a Banach space, where the operator A generates an analytic and compact resolvent family $$\{S_{\alpha }(t,A)\}_{t\ge 0}$$ and the function $$f(\cdot , \cdot )$$ is strongly continuous. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.

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半线性分数柯西问题的近似：II

我们考虑柯西问题 $$\begin{aligned} ({\mathbf {D}}_{t}^{\alpha }u)(t) = A u(t) +f\big (t ,u(t)\big ), \quad t \in (0,T],\ u(0) = u^0,\ 0< \alpha <1, \end{aligned}$$ 在 Banach 空间上，其中算子 A 生成一个解析的和紧凑的解析器族 $$\{S_{\alpha }(t,A)\}_{t\ge 0}$$ 和函数 $$f(\cdot , \cdot )$ $ 是强连续的，我们分析了一个一般的近似方案，包括有限差分和投影方法。