We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^2(D)$ in a bounded domain $D\subset {\mathbb{R}}^d$ with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient ${\kappa }^{ϵ}(x)$ is smooth and periodic with the period $ϵ>0$ being sufficiently small. We derive that its first order approximation measured by both pointwise-in-time in $L^2(D)$ and $L^p((\theta ,T);H^1(D))$ for $p\in [1,\infty )$ and $\theta \in (0,T)$ has a convergence rate of $\mathcal{O}({ϵ}^{1/2})$ when the dimension $d\le 2$ and $\mathcal{O}({ϵ}^{1/6})$ when $d=3$. Several numerical tests are presented to demonstrate the performance of the first order approximation.

考虑具有齐次Dirichlet边界条件和不均匀初始数据的时间分数阶扩散方程的初始边界值问题 $\mathrm{一种}（X）\in {\mathrm{大号}}^2（d）$ 在有界域中 $d\subset {\mathbb{[R}}^d$具有足够平滑的边界。我们在假设扩散系数的情况下分析均化解${\kappa }^{ϵ}（X）$ 平稳且定期 $ϵ>0$足够小。我们推导它的一阶近似由两个时间点${\mathrm{大号}}^2（d）$ 和 ${\mathrm{大号}}^p（（\theta ，Ť）;H^{\mathrm{1个}}（d））$ 对于 $p\in [\mathrm{1个}，\infty ）$ 和 $\theta \in （0，Ť）$ 收敛速度为 $\mathcal{Ø}（{ϵ}^{\mathrm{1个}/2}）$ 当尺寸 $d\le 2$ 和 $\mathcal{Ø}（{ϵ}^{\mathrm{1个}/6}）$ 什么时候 $d=3$。提出了几个数值测试来证明一阶近似的性能。