  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
$\kappa^{\epsilon}(x)$ is smooth and periodic with the period $\epsilon>0$
being sufficiently small. We derive that its first order approximation has a
convergence rate of $\mathcal{O}(\epsilon^{1/2})$ when the dimension $d\leq 2$
and $\mathcal{O}(\epsilon^{1/6})$ when $d=3$. Several numerical tests are
presented to show the performance of the first order approximation.
