Homogenization of time-fractional diffusion equations with periodic coefficients
This provides theoretical convergence rates for homogenization of time-fractional diffusion equations, which is incremental for researchers in fractional PDEs.
The authors analyze homogenization of time-fractional diffusion equations with periodic coefficients, deriving convergence rates of O(ε^{1/2}) for d≤2 and O(ε^{1/6}) for d=3, supported by numerical tests.
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 $κ^ε(x)$ is smooth and periodic with the period $ε>0$ being sufficiently small. We derive that its first order approximation has a convergence rate of $\mathcal{O}(ε^{1/2})$ when the dimension $d\leq 2$ and $\mathcal{O}(ε^{1/6})$ when $d=3$. Several numerical tests are presented to show the performance of the first order approximation.