# Homogenization of time-fractional diffusion equations with periodic coefficients

## TL;DR

This paper studies the homogenization of time-fractional diffusion equations with periodic coefficients, deriving convergence rates for the approximation and validating results through numerical tests.

## Contribution

It provides the first order approximation and convergence rates for homogenized solutions of time-fractional diffusion equations with periodic coefficients.

## Key findings

- Convergence rate of O(ε^{1/2}) for dimensions d≤2
- Convergence rate of O(ε^{1/6}) for dimension d=3
- Numerical tests confirm the effectiveness of the approximation

## Abstract

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.