Homogenization of time-fractional diffusion equations with periodic coefficients

Abstract

© 2020 Elsevier Inc. 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)∈L2(D) in a bounded domain D⊂Rd with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient κϵ(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 L2(D) and Lp((θ,T);H1(D)) for p∈[1,∞) and θ∈(0,T) has a convergence rate of O(ϵ1/2) when the dimension d≤2 and O(ϵ1/6) when d=3. Several numerical tests are presented to demonstrate the performance of the first order approximation.