Homogenization of time-fractional diffusion equations with periodic coefficients

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 $d\le 2$ and $\mathcal{O}({\epsilon }^{1/6})$ when $d=3$. Several numerical tests are presented to show the performance of the first order approximation.

References in corpus (3)

• Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
• Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins
• Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

Cited by in corpus (1)

• Partially Explicit Time Discretization for Time Fractional Diffusion Equation