Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order

Abstract

Variable-order time-fractional diffusion equations (tFDEs), in which the variable fractional order accommodates the fractal dimension change of the surrounding medium via the Hurst index, provide a competitive instrument to describe anomalously diffusive transport of particles through deformable heterogeneous materials. We analyze the mapping properties of the fractional integral with a general time-dependent variable order, based on which we prove the well-posedness and smoothing properties of corresponding variable-order tFDE model in multiple space dimensions. We then derive and analyze a fully discretized finite element approximation, in which we develop a novel decomposition of L-1 discretization coefficients to prove an optimal-order error estimate of the numerical scheme based only on the regularity assumptions on the data. Numerical experiments are performed to substantiate the theoretical findings.