A variably distributed-order time-fractional diffusion equation: Analysis and approximation

Abstract

We prove the wellposedness and smoothing properties of the initial–boundary value problem of a variably distributed-order time-fractional diffusion partial differential equation in multiple space dimensions, which models the subdiffusive transport of solutes traveling through heterogeneous porous media. We accordingly derive a finite element approximation to the problem and prove its optimal-order error estimate, only under the regularity assumptions of the variably distributed order, coefficients and the source term, but without any regularity assumption on the true solution of the problem. Numerical experiments are presented to verify the mathematical and numerical analyses.