Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grunwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.