Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

This article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with O (tau(2) + h(4) + sigma(4)) convergence in the L-1(L-infinity)-norm for the one-dimensional case, where tau, h and sigma are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and O (tau(2) vertical bar ln tau vertical bar + h(1)(4) + h(2)(4) + sigma(4)) convergent in the L-1(L-infinity)-norm, where h(1) and h(2) are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.