Some high-order difference schemes for the distributed-order differential equations

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L 1 ( L ∞ ) norm with the convergence orders O ( ¿ 2 + h 2 + Δ α 2 ) and O ( ¿ 2 + h 4 + Δ α 4 ) , respectively, where ¿, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.