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 L1(L∞) norm with the convergence orders O(τ2+h2+Δα2) and O(τ2+h4+Δα4), respectively, where τ, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.