Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations

The Grünwald formula is used to solve the two-dimensional distributed-order differential equations. Two alternating direction implicit (ADI) difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(τ+h12+h22+Δα2) and O(τ+h14+h24+Δα4) in H1H1 norm, respectively, where τ,h1,h2τ,h1,h2 and ΔαΔα are the step sizes in time, space in xx- and yy-direction, and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O(τ2∣lnτ∣2+h12+h22+Δα2) and O(τ2∣lnτ∣2+h14+h24+Δα4), respectively. Several numerical examples are given to confirm the theoretical results.