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

摘要

The Grunwald 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 0(tau + h(1)(2) + h(2)(2) + Delta alpha(2)) and 0(tau + h(1)(4) + h(2)(4) + Delta alpha(4)) in H-1 norm, respectively, where tau, h(1), h(2) and Delta alpha are the step sizes in time, space in x- and y-direction, and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders 0(tau(2) vertical bar In tau vertical bar(2) + h(1)(2) + h(2)(2) + Delta alpha(2)) and 0(tau(2) vertical bar In tau vertical bar(2) + h(1)(2) + h(2)(2) + Delta alpha(4)), respectively. Several numerical examples are given to confirm the theoretical results. (C) 2015 Elsevier Ltd. All rights reserved.