Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations

The Grunwald formula is used to solve the one-dimensional distributed-order differential equations. Two difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O ( + h 2 + 2 ) and O ( + h 4 + 4 ) in maximum norm, respectively, where , h and are step sizes in time, space and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O ( 2 + h 2 + 2 ) and O ( 2 + h 4 + 4 ) , respectively. An illustrative numerical example is given to confirm the theoretical results. (c) 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 591-615, 2016