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Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions
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Year:    2016

Author:    Rui Du, Zhao-Peng Hao, Zhi-Zhong Sun

East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 2 : pp. 131–151

Abstract

This article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with $\mathscr{O}(τ^2+h^4+σ^4)$ convergence in the $L_1$($L_∞$)-norm for the one-dimensional case, where $τ$, $h$ and $σ$ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and $\mathscr{O}(τ^2|lnτ|+h^4_1+h^4_2+σ^4)$ convergent in the $L_1$($L_∞$)-norm, where $h_1$ and $h_2$ are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.020615.030216a

East Asian Journal on Applied Mathematics, Vol. 6 (2016), Iss. 2 : pp. 131–151

Published online:    2016-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    21

Keywords:    Distributed-order time-fractional equations Lubich operator compact difference scheme ADI scheme convergence stability.

Author Details

Rui Du

Zhao-Peng Hao

Zhi-Zhong Sun

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