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

Published online by Cambridge University Press:  12 May 2016

Rui Du ,
Zhao-peng Hao  and
Zhi-zhong Sun
Rui Du*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Zhao-peng Hao
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Zhi-zhong Sun
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
*
*Corresponding author. Email addresses:rdu@seu.edu.cn (R. Du), haochpeng@126.com (Z. Hao), zzsun@seu.edu.cn (Z. Sun)
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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 convergence in the L1(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 convergent in the L1(L∞)-norm, where h1 and h2 are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

Keywords

Distributed-order time-fractional equationsLubich operatorcompact difference schemeADI schemeconvergencestability
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 131 - 151
Copyright
Copyright © Global-Science Press 2016 

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