Volume 6, Issue 2
Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

East Asian J. Appl. Math., 6 (2016), pp. 131-151.

Published online: 2018-02

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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 $\mathscr{O}(τ^2+h^4+σ^4)$ 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.

• Keywords

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

65M10

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@Article{EAJAM-6-131, author = {Rui and Du and and 17965 and and Rui Du and Zhao-Peng and Hao and and 17966 and and Zhao-Peng Hao and Zhi-Zhong and Sun and and 17967 and and Zhi-Zhong Sun}, title = {Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {6}, number = {2}, pages = {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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.020615.030216a}, url = {http://global-sci.org/intro/article_detail/eajam/10785.html} }
TY - JOUR T1 - Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions AU - Du , Rui AU - Hao , Zhao-Peng AU - Sun , Zhi-Zhong JO - East Asian Journal on Applied Mathematics VL - 2 SP - 131 EP - 151 PY - 2018 DA - 2018/02 SN - 6 DO - http://doi.org/10.4208/eajam.020615.030216a UR - https://global-sci.org/intro/article_detail/eajam/10785.html KW - Distributed-order time-fractional equations, Lubich operator, compact difference scheme, ADI scheme, convergence, stability. AB -

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.

Rui Du, Zhao-PengHao & Zhi-ZhongSun. (2020). Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions. East Asian Journal on Applied Mathematics. 6 (2). 131-151. doi:10.4208/eajam.020615.030216a
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