East Asian J. Appl. Math., 6 (2016), pp. 131-151.
Published online: 2018-02
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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} }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.