@Article{EAJAM-6-131, author = {Du , RuiHao , Zhao-Peng and Sun , Zhi-Zhong}, 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} }