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 - <p style="text-align: justify;">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.</p>