The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions
The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions
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DOI：
10.1016/j.camwa.2020.11.015
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发表时间：
2021-01
期刊：
影响因子：
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通讯作者：
Xuehua Yang;Haixiang Zhang;Jie Tang
中科院分区：
文献类型：
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作者：
Xuehua Yang;Haixiang Zhang;Jie Tang
提出了一种基于正交样条配置(OSC)方法的高阶方法，用于求解二维矩形域上边平行于坐标轴的四阶次扩散问题，其解在初始时刻呈现典型的弱奇异性。通过引入辅助变量v=Δu，将四阶问题化为一对二阶系统。针对α∈(0，1)阶卡普托分数导数，通过在初始时刻附近插入更多的网格点，考虑了梯度网格上的L1格式。借助于互补离散卷积核和离散分数Grönwall不等式等性质，我们建立了原始未知u和辅助变量v的无条件稳定性和收敛性质，并给出了一些数值实验以进一步验证我们的理论分析。
A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v= Δ u, the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α∈(0, 1) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Grönwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v. Some numerical experiments are provided to further verify our theoretical analysis.