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 $v=\Delta u$, 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 $\alpha \in (0,1)$ 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.

提出了一种基于正交样条搭配（OSC）方法的高阶方法，用于求解矩形平行于坐标轴的二维矩形域上的四阶子扩散问题，其解在初始时表现出典型的弱奇异性时间。通过引入辅助变量$v=\Delta ü$，将四阶问题简化为几个二阶系统。考虑阶次Caputo分数导数的L1渐变网格方案$\alpha \in （0，\mathrm{1个}）$通过在初始时间附近插入更多网格点。借助互补离散卷积核和离散分数Grönwall不等式的某些属性，我们为原始未知数建立了无条件稳定性和收敛性$ü$ 和辅助变量 $v$。提供了一些数值实验以进一步验证我们的理论分析。