In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the non-uniform mesh. Then truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as $\min\{4-\alpha,r\alpha\}$ and $\frac{4-\alpha}{\alpha}$, respectively, where $\alpha\in(0,1)$ is the order of fractional derivative and $r\geq 1$ is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general $r$. Then we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for $r=1$, is analyzed for non-smooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.

翻译：本文针对含初值奇异性的Caputo型时间分数阶扩散方程，提出了一种高阶逼近方法。首先，基于拉格朗日多项式插值构建非均匀网格上的数值算法来近似Caputo导数。随后，在分级网格上获得了该逼近方法的截断误差阶为$\min\{4-\alpha,r\alpha\}$，最优分级常数为$\frac{4-\alpha}{\alpha}$，其中$\alpha\in(0,1)$为分数阶阶数，$r\geq 1$为网格分级参数。利用该新型逼近方法，建立了Caputo型时间分数阶扩散方程在时间分级网格上的差分格式，并证明该格式对一般$r$具有唯一可解性。进一步推导了均匀网格上格式的无条件稳定性。针对非光滑解情形（特别是$r=1$时）分析了格式的收敛性，并推广至光滑解情形。最后通过数值算例验证了格式的精度。