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In this paper, a high-order approximation to Caputo-type time-fractionaldiffusion equations involving an initial-time singularity of the solution isproposed. At first, we employ a numerical algorithm based on the Lagrangepolynomial interpolation to approximate the Caputo derivative on thenon-uniform mesh. Then truncation error rate and the optimal grading constantof 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 themesh grading parameter. Using this new approximation, a difference scheme forthe Caputo-type time-fractional diffusion equation on graded temporal mesh isformulated. The scheme proves to be uniquely solvable for general $r$. Then wederive the unconditional stability of the scheme on uniform mesh. Theconvergence of the scheme, in particular for $r=1$, is analyzed for non-smoothsolutions and concluded for smooth solutions. Finally, the accuracy of thescheme is verified by analyzing the error through a few numerical examples.

High order approximation to Caputo derivative on graded mesh and time-fractional diffusion equation for non-smooth solutions