  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.
