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

Abstract

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 $r\ge 1$ 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.

18 pages, 2 figures and 7 tables