High-Order Approximation to Caputo Derivative on Graded Mesh and Time-Fractional Diffusion Equation for Nonsmooth Solutions

In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations (TFDEs) 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 nonuniform mesh. The truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as $min{4−α,rα}$ and $(4−α)/α$⁠, respectively, where $α∈(0,1)$ is the order of fractional derivative and $r≥1$ is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on the 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 nonsmooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.