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

TL;DR

This paper develops a high-order numerical approximation for Caputo derivatives on graded meshes, enabling accurate and stable solutions to time-fractional diffusion equations with non-smooth initial data.

Contribution

It introduces a novel high-order approximation method for Caputo derivatives on graded meshes, improving accuracy for non-smooth solutions in fractional diffusion equations.

Findings

Truncation error rate is $ ext{min}igrace{4-eta,retaigrace}$.

The difference scheme is unconditionally stable on uniform mesh.

Numerical examples confirm the scheme's high accuracy.

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 $r\geq 1$ 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.