Finite Difference Methods for Fractional Differential Equations on Non-Uniform Meshes

Abstract

The solutions of fractional equations with Caputo derivative often have a singularity at the initial time. Therefore, for numerical methods on uniform meshes it is difficult to achieve optimal convergence rates. To improve the convergence, Liu et al. [10] considered a finite difference method on non-uniform meshes. Following the ideas of [10], we introduce two more sets of non-uniform meshes and show that the corresponding discrete models have higher convergence rates. Besides, we apply the trapezoidal rule in the case of linear fractional partial differential equations. The results of numerical experiments are consistent with the theoretical analysis.