Error analysis of a finite difference method for the distributed order sub-diffusion equation using discrete comparison principle

Abstract

A finite difference method for numerically solving the initial boundary value problem of distributed order sub-diffusion equations with weakly singular solutions is presented, where Caputo fractional derivative is approximated by L1 scheme on nonuniform mesh and the space is discretized by finite difference method. The error analysis for the fully discrete scheme is obtained by using the discrete comparison principle. By carefully choosing a barrier function, the final error bound is β-robust which does not blow up when β→1−, where β (0<β<1) is considered to be the order of the distributed order fractional derivative operator. The numerical results are given and have shown that the error analysis is sharp.