An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel

Abstract

In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit (ADI) method in time, combined with the second-order fractional quadrature convolution rule introduced by Lubich and the classical L1 approximation for Caputo fractional derivative. The detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order O(τmin{1+α,2−α}+h12+h22). Some numerical results are also given to confirm our theoretical prediction.