Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme

Abstract

In this paper, a new numerical scheme based on weighted and shifted Grünwald formula and compact difference operate is proposed. The proposed numerical scheme is used to solve time fractional partial integro-differential equation with a weakly singular kernel. Meanwhile the time fractional derivative is denoted by the Riemann-Liouville sense. Subsequently, we prove the stability and convergence of the mentioned numerical scheme and show that the numerical solution converges to the analytical solution with order O(τ2 + h4), where τ and h are time step size and space step size, respectively. The advantage is that the accuracy of the suggested schemes is not dependent on the fractional α. Furthermore, the numerical example shows that the method proposed in this paper is effective, and the calculation results are consistent with the theoretical analysis.