The finite difference method for the fourth-order partial integro-differential equations with the multi-term weakly singular kernel

This paper is concerned with an efficient numerical method for a class of fourth-order partial integro-differential equations (PIDEs) with weakly singular kernel. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t=0. The proposed method is constructed on the graded meshes, which can achieve the second-order convergence in time for weakly singular solutions. The product integral rule of the Riemann-Liouville fractional integral with the generalized time-stepping is used for the time discretization, and the standard central difference formula is used for the space discretization. The stability and convergence of the method are proved, and the optimal error estimates in the discrete L-2-norm and H-1-norm are obtained. Numerical results show the effectiveness of the proposed method. Further, by the extrapolation method, we improve the convergence order in space and time to four order, respectively.