High Order Numerical Methods For Two Fourth-order Integro-differential Equations With Weakly Singular Kernel

In this paper,we consider the high order Legendre-Galerkin spectral methods for two fourth-order partial integro-differential equations with weakly singular kernel.For the first fourth-order integro-differential equation with first-order partial derivative in time,the Crank-Nicolson scheme is used for time discretization,the integral term is approximated by Jacobi numerical integration and Legendre numerical integration.Legendre-Galerkin spectral method is used in space,which result in the corresponding sparse discrete algebraic system of the equation.Numerical experiments show the effectiveness and large time stability of the discrete scheme.For the second fourth-order integro-differential equation with second-order partial derivative in time,the second order central difference method is used for time discretization,Jacobi numerical integration and Legendre numerical integration are employed to approximate the intergral term.Legendre-Galerkin spectral method is used in space approximation,we then obtain the corresponding sparse discrete algebraic system of the second equation.Numerical experiments show that the method is effective.