Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral-differential equations and comparing with other methods, we find that the two methods are useful.