A new algorithm for nonlinear fourth order multi-point boundary value problems

In this paper, a new algorithm is presented to solve the nonlinear fourth-order differential equations with complicated boundary conditions. The approach combines the Quasi-Newton's method and the simplified reproducing kernel method. It is worth mentioning that the Quasi-Newton's method is proposed for solving the nonlinear differential equations for the first time. Meanwhile, the simplified reproducing kernel method is applied to solve the linear equations which are obtained from the conversion of the Quasi-Newton's method, avoiding the time-consuming Schmidt orthogonalization process. Moreover, the reproducing kernel space and its reproducing kernel are reasonably simple as no considering of the complicated boundary conditions. Furthermore the present scheme is employed successfully on some numerical examples.