A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind

We present two new finite difference methods of order two and four in a coupled manner for the general one-dimensional nonlinear biharmonic equation yIV=f(x,y,y′,y″,y″′) subject to the boundary conditions y(a)=A0, y′(a)=A1, y(b)=B0,y′(b)=B1. In both cases, we use only three grid points and do not require to discretize the boundary conditions. First-order derivative of the solution is obtained as a by-product of the methods. The methods are successfully applied to the problems both in cartesian and polar coordinates. Numerical examples are given to illustrate the methods and their convergence.