Some recent developments of numerovʹs method

This paper is a survey of some recent developments of Numerovʹs method for solving nonlinear two-point boundary value problems. The survey consists of three different parts: the existence-uniqueness of a solution, computational algorithm for computing a solution, and some extensions of Numerovʹs method. The sufficient conditions for the existence and uniqueness of a solution are presented. Some of them are best possible. Various iterative methods are reviewed, including Picardʹs iterative method, modified Newtonʹs iterative method, monotone iterative method, and accelerated monotone iterative method. In particular, two more direct monotone iterative methods are presented to save computational work. Each of these iterative methods not only gives a computational algorithm for computing a solution, but also leads to an existence (and uniqueness) theorem. The estimate on the rate of convergence of the iterative sequence is given. The extensions of Numerovʹs method to a coupled problem and a general problem are addressed. The numerical results are presented to validate the theoretical analysis.