Some recent developments of Numerov's method

This paper is a survey of some recent developments of Numerov's 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 Numerov's 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 Picard's iterative method, modified Newton's 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 Numerov's method to a coupled problem and a general problem a re addressed. The numerical results are presented to validate the theoretical analysis.