Efficient Global Methods for the Numerical Solution of Nonlinear Systems of Two point Boundary Value Problems

In this paper we will be concerned with numerical methods for the solution of nonlinear systems of two point boundary value problems in ordinary differential equations. In particular we will consider the question which codes are currently available for solving these problems and which of these codes might we consider as being state of the art. In answering these questions we impose the restrictions that the codes we consider should be widely available (preferably written in MATLAB and/or FORTRAN) they should have reached a fairly steady state in that they are seldom, if ever, updated, they try to achieve broadly the same aims and, of course, it is relatively inexpensive to purchase the site licence. In addition we will be concerned exclusively with so called boundary value (or global) methods so that, in particular, we will not include shooting codes or Shishkin mesh methods in our survey. Having identified such codes we go on to discuss the possibility of comparing the performance of these codes on a standard test set. Of course we recognise that the comparison of different codes can be a contentious and difficult task. However the aim of carrying out a comparison is to eliminate bad methods from consideration and to guide a potential user who has a boundary value problem to solve to the most effective way of achieving his aim. We feel that this is a very worthwhile objective to pursue. Finally we note that in this paper we include some new codes for BVP's which are written in MATLAB. These have not been available before and allow for the first time the possibility of comparing some powerful MATLAB codes for solving boundary value problems. The introduction of these new codes is an important feature of the present paper.