ABSTRACT
The aim of this chapter is to develop a sense of what kinds of initial value problems can be solved numerically and how to prepare problems for their numerical solution. First we review the “facts of life” about the existence and uniqueness of solutions. It is convenient in the theory of ordinary differential equations (ODEs) to work with problems written in a standard form, and because the codes all expect problems to be presented in this way, we must go into this. Some basic mathematical tools are found in the appendix. The concept of “order” is fundamental to a study of the numerical solution of the initial value problem. Because it may not be familiar to the reader, the elements are developed here. Finally a series of substantial examples are taken up to show how one might be able to deal with problems that do not fit neatly into the standard theory of ODEs and their numerical solution. Some of the examples will be used throughout the book for illustrative purposes.
