Preliminary Theory

In the following lectures we will look at numerical methods for solving ordinary differential equations. We will initially confine ourselves to initial value problems, i.e.

with in and .

From this data, we would like to find the value of at one or more values of .

Later we will also look at extending these methods to solve systems of linear ODE's. To underpin the theoretical aspects of these methods,we will need the following definitions and results:

Definition (Lipschitz Condition)

Definition (convex set)

Theorem 18

Theorem 19 (unique solution)

Definition (well-posed problem)

Theorem 20