#### CS3414 Afterclass Notes --- 6 June, 2002

**Ordinary Differential Equations ** (parts of Chapters 8 and 12)
- Introduction

- ODEs: equations involving a unknown functions x(t) and their
derivatives.

- Some terminology and basic concepts: IVP, BVP, order of an equation.

- A single IVP:

x'(t) = f(t,x)

x(t_{0}) = x_{0},

where
f() and x_{0} are given.

- A numerical solution to an IVP consists of a series of points
(x
_{0},t_{0}),
(x_{1},t_{1}), ...,
(x_{N},t_{N}), where t_{N} is the
`final' time for which the solution is desired, and
x_{k} is my approximation to x(t_{k}).

- Some important themes in numerical ODEs:

- Step size selection: in general, a smaller step size means
better accuracy; a larger step size means less work, i.e., we get to
t
_{final} with fewer function evaluations.

- Adaptive step size selection based on local error estimates,
which are themselves (typically) based on comparing a good
solution and a `better' solution.

- Local vs. global error:

- Global error: x
_{k} - x(t_{k})

- Local error: the error made at step k, assuming the answer
was correct at step k-1,

i.e., x_{k} - z(t_{k}),
where z(t) exactly solves the IVP:

z'(t) = f(t,x)

z(t_{k-1}) = x_{k-1}.