Finite Difference Approximation

Ponder...

A derivative evaluated at a grid point can be approximated by nearby samples,

so discretising an ODE turns it into a discrete equation for the samples .

Example 7.1

This may arise when discretising a 2nd order ODE:

which we can correspond to

From derivatives to shifts

Sampling at equally spaced points with spacing replaces each derivative by a combination of neighbouring sample values, e.g.

A 2nd order ODE therefore becomes a discrete equation relating , and , such as

with constants , , . Such equations are solved with the same complementary-function and particular-integral methods used for ODEs.