In Taylor's theorem let Y be the state vector , and use it to estimate to 2nd order, the value of given , and . Call the value we obtain in this way to indicate that it is the result of the predictor step.
Where for some in ...
As the differential equations only give us the value of the first derivative in terms
of time and value of the state variable, we need an estimate of the second derivative. Now
use Taylor's theorem again to estimate the value of using and
.
Note that this time we use in Taylor's theorem.
Where for some in ...
Which we can solve for ...
Now define .
Inserting this into the predictor...
Predictor
Where...
Which is...