Forward Differencing Equations for Second-Order Iteration

Recall how we used forward finite differencing to simulate the behavior of a function that changed proportional to its current size in Lecture 19.  Consider now a function that changes "how fast it changes" proportional to its current change and its current value--in other words its current acceleration is proportional to its velocity and to its size:

Let's work out how to find a finite differencing function:

This is the current change or approximation to velocity

This is the current change in change or approximation to acceleration

Our model is that the acceleration is proportional to  size of the current function and its velocity, let these proportions be: -α and -β

Solve this for the "latest" values

Replace to find the form of the solution

Created by Wolfram Mathematica 6.0  (16 November 2007)