Topic 4: Ordinary differential equations

Introduction

Since the time of Newton and Leibnitz, ordinary differential equations [ODEs] have been a centrepiece of quantitative science: many scientific laws and phenomena are represented as connections between derivatives in a differential equation.

The central importance of differential equations is associated with an enormous body of knowledge about their solutions.

Computers provide convenient solutions of complex, non-linear, possibly coupled ODEs with a modest level of effort (compared to the effort required for a deep mathematical analysis). Provided one is sceptical and careful about a computer solution, the answers from a numerical solution may be as useful as a precise closed form solution.

Various methods: packages

There are many ways to code a computer to solve ODEs. There is a vast literature dealing with the accuracy and robustness of various algorithms.

While you could probably code up a simple trapezoidal rule in a few minutes, it would generally be safer and probably more convenient to use a package which has been optimised and rendered robust and accurate by its creators.

You can get ODE packages from text books (such as Numerical recipes) or off the web.

4th order Runge-Kutta methods

Many experienced scientific computational experts would use a package based on 4th order Runge-Kutta methods to solve ODEs. You can get convenient versions of this software from several sources.

Tutorial goals

Write a program to solve the ODE describing a driven, anharmonic oscillator:

$$m \frac{d^2\,x}{d\,t^2} = -k \vert x\vert^{p-1}\,\frac{x}{\vert x\vert} + F_{ext}(x,t).$$

If you choose to use a Runge-Kutta integrator, you will want to convert this equation to a pair of coupled 1st order ODEs (the ``Dynamical Form'').