Exploring ODEs with Matlab

Abstract

Many physical processes, both natural and engineered, are best described by ordinary differential equations (ODEs), which relate time derivatives of particular quantities to each other.

In this chapter, we continue our exploration of Matlab in the context of numerical methods. From two well-known physical laws—Newton’s second law of motion (F=ma) and Newton’s law of universal gravitation ( $F = G\frac{Mm}{r^{2}}$)—we develop an ODE to describe the orbits of satellites around planets. We then study and apply various numerical methods to solve numerically for an orbit given a satellite’s initial position and velocity. Our explorations will yield one universal truth of numerical methods: no one method works best on all problems. In order to determine which is the best for this application, we will rely on some common sense reasoning to make predictions about what we expect to see for certain initial conditions.