Differential Equations

Abstract

Many models derived from real-life physical situations result in the need to solve a differential equation. To get an understanding of the basics of this enormous topic we begin with the simplest situation: a single first-order ordinary differential equation with a known initial condition. We begin with Euler’s method which provides a good introduction to most of the more advanced methods. Euler’s method provides a basis for methods based on either a Taylor series or a numerical integration based theory which allows the development of higher order methods. The first higher order methods we discuss here are the Runge-Kutta methods. Runge-Kutta methods are fixed step methods that use intermediate points to gain higher-order behavior. Next we turn to multistep methods which use data from multiple prior data points for explicit methods, or include the current target in implicit methods. The two can be used to advantage as a predictor-corrector pair. Treating the independent and dependent variables as vector quantities allows systems of differential equations to be approached using these same methods. Higher order differential equations can also be recast as systems of first-order equations. Shooting methods provide a good approach to (two-point) boundary value problems. The second initial condition (typically the slope) is an unknown and we solve for that unknown to ensure the final point is on target. The methods of Chap. 5 can be combined with our initial value problem methods to solve the resulting “equation”.