Chapter 20. Solving Differential Equations

Differential equations, those that define how the value of one variable changes with respect to another, are used to model a wide range of physical processes. You will use differential equations in chemistry , dynamics, fluid dynamics, thermodynamics , and almost every other scientific or engineering endeavor. A differential equation that has one independent variable is called an ordinary differential equation or ODE. Examples of ODEs include the equations to model the motion of a spring or the boundary layer equations from fluid dynamics. A partial differential equation (PDE) has more than one independent variable. The Navier-Stokes equations are an example of a set of coupled partial differential equations used in fluid dynamic analysis to represent the conservation of mass, momentum, and energy.

This chapter will focus primarily on how to solve ordinary differential equations and will touch upon the more difficult to solve partial differential equations only briefly at the end of the chapter. We will discuss the difference between initial value and two-point boundary problems. We will write a class that represents a generic ODE and write two subclasses that represent the motion of a damped spring and a compressible boundary layer over a flat plate. We will develop a class named ODESolver that will define a number of methods used to solve ODEs and compare results generated by these methods with results from other sources.

The specific topics covered in this chapter are ”

• Ordinary differential equations

• The ODE class

• Initial value problems

• Runge-Kutta schemes

• Example problem: damped spring motion

• Embedded Runge-Kutta solvers

• Other ODE solution techniques

• Two-point boundary problems

• Shooting methods

• Example problem: compressible boundary layer

• Other two-point boundary solution techniques

• Partial differential equations