The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 9

Differential Equations, Boundary Conditions, and Solutions

9.1. BOUNDARY CONDITIONS AND UNIQUE SOLUTIONS TO DIFFERENTIAL EQUATIONS

We should briefly make an important aside. The main question to consider is this: How many boundary conditions are required to uniquely specify a single solution to a given specific differential equation?

Second-order differential equations, meaning equations that contain derivatives no higher than second derivatives, fit into three types and they are, with examples:

• Elliptic differential equations such as the Laplace equation:

  

• Hyperbolic differential equations, the wave equation:

  

• Parabolic differential equations, of which the heat equation is best known:

  

In each case the ellipsis dots signify other functions and terms not containing derivatives higher than second derivatives.

To classify differential equations in two variables, write them as

Then change variables to find the normal form, i.e., switch to a new coordinate system that makes the coefficients of the second-order derivatives into simple numbers, and three types emerge as follows: ...