Page 21 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

11

CHAPTER 2

The Fourier Method: Separation

of Variables

In this chapter we will work through a few example problems in order to introduce the general
idea of separation of variables and the concept of orthogonal functions before moving on to a
more complete discussion of orthogonal function theory. We will also introduce the concepts
of nondimensionalization and normalization.
The goal here is to use the three theorems stated below to walk the student through the
solution of several types of problems using the concept of separation of variables and learn some
early lessons on how to apply the method without getting too much into the details that will
be covered later, especially in Chapter 3.
We state here without proof three fundamental theorems that will be useful in finding
series solutions to partial differential equations.

Theorem 2.1. Linear Superposition: If a group of functions u n ,n = m through n = M are all
solutions to some linear differential equation then

M

c n u n
n=m

is also a solution.

Theorem 2.2. Orthogonal Functions: Certain sets of functions n defined on the interval (a, b)
possess the property that

b

n m dx = constant, n = m
a
b

n m dx = 0, n = m
a