Page 22 - Essentials of applied mathematics for scientists and engineers

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12 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
These are called orthogonal functions. Examples are the sine and cosine functions. This idea is discussed
fully in Chapter 3, particularly in connection with Sturm–Liouville equations.

Theorem 2.3. Fourier Series: A piecewise continuous function f (x) defined on (a, b) can be
represented by a series of orthogonal functions n (x) on that interval as

∞

f (x) = A n n (x)
n=0
where
b f (x) n (x)dx
x=a
b n (x) n (x)dx
A n =
x=a
These properties will be used in the following examples to introduce the idea of solution of partial
differential equations using the concept of separation of variables.

2.1 HEAT CONDUCTION
We will first examine how Theorems 1, 2, and 3 are systematically used to obtain solutions
to problems in heat conduction in the forms of infinite series. We set out the methodology
in detail, step-by-step, with comments on lessons learned in each case. We will see that the
mathematics often serves as a guide, telling us when we make a bad assumption about solution
forms.

Example 2.1. A Transient Heat Conduction Problem

Consider a flat plate occupying the space between x = 0and x = L. The plate stretches out
in the y and z directions far enough that variations in temperature in those directions may be
neglected. Initially the plate is at a uniform temperature u 0 .Attime t = 0 the wall at x = 0
+
is raised to u 1 while the wall at x = L is insulated. The boundary value problem is then

ρcu t = ku xx 0 < x < L t > 0 (2.1)
u(t, 0) = u 1
u x (t, L) = 0 (2.2)

u(0, x) = u 0

2.1.1 Scales and Dimensionless Variables
When it is possible it is always a good idea to write both the independent and dependent
variables in such a way that they range from zero to unity. In the next few problems we shall
show how this can often be done.