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across that end. We will assume the simplest condition: that the temperature at each
0
end of the rod is held at 0 for all t > 0.
2. Initial Conditions.
The initial temperature, that is u(x, 0), must be specified as a function f(x) on the
interval 0 ≤ x ≤ l, where l is the length of the rod. The function f(x) needs to be at
least piecewise continuous on 0 ≤ x ≤ l.

9.1 Homogeneous Dirichlet conditions

The problem we are going to solve can be summarized as follows. Find a function u(x, t)
such that:
2
u t = c u xx (9.1)
u(0, t) = u(l, t) = 0, t > 0 (9.2)

u(x, 0) = f(x), x ∈ [0; l] (9.3)
The method of solution we will use is called the Method of Separation of Variables.
It is first assumed that there exist solutions of the form

u(x, t) = X(x)T(t),

that is, solutions which are products of a function of x and a function of t. There will
turn out to be an infinity of such product solutions

u n (x, t) = X n (x)T n (t),

each of which satisfies the two boundary conditions. Using the fact that the pde is linear,
any linear combination of these,
X
a n X n (x)T n (t)
n
will also be a solution. The constants a n will then be chosen to make the infinite sum
satisfy the initial condition.
If u(x, t) = X(x)T(t), then it is easy to find its partial derivatives. We need u t and
u xx to substitute into the PDE:

00
0
u t = X(x)T (t), u xx = X (x)T(t).
Substituting these derivatives into the PDE (9.1),
0
00
2
X(x)T (t) = c X (x)T(t),
must be true for 00 ≤ x ≤ l and t > 0. The next step is to get everything involving x on
one side and everything involving t on the other. This can be done by dividing both sides
2
of the equation by c X(x)T(t).
00
00
0
0
2
X(x)T (t) c X (x)T(t) T (t) X (x)
= ⇒ =
2
2
2
c X(x)T(t) c X(x)T(t) c T(t) X(x)
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