Page 107 -

3. The Heat Equation
94
1.2
0.8
−−−: N=10
0.6
___: N=100
0.4
0.2 1 −.−: N=3
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
FIGURE 3.3. The first 3, 10, and 100 terms of the sine-series approximation of
f(x)=1.
When allowing infinite series in the initial data, the solution given by
(3.22) also becomes an infinite series. For the present example, we get the
following formal solution:
∞
4 1 −((2k−1)π) t
2
u(x, t)= e sin ((2k − 1)πx). (3.24)
π 2k − 1
k=1
Recall here that this solution is referred to as being formal since we have
not proved that the series and its derivatives converge and satisfy all the
requirements of the heat equation (3.20).
We have plotted the formal solution of this problem as a function of x
at t =0, 0.01, 0.1 in Fig. 3.4. Note that the observations concerning the
qualitative behavior of the solution stated in Example 3.1 also apply to the
present solution.

The key observation of the example above is that finite linear combi-
nations of eigenfunctions are not sufficient to cover all interesting initial
functions f(x). Thus we are led to allow infinite linear combinations of the
form
∞

f(x)= c k sin (kπx). (3.25)
k=1
By letting N tend to infinity in (3.22), we obtain the corresponding formal
solution of the problem (3.20),
∞
2
−(kπ) t
u(x, t)= c k e sin (kπx). (3.26)
k=1