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624 CHAPTER 17 The Heat Equation

2
1.2

1.5 1

0.8
1
0.6

0.4
0.5
0.2

0 0
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
x x
FIGURE 17.7 The solution in Example 17.5 at
FIGURE 17.6 The solution in Example 17.5 at
t = 1.2 and k = 0.3,0.6,1.4, and 2.7.
t = 0.2 and k = 0.3,0.6,1.4, and 2.7.
4

4
3

3
2
2

1 1

0
0 0 0.5 1 1.5 2 2.5 3
x
0 0.5 1 1.5 2 2.5 3
x
FIGURE 17.9 u 2 (x,t) at times t = 0.4,0.9,
FIGURE 17.8 u 1 (x,t) at times t = 0.4,0.9,
1.5, and 3.6.
1.5, and 3.6.

∞ n+1
2
8(−1) − 4 −n t/4
u 1 (x,t) = sin(nx)e .
n 3
n=1
If the ends are insulated the solution is
n
2
2
1 n π (−1) n+1 + 6(−1) − 6 2
∞
3
u 2 (x,t) = π + cos(nx)e −n t/4 .
12 n 4
n=1
Figure 17.8 shows u 1 (x,t), and Figure 17.9 shows u 2 (x,t), at times t =0.4,0.9, 1.5 and 3.6.
3
Both solutions decrease as t increases. However, as t →∞, u 2 (x,t) → π /12, as suggested in
Figure 17.9.
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October 14, 2010 15:25 THM/NEIL Page-624 27410_17_ch17_p611-640