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17.2 The Heat Equation on [0, L] 617

y

z 1 z 2
z

y = –z/AL

y = tan(z)

FIGURE 17.1 Eigenvalues as the problem with
radiating end.

For each positive integer n, we now have a function
z n x 2 2

u n (x,t) = X n (x)T n (t) = sin e −z n kt/L .
L
that satisfies the heat equation and the boundary conditions. To satisfy the initial condition, write
the superposition
∞

z n x 2 2
u(x,t) = c n sin e −z n kt/L .
L
n=1
We must choose the c n ’s to satisfy
∞

z n x
u(x,0) = f (x) = c n sin .
L
n=1
This is not a Fourier sine series, but it is an eigenfunction expansion in the eigenfunctions of a
Sturm-Liouville problem. The coefficients are
L
f (ξ)sin(z n ξ/L)dξ
0
c n = .
L 2
sin (z n ξ/L)dξ
0
The difficulty in computing these numbers is one we frequently encounter in such problems. We
do not have an explicit formula for the z n ’s. However, we can compute approximate values of the
z n ’s for a given L. For the first four values, with L = 1, we obtain the approximate values
z 1 ≈ 2.0288, z 2 ≈ 49132, z 3 ≈ 7.9787, z 4 ≈ 11.0855.
Using these numbers, we can perform numerical integrations to approximate
c 1 ≈ 1.9207,c 2 ≈ 2.6593,c 3 ≈ 4.1457,c 4 ≈ 5.6329.

With just the first four terms, we have an approximation
u(x,t) ≈1.9027sin(2.0288x)e −4.1160kt + 2.6593sin(4.9132x)e −24.1395kt
+ 4.1457sin(7.9787x)e −63.6597kt + 5.6329sin(11.0855x)e −1,228.883kt .
Depending on k, these exponentials may be decaying so fast that these first four terms would be
a good enough approximation for some applications.

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October 14, 2010 15:25 THM/NEIL Page-617 27410_17_ch17_p611-640