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17.5 Heat Conduction in an Infinite Cylinder 637

Case 3: λ> 0
2
2
Write λ = ω with ω> 0. Now T + kω T = 0, with general solution

2
T (t) = ce −ω kt .
This is a bounded function. The equation for F becomes
1
2

F (r) + F (r) + ω F(r) = 0.
r
Write this as
2
2
r F (r) +rF (r) + ω F(r) = 0.

This is Bessel’s equation of order zero. Bounded solutions on [0, R] are constant multiples of
J 0 (ωr).
Thus far we have
2
U ω (r,t) = a ω J 0 (ωr)e −ω kt .
The condition U(R,0) = 0 requires that
J 0 (ωR) = 0.
Let j 1 < j 2 < ··· be the positive zeros of J 0 (x) in ascending order. For J 0 (ωR) = 0, there must
be some positive integer n such that ωR = j n . Denote ω n = j n /R. This gives us the eigenvalues
of this problem:
j 2
2 n
λ n = ω = .
n 2
R
The eigenfunctions are constant multiples of J 0 ( j n r/R).
Now for n = 1,2,···, we have functions

j n r 2 2
e − j n kt/R
U n (r,t) = a n J 0
R
satisfying the heat equation and the boundary condition. To satisfy the initial condition, employ
a superposition
∞
2
j n r − j n kt/R 2
U(r,t) = a n J 0 e .
R
n=1
Now we must choose the coefficients so that
∞
j n r
U(r,0) = f (r) = a n J 0 .
R
n=1
Let ξ =r/R to write
∞

f (rξ) = a n J 0 ( j n ξ)
n=1
for 0 ≤ ξ ≤ 1. In this framework, previous results on eigenfunction expansions apply, and we can
write
1
2 rf (rR)J 0 ( j n ξ)dξ
a n = 0 .
2
J ( j n )
1
With these coefficients, we have the solution for U(r,t).

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October 14, 2010 15:25 THM/NEIL Page-637 27410_17_ch17_p611-640