10—Partial Differential Equations 245functions ofxthat occur whenα > 0 , you will need the circular functions ofx, sines and cosines,
implying thatα < 0. That is also compatible with your expectation that the temperature should
approach zero eventually, and that needs a negative exponential in time, Eq. (10.11).
g(x) =Asinkx+Bcoskx, with k^2 =−α/D and f(t) =e−Dk
(^2) t
g(0) = 0impliesB= 0. g(L) = 0impliessinkL= 0.
The sine vanishes for the valuesnπwherenis any integer, positive, negative, or zero. This implies
kL=nπ, ork=nπ/L. The corresponding values ofαareαn=−Dn^2 π^2 /L^2 , and the separated
solution is
sin
(
nπx/L
)
e−n
(^2) π (^2) Dt/L 2
(10.13)
Ifn= 0this whole thing vanishes, so it’s not much of a solution. (Not so fast there! See problem10.2.)
Notice that the sine is an odd function so whenn < 0 this expression just reproduces the positiven
solution except for an overall factor of(−1), and that factor was arbitrary anyway. The negativen
solutions are redundant, so ignore them.
The general solution is a sum of separated solutions, see problem10.3.
T(x,t) =
∑∞
1ansin
nπx
L
e−n
(^2) π (^2) Dt/L 2
(10.14)
The problem now is to determine the coefficientsan.This is why Fourier series were invented. (Yes,
literally, the problem of heat conduction is where Fourier series started.) At timet= 0you know the
temperature distribution isT=T 0 , a constant on 0 < x < L. This general sum must equalT 0 at
timet= 0.
T(x,0) =
∑∞
1ansin
nπx
L
(0< x < L)
Multiply bysin
(
mπx/L
)
and integrate over the domain to isolate the single term,n=m.
∫L
0dxT 0 sin
mπx
L
=am
∫L
0dxsin^2
mπx
L
T 0 [1−cosmπ]
L
mπ
=am
L
2
This expression foramvanishes for evenm, and when you assemble the whole series for the temperature
you have
T(x,t) =
4
π
T 0
∑
modd1
m
sinmπx
L
e−m
(^2) π (^2) Dt/L 2
(10.15)
For small time, this converges, but very slowly. For large time, the convergence is very fast, often
needing only one or two terms. As the time approaches infinity, the interior temperature approaches
the surface temperature of zero. The graph shows the temperature profile at a sequence of times.