Page 99 - 6099

We determine the coefficients in the series with the initial condition.
∞ r
X 2 nπx
v(x, 0) = a n sin = f(x) − µ(x)
h h
n=1
r
Z h
2 nπx
a n = sin (f(x) − µ(x)) dx
h 0 h
The temperature of the rod is

∞ r
X 2 nπx 2
u(x, t) = µ(x) + a n sin e −κ(nπ/h) t
h h
n=1

9.4 Inhomogeneous Equations with Homogeneous
Boundary Conditions

Now consider the heat equation with a time dependent source, s(x, t).

u t = κu xx + s(x, t), u(0, t) = u(h, t) = 0, u(x, 0) = f(x). (9.10)

In general we cannot transform the problem to one with a homogeneous differential equa-
tion. Thus we cannot represent the solution in a series of the eigen-solutions of the partial
differential equation. Instead, we will do the next best thing and expand the solution in a
series of eigenfunctions in X n (x) where the coefficients depend on time.

∞
X
u(x, t) = u n (t)X n (x)
n=1

We will find these eigenfunctions with the separation of variables, u(x, t) = X(x)T(t)
applied to the homogeneous equation, u t = κu xx , which yields,

r
2 nπx
+
X n (x) = sin , n ∈ Z .
h h
We expand the heat source in the eigenfunctions.

∞ r
X 2 nπx
s(x, t) = s n (t) sin
h h
n=1
r Z h
2 nπx
s n (t) = sin s(x, t) dx,
h 0 h

We substitute the series solution into Equation 9.10.
∞ r ∞ r ∞ r
X 0 2 nπx X nπ 2 2 nπx X 2 nπx
u (t) sin =−κ u n (t) sin + s n (t) sin
n
n=1 h h n=1 h h h n=1 h h

92