ACM 95/100c
April 16, 2004
Problem Set III
0. (2 pts) Write down your gradingsection number.
1. (40 pts)
(a) Solve the initial value problem for the heat equation with timedependent sources
∂u
∂t
=
k
∂
2
u
∂x
2
+
Q
(
x, t
)
u
(
x,
0) =
f
(
x
)
(1)
subject to the following boundary conditions:
(i) (5 pts)
u
(0
, t
) =
A
(
t
),
∂u
∂x
(
L, t
) = 0
(ii) (5 pts)
∂u
∂x
(0
, t
) =
A
(
t
),
∂u
∂x
(
L, t
) =
B
(
t
)
(iii) (5 pts)
∂u
∂x
(0
, t
) = 0,
∂u
∂x
(
L, t
) = 0
(iv) (5 pts)
u
(0
, t
) = 0,
u
(
L, t
) = 0.
(b) (10 pts) Specialize part (iii) to the case
Q
(
x, t
) =
Q
(
x
) (independent of
t
) such that
L
0
Q
(
x
)
dx
= 0. Show that, in this case, there are no timeindependent solutions. What happens to
the timedependent solution as
t
→ ∞
? Explain!
(c) (10 pts) Specialize part (iv) to the case
Q
(
x, t
) =
Q
(
x
) (independent of
t
). Show that, in
this case, the solution approaches a steadystate solution. Explain!
2. (30 pts) Let
u
(
x, y
) and
v
(
r, θ
) be two functions which, for
x
=
r
cos(
θ
) and
y
=
r
sin(
θ
), satisfy
u
(
x, y
) =
v
(
r, θ
)
.
(2)
(a) (10 pts) Show that, for
x
=
r
cos(
θ
) and
y
=
r
sin(
θ
) we have
∂u
∂x
(
x, y
) = cos(
θ
)
∂v
∂r
(
r, θ
)

sin(
θ
)
r
∂v
∂θ
(
r, θ
)
(3)
and
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 Spring '09
 NilesA.Pierce
 Boundary value problem, 5 pts, 2 pts, 10 pts, 15 pts

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