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MATH 412: Sample Questions for Midterm 1
1. Say whether or not the following deﬁnes a vector space (and explain): the set of all
solutions
u
(
x, t
) to the semiinﬁnite diﬀusion equation
u
t
=
ku
xx
for
x
∈
(0
,
+
∞
), with
u
(
x,
0) =
f
(
x
) and
(a)
u
(0
, t
) = 0
(b)
u
x
(0
, t
) = 0
2. Write the Conservation Law equation for the density function
u
(
x, t
) in
R
n
. Specify
what ﬂux function
~
F
leads to the heat equation, and substitute it to obtain the heat
equation in
R
n
.
3. Consider the 1D heat equation between 0 and
L
, with Neumann boundary conditions
speciﬁed:
u
t
=
ku
xx
,
0
< x < L, t >
0
u
x
(0
, t
) =
Q
1
,
t >
0
u
x
(
L, t
) =
Q
2
,
t >
0
u
(
x,
0) =
φ
(
x
)
,
0
< x < L,
where
Q
1
and
Q
2
are given constants. Find the steady state solution(s), and discuss
their dependence on
Q
1
,
Q
2
, and
φ
(
x
).
4. Solve the Cauchy problem for
u
(
x, t
):
±
u
t
+
cu
x
=

λu
3
,
x
∈
R
, t >
0
u
(
x,
0) = exp(

x
3
)
,
with
λ >
0 and
c >
0, and assuming
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