MATH 152, FALL 2004:
MIDTERM
#I
Problem
$1
(15
)
Let u~(x,
t) and u2(x,
t) denote the solutions of the equation
with initial and boundary conditions respectively u1 (x, 0)
=
gl (x), u1 (0, t)
=
fl
(t),
ul(L,t)
=
hl(t) and u2(x,0)
=
gz(x), uz(0,t)
=
f2(t), uz(L,t)
=
112(t). Assume
that gl
I
gz,
fl
<
f2 and hl
5
h2. Prove that u1
5
u2 in the set R
=
[0, L] x [0, m).
Hint:
Let w(x,
t)
=
ul(x,
t)

u2(x,t) and prove that w(x, t)
5
0 in R.
Problem #2
(zd
f
fiS
)
This is an example of a heat problem with internal heat source for which the
inaxiinum principle does not hold. Consider
i
ut =uz, +2(t
+
1) +x(12)
(1)
u(0,
t)
=
0, u(1, t)
=
0
u(x,O)
=
x(1 x)
for
0
<
x
<
1 and t
>
0.
a)
Verify that u(x,
t)
=
(t
+
l)x(l

x) is a solution for (1).
$
b)
Find the maximum M and the minimum m of the initial and boundary data.
f
c)