CHEE 3334
Fall ’97
Michael Nikolaou
FINAL EXAM
Openbook, opennotes.
Total points 100.
Please, budget your time
!
Do NOT try to read the book during the
exam!
Write neatly!
Do NOT use your programmable calculator to find results you are asked to compute in a
certain wayafter all, the calculator may find the wrong answer or not all answers!
Show all work.
Return
this sheet with your exam
.
GOOD LUCK and HAPPY HOLIDAYS!
The concentration
u
(
x
,
t
) of a species diffused through a membrane is described by the partial differential
equation (PDE)
2
2
2
∂
∂
+
∂
∂
=
∂
∂
x
u
x
u
t
u
0
,
1
0
<
<
t
x
(1)
with initial condition
)
1
(
)
0
,
(
x
x
x
u

=
,
for all
x
in the interval (0,1)
(2)
and boundary conditions
0
)
,
0
(
=
t
u
,
for all
0
t
(3)
t
e
t
u


=
1
)
,
1
(
,
for all
0
t
(4)
(10 pts.)
1.
Is the above PDE (equation (1)) linear or nonlinear?
Justify your answer.
An equation of the form
L(u) = f, (where
u is the unknown function and f is a forcing function that does
not depend
on u) is linear if the mapping
L is linear.
A mapping
is linear
if L(u
1
+ u
2
) = L(u
1
) + L(u
2
)
for any u
1
, u
2
and L(cu) = c L(u)
for any u and any number c.
The above
equations can be written as
2
2
2
0
u
u
u
t
x
x


=
.
It is
straightforward that L(u
1
+ u
2
)
≠
L(u
1
) + L(u
2
),
i.e.
2
2
2
2
2
2
1
2
1
2
1
2
1
1
1
2
2
2
2
2
2
(
)
(
)
(
)
( )
( )
( )
(
)
(
)
(
)
u
u
u
u
u
u
u
u
u
u
u
u
t
x
t
x
t
x
x
x
x
+
+
+


³


+


±
ς
³
÷♠
,
and L(cu)
≠
c L(u),
i.e.
2
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 Fall '06
 nikolau
 Numerical Analysis, truncation error, UIL, Michael Nikolaou, finite divided differences

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