MAE105
First Midterm Exam
(open book; closed notes; no computers, no calculators, no cell phones)
Name:_________________________
Time: 3:50 to 4:50pm
Date: April 17, 2008
Problem 1:
Consider the following diffusion PDE:
∂
u
∂
t
−
∂
2
u
∂
x
2
=
0,
t
>0,
0<
x
<
π
,
(1)
with the following boundary conditions:
∂
u
∂
x
(0,
t
)
=
0,
u
(
π
,
t
)
=
0.
(2)
(a) (0.5 Point) Based on the method of "separation of variables", set
u
(
x
,
t
)
=
φ
(
x
)
G
(
t
), differentiate as necessary,
and substitute into the PDE (1).
(b) (0.5 Point) Collect terms such that a function of only
x
is set equal to a function of only
t
, and use this fact to
find two ODE’s, one for
φ
(
x
), and the other for
G
(
t
), in such a way that the solution for
φ
(
x
) would be periodic.
(c) (0.5 Point) Write down the general solution of the ODE for
φ
(
x
). Is your solution periodic?
(d) (0.5 Point) Write down the general solution of the ODE for
G
(
t
). Does your solution decay exponentially in
time?
(e) (1 Point) Use the boundary conditions (2) to obtain the values of
φ
′
(
x
) at
x
=
0 and
φ
(
x
) at
x
=
π
.
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 Spring '07
 NeimanNassat
 Calculus, Ode, Constant of integration, Boundary conditions

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