MATH 257/316: SAMPLE MIDTERM 2
Closed Book and Notes. 55 minutes. Total 50 points
PROBLEM 1: Consider the heat equation for u(x, t) with a source term:
ut = uxx + sin(x) ,
u(0, t) = 0,
u(π, t) = 1 ,
u(x, 0) = sin(3x) ,
for
for
for
0 ≤ x ≤ π,
t > 0,
t > 0,
0 ≤ x ≤ π.
i) Find the steady-state solution us (x) to this problem.
ii) Expand this steady-state us (x) in a Fourier sine series us (x) =
show that
b1 = 1 +
2
,
π
bn = −
2
(−1)n ,
nπ
for
P∞
n=1 bn
sin(nx) to
n ≥ 2.
iii) By evaluating the Fourier sine series above at x = π/2 obtain an explicit formula for
P∞
m
m=1 (−1) /(2m + 1).
iv) Find an infinite series representation for the solution u(x, t) to the PDE.
PROBLEM 2: Solve the following wave equation for u(x, t) modeling forced vibrations
of a string where the string is held in a deflected state at the left wall:
utt = 9uxx + e−t ,
u(0, t) = 1 ,
u(x, 0) = −x/π ,
u(π, t) = 0 ,
ut (x, 0) = 0 ,
0 ≤ x ≤ π,
for
t > 0,
t > 0,
for
0 ≤ x ≤ π.
(Hint: Write u(x, t) = w(x) + v(x, t) where the w(x), with w′′ = 0, is chosen to ensure
homogeneous boundary conditions for v at x = 0, π.)
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