Math 257/316, Midterm 2, Section 103
17 November 2006
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
Maximum score 100.
1. Let f (x) be a function that has a period of 2π having the values:
½
0 when − π ≤ x < 0
f (x) =
π − x when 0 < x ≤ π
(a) Determine the Fourier series expansion for f (x).
(b) To what value does this series converge when x = 0? Use this value and the series in (a) to show
that:
π2
1
1
1
= 2 + 2 + 2 + ...
8
1
3
5
Hint: It may be useful to know that for n 6= 0:
Zπ
(π − x) cos(nx)dx =
1 − cos(nπ)
n2
0
Zπ
(π − x) sin(nx)dx =
π
n
0
[20 marks]
2. Solve the following inhomogeneous initial boundary value problem for the heat equation:
ut = uxx + x, 0 < x < 1, t > 0
u(0, t) = 0 and u(1, t) = 0
1
u(x, 0) = sin(πx) − (x3 − x)
6
[40 marks]
3. Determine the solution to the following boundary value problem:
1
1
urr + ur + 2 uθθ
r
r
= 0, 0 < r < a, 0 < θ < π
r→0
u(r, 0) = 0 = u(r, π), u(r, θ) < ∞
u(a, θ) = sin(2θ)
[40 marks]
1