Math 257/316 Section 201
Total = 50 points
Midterm 2
Mar 8
[There are 2 questions.]
Problem 1.
a) [10 points] For the function f (x) = x2 on [0, 1], sketch (roughly) its odd, and even
2-periodic extensions, and find its Fourier sine series, and its Fourier cosine series,
using (if you need it) for k = 1, 2, 3, . . .,
Z
0
1
2(−1)k
x cos(kπx)dx = 2 2 ,
k π
2
1
Z
x2 sin(kπx)dx =
0
2((−1)k − 1) (−1)k
−
.
k3 π3
kπ
b) [15 points] Solve the following problem for the heat equation with non-zero derivative
(flux) BCs:

ut = uxx
0 < x < 1, t > 0

ux (0, t) = 0, ux (1, t) = 2
t>0

u(x, 0) = 0
0≤x≤1
1
(Blank page)
2
Problem 2.
a) [19 points] Solve the following equation describing diffusion with growth, subject to
non-zero BCs:

ut = α2 uxx + u
0 < x < L, t > 0

u(0, t) = 0, u(L, t) = 1
t>0

u(x, 0) = 0
0≤x≤L
but leave any Fourier coefficients in terms of integrals (i.e. do not take time to
evaluate these integrals). Hint: first find the steady-state, then use separation of
variables to find the remainder.
b) [6 points] Determine the long-time (t → ∞) behaviour of the solution (considering
all possible positive values of the diffusion rate α2 ).
3
(Blank page)
4