Math 257/316 Assignment 7
Due Monday Mar. 9 in class
1. The concentration u(x, t) of a reactive chemical diffusing in one dimension satisfies

0 < x < 2, t > 0
 ut = uxx − u,
u(0, t) = −1, u(2, t) = 1
.

u(x, 0) = 0
where the ‘loss’ term represents a reaction which consumes the chemical. Find u(x, t)
and sketch the solution for 3 different values of t. Hint: first find the steady-state,.
You may find it helpful to know that (you can check this by integrating by parts) the
Fourier sine series coefficients for sinh(x − 1) on [0, 2] are:
0
n odd,
bn =
−4nπ
sinh(1)
n even.
4+n2 π 2
2. There is a gas leak at the end x = 1 of a corridor 0 ≤ x ≤ 1. The concentration of
gas satisfies
ut = uxx ,
0<x<1
ux (0, t) = 0,
ux (1, t) = 1
u(x, 0) = 0
(a) Find the solution for u(x, t) by first finding a particular solution of the form
v(x, t) = ax2 + bx + ct that satisfies the PDE and BCs; then write u(x, t) =
v(x, t) + w(x, t), and find and solve the homogeneous problem for w.
(b) An alarm in the middle of the corridor (x = 1/2) is triggered when the gas
concentration reaches 1. By considering the size of the different terms in your
solution, deduce that the alarm goes off at approximately t ≈ 25/24.
3. Solve the following non-homogeneous problem for the heat equation.
ut = uxx + sin(πx) t e−π
u(0, t) = 0,
u(1, t) = 5
u(x, 0) = 5x.
1
2t
0 < x < 1, t > 0