Homework 4
Solutions
1. (#3.1.1 in Strauss) Solve ut = kuxx ; u(x, 0) = ex ; u(0, t) = 0 on the half-line 0 < x < .
Solution:
Using the solution formula for the Dirichlet problem associated with the heat equation on the
half-line, we get
Z h
i
1
2
2
u(x,
Homework 2
Solutions
1. (#1.3.5 in Strauss) Derive the equation of one-dimensional diffusion in a medium that is
moving along the x axis to the right at constant speed V .
Solution:
Let u(x, t) be the concentration of a dye diffusing in the narrow pipe wi
Homework 3
Solutions
1. Solve the initial value problem
utt = 4uxx + xt,
u|t=0 = x2
ut |t=0 = x.
t > 0, x R
Solution:
Make the equation homogeneous: a particular solution to the equation is xt3 /6 because
3
3
xt
xt
4
= xt.
6 tt
6 xx
Hence
v(t, x) = u(t,