1. (#3.1.1 in Strauss) Solve ut = kuxx ; u(x, 0) = ex ; u(0, t) = 0 on the half-line 0 < x < .
Using the solution formula for the Dirichlet problem associated with the heat equation on the
half-line, we get
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 .
Let u(x, t) be the concentration of a dye diffusing in the narrow pipe wi
1. Solve the initial value problem
utt = 4uxx + xt,
u|t=0 = x2
ut |t=0 = x.
t > 0, x R
Make the equation homogeneous: a particular solution to the equation is xt3 /6 because
v(t, x) = u(t,