Homework #4
Due: 11:59pm on Saturday February 15. Submission is through Gradescope (link in
the Canvas page).
You are strongly encouraged to collaborate with each other, but each student must
submit their own work. You are also encouraged to talk about the homework in any of the
office hours.
Please submit either photocopies of paper solutions or electronically written/typed files.
Always write full solutions and remember to justify your solutions and arguments carefully.
Good luck, and don’t hesitate to reach out if you have any questions!
1. Solve the diffusion equation ut = kuxx on the whole line, with inital condition u(x, 0) =
ϕ(x) given by
®
1, |x| < l,
ϕ(x) =
0, |x| > l,
where l > 0 is a constant. Write the solution in terms of the special function Erf(x) (c.f.
Section 2.3 of the book), defined by
Z x
2
2
e−z dz.
Erf(x) = √
π 0
R∞
2
2. (a) Compute 0 e−x dx. (Hint: There is no analytic expression
ofäthe
ÄR ∞
ÄR ∞antiderivative.
ä
−x2
−y 2
Instead use the following trick: View the product 0 e dx
e
dy
as a
0
double-integral, and use polar coordinates to compute it.)
R∞
(b) Use part (a) to show that −∞ K(t, x)dx = 1 for all t > 0, where S is the heat
kernel.
(c) Let u be the solution to the heat equation
®
ut = kuxx ,
−∞ < x < ∞, t > 0,
u(0, x) = ϕ(x), −∞ < x < ∞.
R∞
R∞
where −∞ ϕ(x)dx < ∞. Use part (b) to show that the total heat −∞ u(t, x)dx is
constant with respect to t.
3. (a) Solve the Neumann problem for the wave equation on the half-line


utt = c2 uxx
0<x<∞



u(0, x) = ϕ(x) 0 < x < ∞

ut (0, x) = ψ(x) 0 < x < ∞



u (t, 0) = 0
0<t<∞
x
(b) Let c = 1, ψ(x) = 0 and let ϕ be the function illustrated below. Sketch the solution
u(t, x) of (a) at t = 0, t = 1, t = 2 and t = 3.
ϕ(x)
1
1
3
x
4. Consider the hyperbolic PDE uxx + uxy − 2uyy = 0.
(a) Find the general solution of this equation. (Hint: you can factorize the equation
and solve two first-order equations).
(b) Find the solution to the initial value problem


uxx + uxy − 2uyy = 0 −∞ < x < ∞, 0 < y < ∞
u(x, 0) = ϕ(x)
−∞ < x < ∞


uy (x, 0) = ψ(x)
−∞ < x < ∞
(c) Use the method of reflections to solve the Dirichlet problem on the half-line


uxx + uxy − 2uyy = 0 0 < x < ∞, 0 < y < ∞



u(x, 0) = ϕ(x)
0<x<∞

uy (x, 0) = ψ(x)
0<x<∞



u(0, y) = 0
0<y<∞
(Hint: the extended functions need to be slightly different in this situation. You
might need to “reverse-engineer” them in order to make the extended solution
satisfy the Dirichlet condition.
®
f (x)
x>0
nd
2 hint: For both ϕ and ψ, try extensions of the form fext (x) =
af (−bx) x < 0
and determine a and b!)
Remark: Problem 4 shows that we can extend d’Alembert’s formula and the method of
reflections to almost any hyperbolic equation!
5. Solve the inhomogeneous heat equation on the half-line with Dirichlet boundary conditions


ut − kuxx = f (t, x), 0 < x < ∞, 0 < t < ∞,
u(0, x) = ϕ(x),
0 < x < ∞,


u(t, 0) = 0,
0 < t < ∞.
(Hint: you can use the method of reflections to extend both ϕ and f .)