Math 257 – Assignment 4. Due: Wednesday, February 2
1. Consider the heat conduction problem for a rod of length 1 with conductivity α2 = 100:
∂u
∂ 2u
= 100 2 ,
∂t
∂x
with homogeneous boundary conditions
0 < x < 1, t > 0,
u(0, t) = u(1, t) = 0,
t > 0.
Find the solution for each of the following initial conditions:
a) u(x, 0) = 0.
b) u(x, 0) = sin 5πx − 10 sin 8πx.
c) u(x, 0) = 1.
2. Let the function f (x) be given by
(
f (x) =
1,
0 ≤ x < 1,
−1,
1 ≤ x ≤ 2.
Solve the following heat condition problem with the given initial-boundary condition.
∂ 2u
∂u
=
,
0 < x < 2, t > 0,
∂t
∂x2
u(0, t) = u(2, t) = 0,
t > 0,
u(x, 0) = f (x),
0≤x≤2
3. Evaluate the integrals
Z L
mπx
nπx
cos
dx,
cos
L
L
−L
Z
L
nπx
mπx
sin
sin
dx,
L
L
−L
Z
L
cos
−L
nπx
mπx
sin
dx,
L
L
for integers n and m, in each of the cases n 6= m and n = m.
4. Suppose that the function f (x) is 2L-periodic, namely f (x+2L) = f (x) for any real number
x, where L is a fixed positive constant: e.g. sin x is 2π-periodic. Show that
Z L
Z a+2L
f (x) dx =
f (x) dx
−L
a
for any value of a.
5. Spreadsheet (you do not need to hand in anything for this question) Familiarize yourself
with the Excel Tutorials on the webpage
http://www.math.ubc.ca/˜yhkim/yhkim-home/teaching/Math257/notes.php
and read through ‘Using Excel to evaluate Fourier series’. (Those links are found at the
bottom of the above webpage.)