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moienMath 21b: Linear Algebra Spring 2016
Homework 31: Partial differential equations
This is the last homework. It is due on the last day of classes Wednesday April 27, respectively
on Tuesday, April 26, 2016.
1 Solve the heat equation ft = 5fxx on [0, π] with the initial condi-
tion f (x, 0) = max(cos(x), 0).
2 Solve the partial differential equation ut = uxxxx + uxx with initial
condition u(0, x) = x3.
3 A piano string is fixed at the ends x = 0 and x = π and is initially
undisturbed u(x, 0) = 0. The piano hammer induces an initial
velocity ut(x, 0) = g(x) onto the string, where g(x) = sin(3x)
on the interval [0, π/2] and g(x) = 0 on [π/2, π]. How does
the string amplitude u(x, t) move, if it follows the wave equation
utt = uxx?
4 A laundry line is excited by the wind. It satisfies the differential
equation utt = uxx + cos(t) + cos(3t). Assume that the amplitude
u satisfies initial condition u(x, 0) = 4 sin(5x) + 10 sin(6x) and
that its initial velocity is zero. Find the function u(x, t) which
satisfies the differential equation.
Hint. First find the general homogeneous solution uhomogeneous of
utt = uxx for an odd u then a particular solution uparticular (t)
which only depends on t. Finally fix the Fourier coefficients.
5 In this course we have looked at four different types of differen-
tial equations. Systems of linear differential equations x0 = Ax,
nonlinear equations x0 = f (x, y), y 0 = g(x, y), inhomogeneous
equations p(D)f = g as well as partial differential equations like
ut = D2u and wave equation utt = D2u. Give a concrete example
of each type, in each case an example which has never appeared
in a homework or handout.
Partial differential equations
In all these PDE problems, we look at functions f on the interval
[0, π] and write them as a sin-series which means that we only
need to compute the bn using the formula
2Zπ
0
f (x) sin(nx) dx .
π
This is justified as we can think of f continued as f (−x) =
−f (x) on [−π, 0] The temperature distribution f (x, t) in a
metal bar [0, π] satisfies the heat equation
ft(x, t) = µfxx(x, t) = D2f (x, t)
Here µ is a positive constant which depends on the material.
The height of a string f (x, t) at time t and position x on [0, π]
satisfies the wave equation
ftt(t, x) = c2fxx(t, x) = c2D2(f, t)
Here c is a positive constant which tell how fast the waves
move. All these problems are solved by diagonalizing D2 using
a Fourier basis. In the heat equation, write the initial condition
as a Fourier series and write down the solution. In the wave
equation, write both the initial condition f (0, x) as well as the
initial velocity ft(0, x) as a Fourier series and write down the
solutions.