A Mathematica notebook with demonstrations of the example in section 2.3.7, the heat equation with 0 temperature boundary conditions.

A Mathematica notebook with a demonstration of problem 2.3.2g, and a possible project.

Problem 2.3.3(d)

First of all, here are
still pictures (about 2K) of the approximation to the initial condition
using a finite Fourier series with
50 terms and
100 terms.

Now, here are some animations (about 200K) of this problem, with time advancing
slowly and
quickly.

**Problem 2.4.1(a)**

Here are animations (about 200K)
with time advancing
slowly and
quickly.

**Problem 2.4.1(c)**
Here are animations (about 200K)
with time advancing
slowly and
quickly.

**The 2D Heat Equation**

Here is a DPGraph of the solution to the
heat equation on the square
with fixed temperature u=0 on the
boundary, and initial condition u(x,y,0) = 1.

**Unusual Coordinate Systems**

There are 5 coordinate systems in the plane in
which the Laplacian is separable. Aside from rectangular
and polar coordinates, there are three more shown
here.

**Problem 2.5.1(g)**

Laplace's equation on the rectangle
0 < x < L, 0 < y < W, with the sides and top insulating boundaries,
and u(x, 0) = 0 if x > L/2 and u(x, 0) = 1 if x < L/2:

Here are contour maps for L/W =1,
L/W =2,and
L/W =3.
Here are contour maps showing just the contour that hits the corner and two nearby contours, for
L/W =1,and
L/W =3.

Contour plot of problem 4 on MT 1.

**Laplace's Equation in the disk**

The contour map of the
solution to Laplace's equation inside the disk of radius 1 with boundary conditions

u(1, θ) = -1 if -π < θ < 0

**Fourier Series**

Here are some Mathematica notebooks, showing the Gibbs Phenomenon, a
FSS that diverges, and a FSS that appears to converge
even though f(x) is not piecewise smooth.

Here are simple Mathematica notebooks for computing the
Fourier Series,
Fourier Sine Series, and
Fourier Cosine Series.

**d'Alembert Solutions to the 1-D Wave Equation (using DPGraph)**

Fig. 1
A standing wave is the sum of two traveling waves
with the same amplitude.

Fig. 2
Two traveling waves with different amplitudes.

Fig. 3
Initial position is a gaussian, initial velocity is zero.

Fig. 4
Initial position is zero, initial velocity is the derivative of a gaussian.

Fig. 5
Collision of two pulses.

Fig. 6
Collision of a pulse and an "antipulse."

Guitar String
d'Alembert solution for a plucked guitar string.

Bowed String
d'Alembert solution for a bowed violin string.

**d'Alembert Solutions to the 1-D Wave Equation (using Mathematica)**

The initial conditions,
u(x,0) = f(x) and u_{t}(x,0) = g(x), can be changed in the notebook and re-run.
A few ICs are saved:

f(x) = exp(-x^2), g(x) = 0 (Collision of two pulses.)

f(x) = 0, g(x) = exp(-x^2) (Collision of two steps.)

f(x) = arctan(x), g(x) = -c/(1+x^2) (A traveling step.)

guitarString.nb. (d'Alembert solution for a plucked guitar string.)

**Normal Modes of a Square Drum**

Mode 1,1

Mode 2,1
Modes 1,2 and 2,1 in phase
Modes 1,2 and 2,1 combined in a
rotating wave

Mode 2,2

linear combination of modes

**Normal Modes of a Rectangular Drum**

Modes 2,1 and 1,2 on [0, 1.1] x [0, 1].
The frequencies of the normal modes are slightly different.

Modes 1,3 and 3,2 on [0, 4] x [0, 3].
The frequencies of the normal modes differ by about 6%.

Modes 2,2 and 3,1 on [0, 4] x [0, 3].
The frequencies of the normal modes differ by about 3%.

**Page on **Vibrating Drumheads.

** Normal Modes of a Fractal Drumhead**

Nandor Sieben's
page.

Jim Swift's
page.

See also the poster display in the halls of the math building.

**Mathematica notebooks for Lab day**

Laplace Equation in a wedge

Schroedinger's Equation

Instructor Information
Jim Swift's home page
Dept of Mathematics and Statistics
NAU Home Page

e-mail: Jim.Swift@nau.edu