# Applied Math Research Seminars

## 2006-03-03 A Water Drop Computation

Code: drop1.m.
Animation: drop_movie.gif (warning: 4.8MiB gif). Also, my apologies for the colors, I should read up on caxis for how to add some meaningful color information that is not autoscaled at each time step.
Script to make gif: make_movie.sh

## 2006-02-10 Cavity Flow (Part II)

photos of board: on Ben's seminar page.

codes: iter_driver.m, iter_f_lam.m, and iter_f_psi.m.

In the following pictures, the initial conditions are the same (k=2,l=2) except for the amplitude with epsilon=1/16 and lambda0=-19.7392

lambda: -19.773
initial amplitude=1
N=16

lambda: -33.057
initial amplitude=20
N=16

lambda: -73.01
initial amplitude=40
N=20

lambda: -207.02
initial amplitude=75
N=36

lambda: -209.53
initial amplitude=75.5
N=36

lambda: -212.05
initial amplitude=76
N=36

lambda: -217.14
initial amplitude=77
N=36

## 2004-04-01 Schrödinger's equation

### Maple animation

The following animation shows the evolution of an initial Gaussian pulse under zero potential:

Here is a MNG animation of the same thing if your browser supports it.

### How-to make a animated .gif with maple

• Make the animation in maple (see the worksheets above for how to do this)
• Right-click on the animation, selected export-as->GIF
• Use the unix command `convert` (part of the ImageMagick package like this:
`convert test.gif test-c.gif`
This will half the size of your animation. You may need recent ImageMagick, possibly one released after 2003-06-20 when the LZW patent expired in the USA. Note this patent is still valid in Canada, use gifs at your own risk.
• You can also make a MNG like this:
`convert test.gif test-c.mng`
but browser support for .mng seems minimal without a plugin.

## 2004-03-04 session

Philip’s codes: `wave1.m`, `wave1_f.m`, `wave1.m`, `wave1_f.m`, and `convtest.m`.

## 2004-02-26 middle singular point

Solving backwards numerically starting in the right-hand singular point, we compute the value of η such that w(η)=0. Example outputs from the code for three difference tolerances are:

This process seems sensitive to the tolerances specified to `ode15s`; the following plot shows the stopping value against the relative tolerance (red circles show solutions which diverged and never reached zero):

Clearly, we cannot trust this code to capture the correct solution as it passes through the middle singular point. Also, if you look very closely at the last plot, it actually resembles a batch of chocolate chip cookies.

The codes can be downloaded here: `driver3.m`, `odef3_rhs.m`, `odef3_mass.m`, and `odef3_events.m`.

My earlier code also has trouble crossing the middle singular from the left-hand side. See plots of the 2nd derivative and the 3rd derivative.

Download a `maple` code `subs_first_terms.mw` to sub in the first two terms of the w1 solution at the middle singular point.

## 2004-02-19 right-hand singular point

My first code (seems to integrate across middle singularity): driver1.m, odef1_rhs.m, odef1_mass.m, odef1_jac.m.

The code we wrote during the Thursday session (starts in the right-hand singularity and integrates forward and backward): driver2.m, odef2_rhs.m.

## 2004-02-12 Frobenius method

Photographs of the board:

• Side boards:
• The front board:

## 2004-02-05 a linearized example of 2D, rotating & stratified flow

Photographs of the board from Thursday afternoon:

## 2004-01-16 Lagrange points

Photograph of the board from the Thursday session (first-order corrections for L1, L2):

Friday afternoon (leading order and first-order corrections for L3, L4, and L5:

## 2003-11-14 more van der Pol oscillator

Photographs of the board:

### codes

Based on cbm’s codes from last week.

## 2003-11-07 van der Pol oscillator

MathML test: $x\text{'}\text{'}+\mu \left({x}^{2}-1\right)x\text{'}+x$.

### djm’s code

This code computes solutions to the van der Pol oscillator and displays them in the phase plane.

### cbm’s code

For various values of mu, this code numerically estimates the period of the van der Pol oscillator. The results are compared with the leading order analytic results and the second order correction is estimated.

## 2003-10-17

Various photographs of the board: