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:

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

lambda: -33.057
initial amplitude=20

lambda: -73.01
initial amplitude=40

lambda: -207.02
initial amplitude=75

lambda: -209.53
initial amplitude=75.5

lambda: -212.05
initial amplitude=76

lambda: -217.14
initial amplitude=77

2004-04-01 Schrödinger's equation

Maple animation

The following animation shows the evolution of an initial Gaussian pulse under zero potential:
animation of solution
Here is a MNG animation of the same thing if your browser supports it.

How-to make a animated .gif with maple

2004-03-04 session

Follow this link for photographs of the blackboards.

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


2004-02-26 middle singular point

Follow this link for photographs of the blackboards.

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:
graph graph graph
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 to sub in the first two terms of the w1 solution at the middle singular point.

2004-02-19 right-hand singular point

Follow this link for photographs of the blackboards.

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.

ODE solution ODE solution

2004-02-12 Frobenius method

Photographs of the 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:


Based on cbm’s codes from last week.


period versus mu values period versus mu values comparing the numerical estimated second correction with the analytic work the residual error: an estimate of the third order correction estimate of the exponent of the third order correction

2003-11-07 van der Pol oscillator

MathML test: x'' + μ(x2 - 1)x' + x .

djm’s code

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

van der Pol solution phase portrait van der Pol solution

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.

van der Pol solution showing periods period versus mu values estimate of second order correction estimate of the exponent of the second order correction


Various photographs of the board: