75048 Mathematical modeling 1998
Lectures

19980824/25

Introduction. The slime mold example [L&S 1.3]. Dimensional analysis started off with a few simple examples (simple pendulum [L&S 2.2], atomic bomb explosion)

19980831/0901

Dimensional analysis continued, with a more involved example (pipe flow and the Moody diagram) and a proof of Buckingham's pitheorem [Bpi]

19980907/08

Nondimensionalisation, scaling and regular perturbation theory [L&S 6], with applications to the projectile problem [L&S 7.2] and the circular pendulum [L&S 7.1].

19980914/15

We are finished with the circular pendulum, having gotten rid of the secular term t·sin(t) in the perturbation expansion [L&S 2.2]. I spent quite a bit of time talking about the heat/diffusion equation, with application to the roast problem [no reference (yet)].

19980921/22

Lectured on the physiological flow problem [L&S 8] and [Os].

19980928/29

Singular perturbations [L&S 9]. Also got a good start on the problem from biochemical kinetics [L&S 10]: Posed the problem, scaled it, found outer solution (the MichaelisMenten approximation).

19981005/06

Finished our discussion of the biochemical kinetics problem. (The problem is further analyzed in [SS]. I also wrote up an extremely terse summary of the problem in [Enz].) I also did a phase plane analysis of this problem, partly supported by a Maple worksheet.
A printout of the worksheet is also available as a PostScript file, but it is better to get the work sheet and run Maple on it. You can download it and run it in Maple, or you can set up Netscape to automatically start Maple when it sees a Maple worksheet: Go to the Edit>Preferences dialog, select Navigator>Applications, hit New, and put application/maplevr5 in the MIMEType field and xmaple %s in the Application field (remember to check the radio button next to it). Sorry, I don't know how to do this with Inernet Explorer or other browsers.
(In an earlier version of this advice, I wrote application/maplevr4, but these are for Maple Vr5, so I changed their type designation. It is best to have both versions registered in your Netscape setup.)
Finally, I introduced the classical LotkeVolterra equations and performed an initial phase plane analysis, just looking at the question of stability of the nonzero equilibrium point. We found that this equilibrium is a center for the linearized system, which does not settle the question of stability for the nonlinear system.

19981012/13

Finished the analysis of the LotkeVolterra system. I showed a Maple worksheet displaying the main features of the system (the version here also includes some details of the invariant of the system).
A printout of the worksheet is also available as a PostScript file

More on phase plane analysis; more precisely, the classification of singular points, and a little bit about limit cycles. See [L&S 11.3, pp.334344].
Here is yet another Maple worksheet containing some plots for the damped pendulum.
A printout of the worksheet is also available as a PostScript file

19981019/29: Modeling seminar

No lectures these two weeks.

19981102/03

Started on conservation laws, the method of characteristics, and shocks. [CL] and [Wh Chapter 2]

19981109/10

More on conservation laws, and the conditions for "good" versus "bad" discontinuities.

19981116/17

A bit on the derivation of the basic principles of continuum mechanics, including Reynold's transport theorem and the DuboisReymond lemma [L&S 12.1, 14.13]. We finish with talking about the swing problem and the solutions that have come in.
References
[L&S]: Lin & Segel's book Mathematics applied to deterministic problems in the natural sciences.
[Bpi]: the handout on Buckingham's pitheorem.
[Enz]: Enzyme kinetics ala Lin & Segel, a brief note made for my own benefit. Grab a copy if you wish.
[Os]: the handout on Lin & Segels physiological flow problem.
[SS]: Lee A. Segel & Marshall Slemrod: The quasisteady state assumption: A case study in perturbation. SIAM Review 31 #3 (1989), pp. 446477 (available at the mathematics library).
[CL]: the handout on conservation laws.
[Wh]: G.B. Whitham's book Linear and nonlinear waves.
Harald HancheOlsen
19981117 21:47:53 UTC