Introduction to Continuous Mathematical Modeling

Course Description

Textbook

Lecture notes for download. They will be posted as they become available:

• Lecture Set 1 (PDF). Introduction to modeling with differential equations: a discussion of the philosophy of modeling, communicating with mathematics, and some examples of creating mathematical models.
  • Lecture Set 1 (PDF).
  • Supplement on types of graphs (PDF).
  • Supplement on nondimensionalization (PDF).
  • Supplement on the chemostat (PDF) example. I meant to use this version in the lecture notes, but it slipped by me.
• Lecture Set 2 (PDF). Population models. Exponential growth, logistic growth, allee growth. Introduction to phase line, stability, and bifurcations. Introduction to qualitative analysis.
  • Lecture Set 2 (PDF).
• Lecture Set 3 (PDF). Interactions. Linear models (Romeo and Juliet), competition models, combat models, predator-prey models. Linear stability analysis (elementary linear algebra), phase plane analysis, qualitative plotting of trajectories.
  • Lecture Set 3 (PDF).
  • Read the original paper about the Iwo Jima model: A Verification of Lanchester's Law by J. H. Engel, Journal of the Operations Research Society of America, Vol. 2, No. 2. (May, 1954), pp 163-171. (This link only works with authorized computers--a University network connection, such as in the library, will work.)
  • Predator-Prey web-site: Duke University has an online section about the predator-prey model. They motivate the problem and suggest some activities to explore the model.
• Lecture Set 4 (PDF). Physical systems. Newton's second law, vibrations, harmonic oscillators, pendulums, and rotating bead on hoop. Conservative systems, potential energy, phase plane plotting, and energy dissipation.
  • Lecture Set 4 (PDF).
• Lecture Set 5 (PDF). Traffic flow. Density models, integro-differential equations, partial differential equations, method of characteristics, and occurrence of shocks.

The following additional books include many of the topics we will address, and may be consulted as additional resources:

• Haberman, R. Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow. SIAM, 1998. Available at the University Bookstore.
• Braun, M. Differential Equations and Their Applications. Springer-Verlag, 1993 (or later). Includes discussion of combat models, predator-prey models, competition models, and several other interesting applications.
• Strogatz, S.H. Nonlinear Dynamics and Chaos. Addison Wesley, 1995. Includes in depth discussion of bifurcations, phase plane analysis, conservative systems, and the Romeo and Juliet model of love affairs, as well as the topic of chaos. (This is written at a level slightly above the level the instructor will take for the course.)

Online Resources

Java Resources:
There are many mathematical educational or exploratory tools written in Java. Some of these are good for differential equations. The biggest challenge to using Java is that there is often not a good way to save the graphs that are created due to the security features of Java. To save a picture, one of the easiest methods is to use a screen capture program to grab an image from the screen.

• Tools for drawing slope fields (1-d, 1st order ODE):
  • DField Applet at Rice University based on his Matlab tools (see below).
• Tools for drawing direction field (2-d, 1st order system of ODEs):
  • PPlane Applet at Rice University based on his Matlab tools (see below).
  • Phase Portrait Applet at Penn. State University. This has the advantage that it also plots the trajectories when it plots a particular orbit.
  • Phasor applet at the University of Colorado.

Matlab Resources:

• John Polking from Rice University makes his DField and PPlane Matlab programs available for educational institutions. You can learn about them at his website.

Other Resources:

• If you don't mind a poor user-interface, you might be interested in looking at XPP/XPP-Auto which allows you to explore rather complicated systems of equations, including tracking equilibria and periodic solutions using the tool Auto. (This is designed as a research tool, not as a teaching tool--not for the faint of heart.)

Course Objectives

• Course Policy and Syllabus (PDF)
• Term Project Handout (PDF)

• Quiz Objectives Listing.

Schedule and Homework

Follow links in the table below to obtain a copy of the homework in PostScript (.ps) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. An item shown below in plain text is not yet available. For additional information regarding viewing and printing the homework and solution sets, click here.

Grading

Tutorials