MTH 323 - Sec 001

MWF 11:00-11:50
STAG 107
Spring 2010


Professor:

 

Dr. Nathan Louis Gibson  

Office:

 

Kidd 312

Office Hours:

 

MWF 1:00-1:50PM

Course Website:

 

http://www.math.oregonstate.edu/~gibsonn/Teaching/MTH323-001S10

Text Book:

 
  Title: Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow
Series: Classics in Applied Mathematics, Vol. 21
Author: Rich Haberman
Year: 1998
Bookstore Price: $57.04
SIAM Member Price: $43.40

Contents

Course Description

A variety of mathematical modeling techniques will be introduced. Students will formulate models in response to practical problems drawn from the literature of ecology, environmental sciences, engineering or other fields. Informal writing assignments in class and formal written presentation of the models will be required. (Writing Intensive Course) PREREQS: MTH 256 and MTH 341 or instructor approval.

Details

Your goals in this course are to learn how to interpret the mathematical models published in contemporary texts and journals, how to formulate your own mathematical models, and how to present your modeling efforts in a well-written paper.

An approximate outline of topics follows.

Topic Applications Models Concepts
Mechanical Models Spring-mass system
Pendulum 		Ordinary differential equations Linearization
Population Dynamics Discrete one-species systems
Harvesting 		Difference equations
Leslie matrix models
Models with time delays 		Equilibria and Stability
Chaos
Stochasticity
Particle Flow Traffic flow
Heat transfer 		Partial differential equations Waves and shocks

Reading Assignments

Reading assignments will typically involve chapters in the text, but will also include chapters from outside sources, journal articles, and peer-written reports. You may be asked to provide a written summary and/or critique, or participate in classroom/online discussions. Your grade for this component will primarily be based on active participation.

  1. Read for Friday Apr 2: Dynamic Models in Biology, Chapter 1
  2. Read for Friday Apr 23: Coupled Oscillators and Biological Synchronization Peer Reviewing Instructions
  3. Read for Wednesday May 26 the Introduction (and, if you like, the Conclusion) of: Resurrection of "Second Order" Models of Traffic Flow Post to digital dropbox by 5PM Wednesday May 26.

Writing Assignments

This is a Writing Intensive Course (WIC), thus you will be required to write at least 5000 words, at least 2000 of which must be polished papers that you have revised after peer review and instructor feedback. This formal writing requirement will be satisfied by producing a term paper, roughly 5 pages in length (not counting figures). See the calendar for deadlines pertaining to this project, and Term Paper for a description of what is expected.

The remaining portion of the writing requirement will be comprised of homework assignments and informal in-class assignments pertaining to lecture.

For resources on writing, see links below.

  1. Assignment for Friday Apr 9:
    Describe a system (object, situation, etc.) whose dynamics could reasonably be modeled as a harmonic oscillator, i.e., by using a second order, constant coefficient ordinary differential equation (with or without some "damping"). In your description, be sure to include the following:
    1. Convince me why the dynamics can be modeled as a harmonic oscillator
    2. Discuss the advantages/benefits to using such a model for your system
    3. Discuss the disadvantages/drawbacks to using such a model, e.g., limitations of the model, necessary assumptions.

    Your description may be typed or hand-written, should be brief (readable in less than 10 mins), and written for a non-expert in the corresponding application field (e.g., explain all non-mathematical concepts and terms).

  2. Read for Friday May 14: Sections 43, 48, 49. Answer Exercise 49.1. (Hint: consider what would happen to the other two if one species were removed). Additionally, try to think of an example in nature that could be reasonably modelled by the equations in the exercise, be specific. Upload a pdf to the digital dropbox by Monday May 17 5PM.

  3. Read for Monday May 17: Sections 56-58.
    Describe a system (object, situation, etc.) whose dynamics could reasonably be represented using the traffic flow PDE model. Bring a hard copy to class Wednesday May 19.

Computer Assignments (Labs)

This is not a programming course, however many topics are more easily understood by computational experimentation. MATLAB codes will be provided for your use. Your grade for this component will primarily be based on written explanations of what you observe from running the simulations.

For resources on MATLAB, see the section below.

  1. Assignment for Friday Apr 16: Lab 1

  2. Assignment for Wednesday May 5: Lab 2
    1. Complete the Matlab Tutorial that we started earlier (Lab 0), beginning now at Age-Class Population Model.
    2. Before doing Modifying the Leslie Model, do the following:
      • In addition to the fact that the total population is changing, the distribution (proportion) of the population is also changing. Using the hints from the "Tutorial Tasks", determine the end behavior of the total population as well as that of the population distribution. Do this for the original Leslie matrix, the Leslie matrix incorporating a fertility drug, and also a modified leslie matrix which has slightly higher birth and survival rates than the two previous examples (convince yourself that the end behavior is independent of the initial population distribution).
      • Run the following commands for each of the above examples and explain the relationship between the output of the commands and your answers to the previous question:
        • [V,D] = eig(A);
        • [d,i] = max(diag(D))
        • vi = V(:,i)/sum(V(:,i))
    3. Do the section labelled Modifying the Leslie Model, however, rather than performing "tutorial tasks from the section on Leslie models using, instead, Model X", answer the questions in (b) instead.
    4. Type your answers to all questions (no need to include a script, output or plots) and upload the PDF file to the Digital Dropbox in Blackboard by 5PM Wednesday, May 5.

  3. Assignment for Wednesday Jun 2: Lab 3

Term Paper

This is a Writing Intensive Course (WIC), thus you will be required to write at least 5000 words, at least 2000 of which must be polished papers that you have revised after peer review and instructor feedback. This formal writing requirement will be satisfied by producing a term paper, roughly 5 pages in length (not counting figures).

Deliverables in preparation for the final paper are as follows:

  1. Topic (Due Apr 23): 10 points This should be roughly a one paragraph description of the problem or application you intend to model. It is fine to list two competing topics in order to get feedback on both. References are not necessary at this point, although they would be helpful. Topics should be typed and uploaded as a pdf file to the digital dropbox.

  2. Proposal (Due Apr 30): 20 points The project proposal should be roughly one page (single spaced, 1 inch margins). References may be included. In fact, I suggest you find at least two published papers related to the topic to get a feel for what has been done/what would be involved in modeling. The purpose of the proposal is to clearly present a question regarding your application that you intend to answer using a mathematical model, and to describe why answering that question is important. For the proposal, consider that you are applying for funding/permission to pursue this research topic. As with many funding requests, your proposal will be peer-reviewed before being evaluated on its merits. You do not need to have identified the precise methods that you will employ (compare to Introduction section below), however you should try to describe at least what type of equation will be used in the model (e.g., difference vs. differential, ordinary vs. partial, linear vs. nonlinear, stochastic vs. deterministic, etc.). Lastly, please define all uncommon terms and concepts as if the reader is not an expert in the application, but has a mathematical background.

    Please see this sample proposal which is much longer and more detailed than you need to be, but demonstrates the structure and layout of a proposal.

    Instructions for submitting proposal: Your proposal should be typed and exported to PDF format. In order to upload to Blackboard, please follow these instructions:

    Instructions for peer-reviewing proposal

  3. Draft of Introduction (Due May 7): 20 points By this point you need to have identified the methods that you will employ in modeling your problem or application. The introduction should include most of the content of the proposal, but in more detail. A background paragraph or two must list previous work in this area, with citations to references. You should make a particular effort to distinguish the current work from previous efforts (e.g., yours is a simplification/generalization of so-and-so's work). Although you likely do not yet have results, it would be a good idea to describe what you expect to happen so as to have had the practice in describing results.

    Please see this sample paper which is much longer and more detailed than you need to be, but demonstrates the structure and layout of each section of a research paper.

    Instructions for submitting Draft of Introduction: follow directions above for Proposals.

    Instructions for peer-reviewing proposal

  4. Rough Draft (Due May 21): 40 points See sample above under Introduction. Your rough draft should include an abstract and a bibliography. The introduction of the draft should outline the entire paper. It is appropriate to describe tasks not yet completed and to state hypothesis not yet tested. However, some results are expected for this draft; it should not simply be a longer proposal.

    Follow directions from Proposals to upload your Rough Draft to the Discussion Board. Please make use of Writing Resources under Links below.

  5. Final Paper (Due June 4): 80 points See sample above under Introduction. Your final draft should include an abstract and a bibliography, possibly figures and tables, and appendices if necessary (could include code used if short, or lenghtly derivations of equations which interrupt flow of narrative). The introduction of the final draft should outline the entire paper. Any tasks not yet completed should be left out (may be moved to a paragraph in the Conclusions section detailing future work possibilities).

    Follow directions from Proposals to upload your Final Paper to the Discussion Board. Please make use of Writing Resources under Links below.

    Exam

    There will be an in-class final exam covering material from lectures.

    Grades

    Grades for each assignment will be posted to the Blackboard Site.

    Grade Distribution

    Reading Assignments 10%
    Writing Assignments 20%
    Computer Assignments 10%
    Term Paper 40%
    Final 20%
    Total 100%

    Matlab

    The following are options for accessing Matlab at OSU:
    1. The Mathematics Department computer lab is located in the Math Learning Center, Kidder 108.
    2. The computer lab in the Milne basement.
    3. You may have access to Matlab through a computer lab or network of your department. In particular, students can download from here.
    4. If you would like to have Matlab at home, consider purchasing the MATLAB Student Edition.

    The following are online resources for learning Matlab:

    Links

    Writing Resources:

    Modeling Resources:

    Reference Books:

    Blackboard Site

    Last updated: Sun May 23 15:51:46 PDT 2010