# Mathematical Modeling MTH 323 - Sec 010

### Text Book:

 Title: Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Author: Rich Haberman Year: 1998 SIAM Member Price: \$43.40 PDF posted on Canvas (free).

### 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.

Learning Outcomes: After completing this class, successful students will be able to:

• Apply fundamental principles to new problems
• Put together chains of reasoning
• Design experimental investigations to investigate phenomena, to test hypotheses, or to solve practical problems
• Identify the assumptions a model, equation, or claim relies upon, and make a judgment about the validity of the assumptions
• Conduct unit/dimensional analysis to test the self-consistency of an equation
• Make a reasonable prediction from and devise a test of a proposed hypothesis
• Devise multiple explanations for observations and modify them in light of new data
• Analyze a relevant limiting/special case for a given model, equation, claim
• Organize information, ideas, and solutions
• Communicate explanations and results both in writing and verbally
• Exhibit other communication skills: peer review, presentation, paper
• Attain scientific literacy: making sense of scientific news and journal articles
• Perform programming and computational modeling
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 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 of week 1: Dynamic Models in Biology, Chapter 1
• Read exercises also, but do not solve.
• We will not be reading other chapters of this book, but pay attention to what this chapter says about the rest of the book because it will also pertain to the required text for the course.
• This text is primarily intended for biological models. We will discuss how the concepts and techniques apply to other disciplines.
• It would be helpful (but is not necessary) to make an outline of important topics.

Answer the following and bring to class:
1. What is a model?
2. What is a dynamic model?
3. When is a dynamic model preferred over the alternative?
The rest will be posted to Canvas.

### 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 a polished paper 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 Section 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 Section below. Writing Assignments will be posted to Canvas.

### 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.

### 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 a polished paper 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 Week 4) 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 Canvas.

2. Proposal: (Due Week 5) 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. 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.

3. Draft of Introduction: (Due Week 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.

4. Rough Draft: (Due Week 9) 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 Introduction.

5. Final Paper: (Due Finals Week) 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).

### Exam

There will be an in-class final exam covering material from lectures. Sample exam problems will be posted on Canvas.

Grades for each assignment will be posted on Canvas

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

### Matlab

A scientific programming language is required for this course. Matlab is preferred due to the integration of computation and visualization.

The following are online resources for learning Matlab:

Matlab demonstrations
cc2plot.m -- This is a Matlab code that I have written to make it easier for you to visualize solutions to 2nd order linear ODEs with constant coefficients. You should have access to Matlab at many of the computer labs on campus. Simply download this file, run Matlab, and at the prompt type help cc2plot. Copy and paste one of the examples to see how it works. Change the input values to try your own examples, or plot those from the book when plots are not provided. See also cc2plotdemo.m, cc4plot.m, coupledspring.m and coupledspring2.m
pensimulate.m -- Damped, driven pendulum
nonlinearpendulum.ppg -- Nonlinear pendulum gallery for use with pplane8 (2014b, 2016b and 2017b versions), originally from Rice University
.
cobweb.m -- Graphical display of fixed point iterations
logweb.m -- Graphical display of fixed point iterations for logistic map (see paper on period three below)
logphase.m -- Graphical display of fixed point iterations for logistic map (see paper on period three below)
logdelay.m -- Graphical display of solutions for Discrete Logistic with delay.
PDE code

Writing Resources:

Modeling Resources:

Reference Books:

Canvas Site

### Disclaimers

While it may not be stated explicitly each day, students are expected to read each section to be covered before class. Students are responsible for any material missed due to absence. Questions not addressed during class time should be asked in office hours.

“Accommodations for students with disabilities are determined and approved by Disability Access Services (DAS). If you, as a student, believe you are eligible for accommodations but have not obtained approval please contact DAS immediately at 541-737-4098 or at http://ds.oregonstate.edu. DAS notifies students and faculty members of approved academic accommodations and coordinates implementation of those accommodations. While not required, students and faculty members are encouraged to discuss details of the implementation of individual accommodations.”