Department of Mathematics
Pomona College
Math 183. Mathematical Modeling
Course Outline
Fall 2016
Time and Place: MWF 9:00 am - 9:50 am, Millikan 2141
Instructor: Dr.
Adolfo J. Rumbos
Office: Andrew 2287
Phone/e-mail: ext. 18713 / arumbos@pomona.edu
Course Website: http://pages.pomona.edu/~ajr04747/
Office Hours: MWF
10:05 am -10:50 am, TR 10:00 am
-11:00 am,
or by appointment.
Prerequisites: Linear
Algebra and Ordinary Differential Equations
Course
Description. The main goal of this course is to provide
opportunities for students to construct and analyze mathematical models that
arise in the physical, biological and social sciences. Mathematical models are usually created in
order to obtain understanding of problems and situations arising in the real
world; other times, the main goal is to make predictions or to control certain
processes; finally, some models are created in order to aid in decision making.
Construction
of a mathematical model consists of translating a real world problem into a
mathematical problem involving parameters, variables, functions, equations
and/or inequalities. Analysis of the
model involves the solution (if possible) of the mathematical problem through
logical, algebraic, analytical or computational means, and assessing what the
solutions imply about the real situation under study. If an analytical or
computational solution is not possible, computer simulations can sometimes be
used in order to study various scenarios implied or predicted by the model.
Analysis
techniques can be drawn from many areas of mathematics. In this course, it is assumed that students
have a good working knowledge of Calculus, Linear Algebra and Ordinary
Differential Equations. These areas are
adequate for the analysis of some models. However, many modeling situations require the
use of some probability theory and optimization techniques. These mathematical topics will be covered in
the course. In calculus and differential
equations courses, students have been exposed to some continuous models. In this
course, we will also introduce students to discrete
models as well.
Course Structure and Requirements. The course will be structured around a series of case studies that will provide ample opportunity for students to learn about (and to practice) the development and analysis of models raging from the discrete to the continuous, and from the deterministic to the stochastic (or probabilistic), and in many cases involving mixed-type modeling.
Homework problems will be assigned at every meeting and collected on an alternate basis. There will be two midterms. Students will also be required to work in teams of two or three on a modeling project in the last part of the course. The project consists of a term paper describing the construction and analysis of the model. In addition, students will be required to give a formal presentation on the modeling project at the end of the semester.
Grading
Policy. Grades will be based solutions to assigned
problems, a midterm, the term paper and the modeling project presentation. The overall score will be computed as follows
according to the following distribution:
Homework
20%
Exams 50%
Presentations 15%
Modeling term project 15%
Math
183 Fall
2016
Tentative Schedule of Topics and Presentations
Date Topic
W Aug. 31 Introduction
to the process Mathematical Modeling
F Sep.
2 Case Study: Bacterial
Growth in a Chemostat
M Sep. 5 Nondimensionalization
W Sep.
7 Nondimensionalization
(continued)
F Sep.
9 Problems
M Sep. 12 Case
Study: Modeling Traffic Flows
W Sep.
14 Traffic flow models
(continued)
W Sep.
16 Problems
M Sep. 19 Analysis
of a traffic flow model
W Sep.
21 Method of
characteristics
W Sep.
23 Method of
characteristics (continued)
M Sep. 26 Shock
waves
W Sep.
28 Shock waves
(continued)
W Sep.
30 Problems
M Oct. 3 Problems
W Oct.
5 Review
W Oct.
7 Exam 1
M Oct. 10 Case Study: Modeling bacterial mutations
W Oct.
12 Stochastic models
F Oct.
14 Probability
M Oct. 17 Fall Recess.
W Oct.
19 Random variables and distributions
F Oct.
21 Random variables and distributions
(continued)
M Oct. 24 Random
processes
W Oct.
26 Random processes
(continued)
F Oct.
28 Random processes
(continued)
M Oct. 31 Modeling
diffusion
W Nov.
2 Modeling diffusion (continued)
F Nov.
4 Modeling diffusion (continued)
M Nov. 7 Problems
W Nov.
9 Review
F Nov. 11 Exam 2
M Nov.
14 Modeling Project
W Nov. 16 Modeling
Project
F Nov. 18 Modeling Project
M Nov.
11 Modeling Project
W Nov. 23 Modeling
Project
F Nov. 24 Thanksgiving Recess
M Nov.
28 Modeling Project Presentations
W Nov. 30 Modeling
Project Presentations
F Dec.
2 Modeling Project
Presentations
M Dec. 5 Modeling
Project Presentations
W Dec.
7 Modeling Project Presentations