MATH 5270 MODELING WITH DYNAMICAL SYSTEMS

Instructor: Professor H. Stech Telephone: 726 - 8272

Office: 156 Campus Center e_mail: hstech@d.umn.edu

Office Hours: 10:00-11:00 MWF and by appointment

Text: None. However, you are expected to maintain a class notebook that includes all class hand-outs and assignments.

Prerequisites: Calculus and elementary differential equations. Familiarity with a symbolic manipulation program (Mathematica, Maple, Matlab, ...) is desirable.

Objectives: The effective use of "process-based" mathematical modeling requires an understanding of analytic and computer simulation techniques, as well as the ability to extract from the "real world" problem under study a reliable, yet tractable, mathematical representation. Model formulation, of course, requires a familiarity with the basic physical, biological, sociological, … principles in operation.

It is a goal of this course to provide an introduction to the overall modeling process by considering a variety of modeling examples taken from a wide range of disciplines. Mathematical techniques for ordinary differential equations will be reviewed or introduced as each application dictates.

Grading: There will be approximately 10-12 problem sets/coding assignments and a modeling project. There will be no formal in-class exams. The modeling project will be graded according to a class presentation (rendered in PowerPoint.) Graduate students will additionally be required to prepare a short written project report; undergraduate students may submit a written report for extra credit. The final course grade will be determined by a weighted average of the problem set average (60%), the modeling project (30%), and your class notebook (10%.)

The course will be graded according to a scale no higher than

A = 90 % or above

B = 80 - 90 %

C = 70 - 80 %

D = 65 - 70 %

F = below 65 %

Points may be deducted for late assignments. Late problem sets will receive no credit if they are submitted after the graded assignments have been returned to students. This class is available to both undergraduate and graduate students. However, undergraduate students will not be disadvantaged by this fact during the allocation of grades for this class. Graduate student work will be graded with the expectation of thoroughness, precision and rigor expected in all graduate-level classes.

Individuals who have a disability, either permanent or temporary, which might affect their ability to perform in this class are encouraged to inform the instructor at the start of the term. Adaptation of methods, materials, or testing may be made as required to provide equitable participation.

Tentative List of Modeling Topics

I. Introductory Remarks

Types of Models

Process-based vs empirical models

Why we model

Model classifications and characteristics

The modeling process

II. Review of Simple First Order Differential Equations

Fundamental Theory

Elementary Solution Methods

The snow shoveler problem

Car deceleration

Growth models

Two snow shovelers

Compartmental Mixing

III. Computer-Assisted Methods

Symbolic Software

Numerical Simulation Methods

IV. Applications of 1^st Order Linear Equations

Radioactive decay (carbon dating)

Heating/Cooling

Beer's law

Falling bodies

Car acceleration/deceleration

V. Applications of 1^st Order Nonlinear Equations

Draining Vats

Population Growth Models

Harvesting/Control

VI. Simple Systems

Basic Epidemic models

Chemical reactions (Law of Mass Action)

Uncoupled systems

VII. Low Dimensional Systems

Fundamental Theory

The Tractrix

Curves of pursuit

Mass-spring Oscillations

Combat models

The Chemostat

The Lorenz system

VIII. Higher Dimensional Models

Multi-compartment Vat Systems

Multi-Species Population Models

Epidemic Models

Litter Decay

Traffic Flow: Car-following models

IX. Models with Time Delays

The shower

Driver reaction times

Population models (Wright's model, gestation periods)

Epidemic models