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: 11:00-12:00 MWF and by appointment

 

Text: None.  However, you are expected to maintain a class notebook that includes all class handouts 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 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 written report and in-class PowerPoint presentation. 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.

 

This course will adhere to UMD's Student Academic Integrity Policy, which can be found at http://www.d.umn.edu/assl/conduct/integrity. This policy sanctions students engaging in academic dishonesty with penalties up to and including expulsion from the university for repeat offenders. You are expected to neither give nor receive aid on the homework, computational assignments and exams. While it is permissible to ask of other students enrolled in the class general questions about the homework and computer labs, what you turn in should reflect what YOU understand about the material. "Team" solutions are not acceptable.

 

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

Model classifications and characteristics

Why we model

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

 

III. Computer-Assisted Methods

            Symbolic Software

            Numerical Simulation Methods

                       

IV. Applications of 1st Order Linear Equations

            Compartmental Mixing

            Radioactive decay (carbon dating)      

            Heating/Cooling

            Beer's law

            Falling bodies

            Car acceleration/deceleration

 

V. Applications of 1st 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 and Bridge Oscillations

            Combat models

            The Chemostat

            The Lorenz system

                         

VIII. Higher Dimensional Models

            Multi-compartment Vat Systems

            Multi-Species Population Models

            Epidemic Models

            Litter Decay

            Stoichiometric Population Growth Models

 

IX. Models with Time Delays

            The shower

            Driver reaction times

            Population models (Wright's model, gestation periods)

            Epidemic models