Ä¢¹½¶ÌÊÓƵ

Mathematical Modeling

Revised: October 2021

Course Description

Mathematical modeling is the process of describing physical phenomena through the use of mathematical concepts and language, and then analyzing the mathematical system to learn more about the physical system. Topics for this course are chosen based on student and instructor interest, and may include: differential equation models, analysis of stability and sensitivity, difference equations, Markov processes, or optimization.

Prerequisites: MATH 320.

Student Learning Objectives

By the end of the course students will be able to read the literature in mathematical modeling, understanding why the investigators used specific methods of analysis and how they give information about the physical process under consideration. Students will be able to:

• Identify a proper modeling technique to apply to specific problems from the life, physical, or social sciences;
• Formulate a model in mathematical language, using methods from both discrete and continuous mathematics;
• Analyze mathematical models;
• Use computer applications to aid in the design of models for simulation, forecasting, and explication; and
• Interpret the mathematical results in the context of the original physical problem.

Text

Beltrami, Edward. Mathematics for Dynamic Modeling. (2nd edition) Academic Press, 1997. Additional supplemental material may be provided.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• Chapter 1: Simple Dynamic Models
Design of models with Newton’s Laws, conservation of mass, law of mass action, equilibrium analysis I, stability.

• Chapter 2: Ordinary Differential Equations
Linear systems, eigenvalue analysis, phase portraits.

• Chapter 3: Stability of Dynamic Models
Systems of differential equations, linearization, equilibrium analysis II, competition models, SIR models, Lyapunov functions, feedback control.

• Chapter 6: Cycles and Bifurcation
Limit cycles, Poincare-Bendixson Theorem.

• Chapter 7: Bifucation and Catastrophe
Parameter dependence, bifurcations.

• Additional Topics as Time Allows.