Math 450 - Mathematical Modeling Course description

Timothy Reluga

The purpose of the course is to introduce modeling, i.e., the construction of mathematical descriptions in order to capture observed phenomena. The course will explore the natural and human world using mathematics. Particular emphasis will be placed on the process of creating a model starting from observation, data, and first principles.

This course aims to convey some of the heuristic, intuitive, and formal math employed in the scientific description of observed phenomena. Once a particular model has been developed, students will use geometry, calculus, and numerical methods to determine the properties of the model, and to make predictions. While initial model hypotheses are often partially satisfactory, many times one also finds new features of the system that are not adequately accounted for in the model. Thus, theory-building is usually a cyclic, iterative process. For a given phenomenon such as ambulation, or flow of water through a flexible hose, several models may be compared and contrasted, and simplifications will be discussed.

Examples of systems considered in the past include: chemical oscillations, fisheries management, traffic flow, water waves, plant growth, epidemic spread, orbital mechanics, percolation, and elastic behavior of polymer solutions, ambulation, and epidemic network dynamics. Student projects have included the modelling of eye-movement during reading, candle burning, geysers, and human memory. Analysis techniques include model fitting, simulation, bifurcations, and asymptotics. Scientific computing, computer simulations, and physical experiments are often included.

The course is open to a wide range of undergraduate as well as graduate students with majors in mathematics, biology, chemistry, economics, engineering, physics, and related fields. The course should be accessible to students with introductory knowledge of mathematical analysis, matrices, and differential equations.