Student - Faculty Seminar Mathematical Modeling

Mathematical modeling is a scientific attempt to describe real life phenomena using mathematical tools. Each model consists of a set of variable parameters and rules of evolution for these parameters. By applying these rules to the model’s parameters, one can learn a lot about the phenomenon in question, and, based on the results, make informed decisions, predict the future, or make the optimal choice between different options. Many mathematical models are implemented on computers; their simulations can run in a matter of minutes, or even seconds, often replacing real life experiments that consume vast amounts of time and resources. Mathematical modeling is widely applied in a variety of fields, ranging from medicine to engineering, from physics to physiology, from economy to image processing. It is essential to almost every science. There is a great variety of mathematical models; the following are some of the main categories. Linear or nonlinear: each mathematical model is defined by its parameters and the relations between these parameters. If the relations are defined by linear operators, the model is linear; otherwise it is considered nonlinear. Typical examples of mathematical models that can be either linear or nonlinear are those defined by linear or nonlinear differential equations. The second important distinction between different models is whether the model is deterministic or stochastic. In deterministic models, the evolution of the system is completely determined by the initial (starting) value of the model; for example, a model of planetary motion is deterministic. In contrast, stochastic models allow for randomness—they are well-suited to describe processes that are not well defined, like the stock market—and the model variables’ values are defined by their probability distribution rather than a unique value. Finally, static models aim to describe real life processes at a given time, while dynamic models describe an evolution in time. Many dynamic models involve time-dependent differential equations. In the fall of 2007, the student-faculty seminar was devoted to different applications of mathematical modeling. The topics presented by students and discussed with faculty included packaging costs, soda bubbles, nuclear arms races, and others. We used the book An Introduction to Mathematical Modeling by E.A. Bender [1].