Education

Abstract

This issue's Education section exhibits two quite different applications of mathematical modeling---different in the types of real-life problems that are examined and in the types of mathematical models that are used to study them---that both make nice instructional modules. The first paper uses partial differential equations to model the behavior of a practical device that is important to our health: a "pulsed amperometric ion sensor." This device is the possible successor to related devices that are used extensively in clinical laboratories, and in hospitals at patients' bedsides, to measure blood electrolytes. "How Do Pulsed Amperometric Ion Sensors Work? A Simple PDE Model," by Eric Bakker and A. J. Meir, is a collaboration between a chemist and a mathematician at Auburn University. Proceeding from a description of the operation of the device, and the relevant physics and chemistry, the authors pose a number of questions that a mathematical model can help to answer. The answers to these questions help provide quantitative information on optimal operating conditions and parameters for the device. The mathematical model constructed to answer these questions is the one-dimensional heat equation posed on a semi-infinite domain. After deriving the model, the paper presents solutions that exhibit a number of important results. The paper is accessible to advanced undergraduates and graduate students taking a first course on partial differential equations. The other paper in this issue involves a very different set of practical pursuits---bicycle riding and suntans! Indeed the paper by Geert Jan Olsder is titled "Bicycle Routing for Maximum Suntan," a topic that surely is of particular interest in the author's home country (The Netherlands), which is not noted for an abundance of sunshine. In the paper, the author presents a sequence of increasingly difficult but accessible optimal control problems related to maximizing exposure to the sun while riding a bicycle. The problems are chosen to have tractable solutions that are easy to interpret. The solutions also illustrate both the fundamental ideas and the subtleties associated with control theory. The paper makes a nice module that can be used to accompany the teaching of basic principles in control theory. I feel compelled to add that in my part of the world (Colorado), where sunshine is overly abundant as well as very intense, faculty may need to ask students to develop a corresponding strategy to achieve bicycle routing for minimal suntan!