Simulating the Spread of the Common Cold

This modeling scenario guides students to simulate and investigate the spread of the common cold in a residence hall. An example floor plan is given, but the reader is encouraged to use a more relevant example. In groups, students run repeated simulations, collect data, derive a differential equation model, solve that equation, estimate parameter values by hand and through regression, visually evaluate the consistency of the model with their data, and present their results to the class

Lost at Sea: Introduction to Numerical Methods through Navigation

Excerpt: The ship, El Perdido, was damaged during a storm which knocked out its main and backup power generators. Before the backup generator failed, Captain Miguel Gomez sent a distress call and the crew have been able to keep El Perdido a oat, but the ship is adrift in the Pacific Ocean off the coast of California. Thankfully, a US Coast Guard rescue operation is underway after receiving the distress call. The Coast Guard has El Perdido\u27s last known position and has mapped out the surface water velocities in this area as slope fields for longitude (x) and latitude (y), which they have updated using historical data and estimated predictions. Since the search grid is small enough, this curved region on the surface of the earth is relatively at

Logistics of Mathematical Modeling-Focused Projects

This article addresses the logistics of implementing projects in an undergraduate mathematics class and is intended both for new instructors and for instructors who have had negative experiences implementing projects in the past. Project implementation is given for both lower and upper division mathematics courses with an emphasis on mathematical modeling and data collection. Projects provide tangible connections to course content which can motivate students to learn at a deeper level. Logistical pitfalls and insights are highlighted as well as descriptions of several key implementation resources. Effective assessment tools, which allowed me to smoothly adjust to student feedback, are demonstrated for a sample class. As I smoothed the transition into each project and guided students through the use of the technology, their negative feedback on projects decreased and more students noted how the projects had enhanced their understanding of the course topics. Best practices learned over the years are given along with project summaries and sample topics. These projects were implemented at a small liberal arts university, but advice is given to extend them to larger classes for broader use.Comment: 27 pages, no figures, 1 tabl

Algae Population Self-Replenishment

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, in 1994. After reading a summary of news articles on the incident, students create an ODE system model from a verbal description of the factors, visualize this system using an executable Java applet (PPLANE) to predict overall behavior, and then analyze the nonlinear system using the Jacobian matrix, eigenvalues, phase plane, and feasibility conditions on parameters to fully describe the system behavior. Students are expected to be familiar with systems of differential equations, equilibria, jacobian matrices, and eigenvalues. Students will learn modeling from qualitative descriptions, nondimensionalization, applying feasibility conditions to parameters, and how to use technology to interactively analyze a system of differential equations

1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, starting in 1994. After reading a summary of articles written on the incidents, students are guided through the process of creating a first order differential equation from a verbal model of the factors and analyze the nonautonomous ODE using direction field, parameter evaluation, and exact solution computation to fully describe the population behavior. Students are expected to be familiar with the separable method and direction fields. Students will learn building and improving a model from qualitative descriptions, nondimensionalization, evaluating parameters, and how to use DFIELD software to interactively analyze a first order differential equation. An alternative modeling investigation of this problem leads to a nonlinear system of equations shown in modeling scenario Algae Self-Replenishmen

An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants

In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the condensation method requires about half as many calculations. Zeros appearing in the interior of a matrix can cause problems, but a way around the issue can usually be found. Overall, Dodgson's condensation method is an interesting and simple way to find determinants. This paper presents an elementary proof of Dodgson's method.Comment: 7 pages, no figure

Lead-Acid Battery Model Under Discharge with a Fast Splitting Method

A mathematical model of a valve-regulated lead-acid battery under discharge is presented as simplified from a standard electrodynamics model. This nonlinear reaction–diffusion model of a battery cell is solved using an operator splitting method to quickly and accurately simulate sulfuric acid concentration. This splitting method incorporates one-sided approximation schemes to preserve continuity over material interfaces encompassing discontinuous parameters. Numerical results are compared with measured data by calculating battery voltage from modeled acid concentration as derived from the Nernst equation

An Evolutionary Method for the Minimum Toll Booth Problem: the Methodology

This paper considers the minimum toll booth problem (MINTB) for determining a tolling strategy in a transportation network that requires the least number of toll locations, and simultaneously causes the most efficient use of the network. The paper develops a methodology for using the genetic algorithm to solve MINTB and presents the algorithm GAMINTB. The proposed method is tested and validated through a computational study with six example networks. Additional numerical test discovers some interesting properties for the proposed method, and provides guidelines for further application of the GAMINTB

Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations

This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations can create infeasible results. Since oscillation-free behavior is not ensured by stability conditions, a more precise condition would be useful for accurate solutions. Using Von Neumann and spectral analyses, we find and explore oscillation-free conditions for several finite difference schemes. Further relationships between oscillatory behavior and eigenvalues is supported with numerical evidence and proof. Also, evidence suggests that the oscillation-free stability condition for a consistent linearization may be sufficient to provide oscillation-free stability of the nonlinear solution. These conditions are verified numerically for several example problems by visually comparing the analytical conditions to the behavior of the numerical solution for a wide range of mesh sizes