5,044 research outputs found
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
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
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
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
Isogrid design handbook
Handbook has been published which presents information needed for design of isogrid triangular integral-stiffened structures. It develops equations, methods, and graphs to handle wide variety of loadings, materials, and geometry. Handbook is divided into seven sections. Handbook may be used by marine and civil engineers and by students and designers without access to computers
Development and Application of Operational Techniques for the Inventory and Monitoring of Resources and Uses for the Texas Coastal Zone
The author has identified the followed significant results. Techniques for interpretation of LANDSAT images were developed, along with a modified land use classification scheme
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
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
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