
5010SMatrixExponential
12 Sep 2019  Technique Narratives  Contributor(s):: Kurt Bryan
The matrix exponential is a powerful computational and conceptual tool for analyzing systems of linear, constant coefficient, ordinary differential equations (ODE's). This narrative offers a quick introduction to the technique, with examples and exercises. It also includes an introduction to...

10001STilingHallway
12 Sep 2019  Modeling Scenarios  Contributor(s):: Rob Krueger, Eric Stachura
Students will investigate difference equations through the context of tiling hallways. Students will observe patterns in the tiling which will lead to a difference equation model. Solutions will be calculated by iteration. Then students will be introduced to the concept of the shift...

1111SSpreadOfInformation
02 Sep 2019  Modeling Scenarios  Contributor(s):: Jeff Pettit
Students perform experiments to model spread of information within a population. Students collect data, determine essential components and parameters and build a mathematical model which can be compared to predicted data, measured data, and modeled data. Part 3 can require students to move from...

1092SDashItAll
01 Sep 2019  Modeling Scenarios  Contributor(s):: Kurt Bryan
This project uses very basic physics, Newton's Second Law of Motion, to model the motion of a sprinter running down a track. The model is driven by data from the world record race of Usain Bolt in the 2008 Beijing Olympic games. In particular, we derive the classic HillKeller model for a...

1084SGoingViral
31 Aug 2019  Modeling Scenarios  Contributor(s):: Bill Skerbitz
Students employ randomization in order to create a simulation of the spread of a viral disease in a population (the classroom). Students then use qualitative analysis of the expected behavior of the virus to devise a logistic differential equation. Finally, students solve the...

3064SGearTrain
31 Aug 2019  Modeling Scenarios  Contributor(s):: Lukasz Grabarek
In this scenario students will model an inputoutput mechanical system of gears with a second order, nonhomogeneous, ordinary differential equation with constant coefficients. The model incorporates friction and moments of inertia of the gear train components. Students should have some...

9015SUnearthingTruth
28 Aug 2019  Modeling Scenarios  Contributor(s):: Kurt Bryan
This project introduces electrical resistivity tomography, a technique of interest for geophysical imaging, used to produce images of underground features or structures by using electrical current. Specifically, a known electrical current is injected into an object (for example, the earth) and...

1083SFallingMeteorites
24 Aug 2019  Modeling Scenarios  Contributor(s):: Lyle Clifford Smith
After introducing the solution to the ordinary differential equation which models a falling object with drag (firstorder, nonlinear, separable), students will consider generalizing the model to a falling and disintegrating meteorite. The focus is on creative identification of factors. Students...

1082SMirrorMirror
21 Aug 2019  Modeling Scenarios  Contributor(s):: Kurt Bryan
This project models the ``Foucault Knife Edge Test,'' an optical test commonly used by amateur astronomers who make their own mirrors for reflecting telescopes. The goal of the test is to estimate the shape of the surface of a mirror from optical reflection data. The model results in a...

9010STravelingWaves
21 Aug 2019  Modeling Scenarios  Contributor(s):: Eric Stachura
In this scenario, students will be taken through a traveling wave analysis of a porous medium model. While the starting point is a nonlinear partial differential equation (PDE) model, after a change of variables, students are led quickly to an ordinary differential equation (ODE) model....

3011SEulerBallThrowing
19 Aug 2019  Modeling Scenarios  Contributor(s):: Chris McCarthy
If a tennis ball is thrown through the air it will eventually hit the ground due to gravity. Using Euler's method, write a short script (Python, Matlab, R, etc.) to find the trajectory of the ball which will maximize the distance the ball lands from the thrower taking into account air...

Ice Coverage in the Arctic Climate over time
17 Aug 2019  Articles and Publications  Contributor(s):: Wisam Victor Yossif Bukaita, Kelcey Meaney, Angie Dimopulos
In this paper, entitled Ice Coverage in the Arctic Climate, from two students, Angie Dimopulos and Kelcey Heaney, and their professor, Dr. Wisam Bukaita, ice coverage in the Arctic is modeled over time. We present their abstract here and include the paper itself.ABSTRACT The Arctic plays a...

5002SPhasePortraitForRelationshipDynamics
15 Aug 2019  Modeling Scenarios  Contributor(s):: Lawrence C Udeigwe
The different possible dynamics of a twoperson romantic relationship are modeled  as a linear two dimensional system of equations  and analyzed. Students are guided to explore how the mathematical model of one relationship type can be obtained by modifying the mathematical model of another....

1068SWaterBottleCooling
14 Aug 2019  Modeling Scenarios  Contributor(s):: Eli E Goldwyn
The scenario uses an Inquiry Based Learning approach to walk the students through the creation and understanding of a differential equation describing how fluid in a water bottle will change its temperature to approach the ambient temperature in a room. The goal is for the students to understand...

3150SItsABlastFurnace
13 Aug 2019  Modeling Scenarios  Contributor(s):: Kurt Bryan
This project uses the steadystate heat equation to model the temperature distribution in an industrial furnace used for metal production, for example, a blast furnace.The heat flow is assumed to be steadystate, so that only an elementary ordinary differential equation (ODE) is needed, and...

1076SClimateBifurcation
08 Aug 2019  Modeling Scenarios  Contributor(s):: Mayer Humi
In this paper we present (for educational purposes) simple zero and one dimensional models for the mean temperature of the Earth. These models can exhibit bifurcations from the present Earth climate to an ice age or a ``Venus type of climate.'' The models are accompanied by Matlab.

9005SInvasiveSpeciesModel
08 Aug 2019  Modeling Scenarios  Contributor(s):: Eric Stachura
This scenario takes students through the development of an invasive species partial differential equation model. Basic models are discussed first, which lead students to eventually develop their own model which takes into account dispersion. Students will explore various Mathematica modules...

Moon and Mars Exploration
07 Aug 2019  General Resources  Contributor(s):: Frank Wattenberg
Description to be provided by Dr. Frank Wattenberg, Department of Mathematical Sciences, United States Military Academy, West Point NY USA.Current materials can be found at the Moon and Mars Exploration web site.

3130SMatterOfSomeGravity
07 Aug 2019  Modeling Scenarios  Contributor(s):: Kurt Bryan
This project introduces the concept of an inverse problem (or parameter estimation), in the context of the simple linearized pendulum ordinary differential equation. The theme of the project is that one often has a physical model for a system but in which the model contains unknown parameters or...

5025SSaltCompatments
19 Jul 2019  Modeling Scenarios  Contributor(s):: Brian Winkel
Model a phenomena in which salt mixtures from two tanks are mixed using several strategies.