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    Technique Narrative
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    1-015-DimensionlessVariables-TechniqueNarrative
    This material introduces the idea of ``rescaling'' for ordinary differential equations (ODE's) by the use of dimensionless variables. In practice this is an extremely common and useful prelude to the analysis and solution of ODE's.
    linearizationdimensional analysistime scalescalingdimensionless variablesspatial scale
    Free Online Textbook
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    2016-Langtangen-Pedersen - Scaling of Differential Equations
    Nowadays, the greatest practical benefit of scaling is related to running numerical simulations, since scaling greatly simplifies the choice of values for the input data and makes the simulations results more widely applicable.
    scalingdimensionlesscontinuum mechanics
    Potential Scenario
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    2016-Mehmet_Pakdemirli-Mathematical design of a highway exit curve
    Using fundamental principles of physics and calculus, the differential equation determining the curve function is derived.
    kinematicscurve designhighway exittangential acceleration
    Technique Narrative
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    1-001-SepartionOfVariables-TechniqueNarrative
    We discuss strategies to solve first order, ordinary differential equations with mathematical models when the variables may be separated.
    solutionsstrategymethodapplications
    Article or Presentation
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    2020-Teaching Module-Applying Separation Of Variables
    This is one of several Teaching Modules prepare by the Principal Investigates of SIMIODE’s National Science Foundation grant and offered in 2020. The material offered is a complete narrative of how a modeling activity can be offered including...
    modulesEbolaseparation of variablesTeaching ModulesSeparation
    Modeling Scenario
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    1-007-AntTunnelBuilding-ModelingScenario
    We pose the prospect of modeling just how long an ant takes to build a tunnel. With a bit of guidance students produce a model for the time it takes to build a tunnel of length x into the side of a damp sandy hill.
    Spanishhormigatúnelanttunnel
    Modeling Scenario
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    1-138-InnerEarDrugDelivery-ModelingScenario
    Students examine local drug delivery to the cochlea. The delivery system is modeled as a liquid mixing problem. Students formulate the differential equation, and solve the equation using separation of variables or integrating factor.
    concentrationdrug deliveryinner earcochleahearing loss
    Article or Presentation
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    2020-TeachingModule-ModelingFallingColumnOfWater
    We discuss how to model a falling column of water empirically and analytically from first principles in physics laws.
    modulecolumn of waterTorricelli's LawTorricelliTeaching Module
    Potential Scenario
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    2017-Anders_Kallen-Single species models in continuous time
    Part of a series of Lecture notes this lecture contains rich materials on logistic harvest models, spruce budworms, and chemostat.
    chemostatlogisticharvestspruce budworm
    Potential Scenario
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    2011-Carl_Leinbach-Beyond Newton's law of cooling - estimation of time since death
    The paper offers an excellent history of temperature-based methods and then proceeds to demonstrate these with some data. Here the history is rich and will support student investigation as well as faculty development.
    time of deathNewton's aw of cooling
    Modeling Scenario
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    1-097-SwimmingPool-ModelingScenario
    This project involves the dynamics of chlorine concentration during regular swimming pool maintenance cycles. Students will have the opportunity to use both analytic and numerical methods.
    concentrationcchlorinenumerical errorswimming poolhorineerreor
    Modeling Scenario
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    4-039-FallingDarts-ModelingScenario
    we develop, solve, and analyze a second order differential equation model for free fall incorporating air resistance. Students solve the model using two methods -- reduction of order and separation of variables, and method of undetermined...
    optimizationresistanceterminal velocitymuzzle velocitydrag coefficientfree fallair drag
    Modeling Scenario
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    9-010-TravelingWave-ModelingScenario
    Students are taken through a traveling wave analysis of a porous medium model. While the starting point is a nonlinear partial differential equation model, after a change of variables, students are led quickly to an ordinary differential equation...
    traveling wavesseepage velocityporous medium
    Modeling Scenario
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    1-037-CommonColdSpread-ModelingScenario
    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.
    diseasecommon coldresidence hall
    Modeling Scenario
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    1-026-Evaporation-ModelingScenario
    We provide data on evaporation of 91% isopropyl alcohol in six different Petri dishes and one conical funnel and on evaporation of water in one Petri dish. We ask students to develop a mathematical model for the rate of change in the respective...
    evaporationconealcoholPetri dish
    Potential Scenario
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    2017-Adolfo_Rumbos-Differential Equations and modeling
    The study of any phenomenon in the physical or biological sciences, in which continuity and differentiability assumptions about the quantities in question can be made, invariably leads to a differential equation, or a system of differential...
    pendulumLotka Volterra equationsexistenceuniqueness
    Potential Scenario
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    2011-Lucas_Pulley-Analyzing Predator-Prey Models Using Systems of Ordinary Linear Differential Equations
    This research focused on applying biological mathematics to analyzing predation relationships, especially the relationship between the Canadian Lynx and the Snowshoe Hare.
    simulationpredator-preyLotka-VolterrapopulationCanadian LynxSnowshoe Harecritical values
    Modeling Scenario
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    1-035-DotseroVolcanoEruption-ModelingScenario
    In this scenario, we use Carbon-14 dating of the Dotsero volcano in Colorado as a way of emphasizing this multistage process of modeling.
    exponential decayhalf-lifedecayEuler's methodcarbon datingdecay rateCarbon-14Dotsero volcanovolcano
    Modeling Scenario
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    1-006-Financing-ModelingScenario
    We describe two situations, one in which we are saving for a purpose and one in which we are borrowing for a purpose. In the first case we ask for discrete and continuous model of the situation and in the second case we ask that the results of...
    financesavingsloaninterest ratepayment
    Modeling Scenario
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    3-052-OptimalProjectileFiring-ModelingScenario
    We offer the opportunity to model a projectile's trajectory in several cases, all without resistance.
    projectile motionprojectilelaunch anglemaximum range