SIMIODE Resource Guide

SIMIODE Resource Guide

Guide to Modeling Scenarios and Technique Narratives

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This Guide is organized to follow the topics found in a traditional differential equations course.

The numbering system reflects chapter sequencing in a standard differential equations text.

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Table of Contents

 

Modeling One  - First Order Differential Equations

Modeling Two - Numerically Solving Differential Equations

Modeling Three - Second Order Homogeneous Differential Equation Models

Modeling Four - Second Order NonHomogeneous Differential Equation Models

Modeling Five - Linear Systems of Differential Equation Models

Modeling Six - NonLinear Systems of Differential Equation Models

Modeling Seven - Changing the Venue for Solution Success - Laplace Transforms

Modeling Eight - Representing Natural Phenomena with Sines and Cosines

Modeling Nine - Modeling with Differential Equations in Higher Dimensions  

Modeling Ten – General Modeling

 

APPENDIX

Strategies for Analytic Solutions and Qualitative Analysis

Technique Narratives

 

First Order Differential Equations

Second Order Differential Equations

Linear Systems of  Differential Equations

NonLinear Systems of  Differential Equations

Fourier Series Development

Partial Differential Equation Solution Strategies

 

Modeling One  -  First Order Differential Equations

<Return to Table of Contents>

 

Narrative and Development

1-001-Text-SepVariables

    Technique narrative on solution method of separation of variables

1-002-Text-Integrating Factor

    Technique narrative on solution method of integrating factors

1-009-Text-Bifurcation

    Early introduction to bifurcation with experimentation.

Activities

1-001-M&MDeathImmigration

    Modeling Death and Immigration with M&M 's and Simulation

1-001A-M&MDeathImmigration-Variation

    Variation on modeling death and immigration with M&M’s

1-001B-M&MDeathImmigrationMystery

    Each student sets own immigration rate and others need to solve this mystery

1-001pgf- BirthDeathImmigration

    Probability generating function approach to simulation.

1-002-Tossing

    Modeling a simulation of a large number of dice tossings

1-003-CollegeSavings

    Saving for Child's College Education

1-004-Microorganism Immigration

    Modeling Immigration in a Petri Dish

1-005-OilSlick

    Modeling the Spread of Oil Slick with Incomplete Data

1-005A-OilSlick

    Same as 1-5-OilSlick but with client setting.

1-005a-ChemDataCollection

    Analysis of incomplete reaction data

1-005b-ChemDataCollection

    Analysis of incomplete reaction data

1-006-FinancingSavingsAndLoan

    Bank Investment Analysis and Bank Loan Analysis

1-007-AntTunnelBuilding

    How long does it take an ant to build a tunnel of length x in soil?

1-009-ICUSpread

     Modeling the spread of ICU’s in US Hospitals from 1958-1974

1-010-LSDAndProblemSolving

    Modeling LSD in the body and correlating amounts with test performance.

1-011-Kinetics

    Chemical Kinetics Models - Zeroth, First, and Second Order Reactions

1-011A-Kinetics

    Same as 1-11-Kinetics but with more guidance and less narrative.

1-012-SublimationCarbonDioxide

    Sublimation of Carbon Dioxide

1-013-Sleuthing

     Situations associated with stopping cars, projectile steel ball, and time of death

1-014-DrainingContainers

     For fixed volume column which radius of cylinder of water drains fastest

1-015-Torricelli

     Modeling  falling column of water

1-016-DogDrugs

    Modeling drugs for anesthesiology

1-017-DiseaseSpread

    Modeling spread of disease using logistic equation

1-018-LogisticPopModeling

    Limited Growth Population Modeling

1-019-RocksInTheirHeads

    Sensitivity Analysis  - Data Collection Experiment Comparing of Rock Masses

1-020-IceMelt

    Analyzing models on which melts first a sphere or cube of ice of the same volume

1-021-FeralCatControl

    Model for three feral cat control policies

1-022-SpreadOfTechnology

    Model the spread of a number of technological advances

1-023-RumorSpread

    Rate of Spread of False and True articles on the Internet

1-024-MalariaControl

    Modeling and numerical methods for first order malaria growth

1-025-MixingItUp

    Modeling more and more complex salt mixing situations.

1-026-Evaporation

    Modeling the evaporation of an alcohol and water mixture in various containers.

1-027-StochasticProcesses

    Modeling randomness with stochastic processes.

1-028-SouthernSweetIcedTea

    Data is offered to model making sweet iced team using luminescence

1-029-ConeToCubeFlow

    Modeling water flowing from cone to cube and out

1-030-IntraocularGasBubbles

    Modeling dissipation of intraocular gas bubbles used in eye surgery.

1-031-CoolIt

    Changing temperature of container of water in a changing environment.

1-032-WordPropagation

    Modeling the rate at which words propagate through English language text

1-033-SouthernBarbeque

    Phases of barbequeing brisket are modeled using real data.

1-034-FishMixing

    Student designed fishing strategies for mix of fish in lake

1-036-NeutralBuoyancy

            Finding depth in water at which an object  settles to neutral buoyancy

1-037-CommonColdSpread

    Students conduct simulation of spread of common cold and model.

1-038-Ebola

    Modeling Ebola epidemic with first order differential equation models

1-040-OutcomeSavings

    Determining monthly deposit rate for long term savings goal

1-041-AirToTop

    Variable ascent rate and air management in SCUBA diving

1-042-Kool-Aid

    Modeling the amount of drink powder in a second tank of flow system.

1-043-CoolingUpAndDown

    Air conditioning cooling modeling.

1-044-CollegeBound

    Planning for full college education costs for the daughter of a friend.

1-045-TimeOfDeath

    Determining time of death given observations and environmental conditions.

1-050-BargeAhead

    Optimizing a barge trip upriver

1-051-OneTankSaltModel

    Build one compartment salt mixing model

1-052-SaltWaterTanks

    Studying tank in which water inflow containing salt increases.

1-054-S-GrowthInFarmland

    Mopdeling the growth of farmland with incomplete data

1-055-WaterFallingInCone

    Modeling the falling of water in a right circular cone

1-057-FiguringFluidFlow

    Evaluating three models of fluid flow from a tank using data

1-058-WaterClocks

    A container is designed so water will fall out at constant rate of change in height.

1-059-ContainerShapeFallingWater

    Modeling column of falling water in different shaped containers.

1-060-SalesMarketing

    Building a model of sales of consumer products from a classic marketing study.

1-064-TorricelliBox

    Modeling falling column of water with a box at the base of the column

1-061-PotatoCooling

    Cooling of a baked potato and compare it to student-collected data.

1-066-USCensusModeling

    Modeling the US Census data with several different models.

1-067-ModelingWithSigmoidCurve

    Modeling using logistic and Gompertz S-shape curves is offered.

1-070-FisheryHarvest

    Modeling harvesting of Atlantic cod fishery

1-071-S-NewtonWatsonTimeOfDeath

    Sherlock Holmes determines time of death

1-073-WaterExitBottle

    Estimating a parameter in Torricelli’s model of water exiting a container

1-074-BottleWaterFlow

    Comparing two models of water flowing out of a container through exit hole

1-079-HomeHeating

    Modeling hos to heat your home while you are away

1-080-DrugAdministration

    Building a simple model for drug administration

1-081-TumorGrowth

    Two different models for growth of cancer tumor.

1-085-DrugBolus

    Modeling intravenous bolus of drug in the body

1-086-MedicinalPill

     Modeling administration of medicinal pills

1-089-SpreadOfDisease

    Spread of disease and applications to Sleeping Beuaty fairy tale

1-090-EmptySphericalTank

    Comparing two ways to empty spherical tank of water

1-091-Slopefields

    Building population models for various situations and using slope fields

1-093-SucroseReaction

    Determining model for sucrose hydrolysis using lab data

1-094-SteepingTea

    Modeling temperature change and dissolution of sugar in brewing fruit tea

1-095-RatingChessPlayers

     Using Elo’s Method for rating chess players and difference equations

1-101-ClassM&MDeathImmigration

    Generating data and using individual model to estimate parameters

1-102-CancerGrowth

    Several models are offered along with data for cancer growth models.

1-105-AnimalFalling

    Comparing terminal velocity for variety of animal’s falling

1-115-ModelingWithFirstOrderODEs

    Several short illustrations and two exercises for modeling

1-120-CircularRollerCoaster

    Modeling a circular roller coaster to determine velocity to stay on track

1-122-SpreadPEV

    Using recent sales data model the spread of plug-in electric vehicles

1-130-AspirinAbsorption

    Pharmacokinetic modeling of absorption of aspirin in body

1-131-CaffeineElimination

   Pharmacokinetic modeling of elmination of caffeine from the body

1-132-DigoxinElimination

   Pharmacokinetic modeling of elmination of digoxin from the body

1-140-LeakyBucket

    Modeling the height of water in a tank with a leak and water pouring in

 

Modeling Two -  Numerically Solving Differential Equations

<Return to Table of Contents>

Narrative and Development

2-001-Numerical Methods Comparisons

    Developing and caring for several numerical methods for first order equations

 

Modeling Three - Second Order Homogeneous Differential Equation Models

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Narrative and Development

    Under development.

Activities

3-001-SpringMassDataAnalysis

    Data on a spring mass system with resistance is given for modeling for analysis

3-002-ModelsMotivatingSecondOrder

    From real data several ways to model spring mass system emerge.

3-004-VanderPol

    Study of van der Pol’s equation with applications and spreadsheet simulation

3-006-Buoyancy

    Data on a bobbing container motivates model and parameter estimation

3-008-HangTime

    Hang Time Modeling

3-010-EnergyInSpringSystems

    Exploring damping and forcing terms to discover energy in system

3-013-WhiffleBallFall

    Using data on whiffle ball fall model resistance and predict the fall position.

3-016-FallingCoffeeFilters

    Using data on stack of coffee filters to build model and estimate parameters.

3-017-StackedCoffeeFilters

    Using data on stacked coffee filter falling from the literature build models.

3-019-ShuttleCock Fall

    Modeling a falling shuttlecock

3-029-FerrisWheelCatch

    We model the throw of an object to a person on a moving Ferris wheel.

3-030-SecondOrderIntro

    Broad introduction to second order homogeneous and nonhomogeneous, linear,

          constant coefficient, differential equations with many applications

3-035-StadiumDesign

    Design a stadium which is fair to home run hitters in all directions.

3-040-FirstPassageTime

    We model and determine the first passage time for underdamped oscillator

3-041-UpDown

    Relate times when projectile passes the same point – up and then down

3-044-DeepWell

            Given total time of pebble fall to sound coming back tell how deep a well

3-045-RampBounce

            Bounce a ball on a tilted ramp for optimal horizontal distance traversed

3-054-Relay

            Place infielder in optimal position for minimum time relay through form outfield

3-065-UpDown

            Modeling vertical projectile motion with resistance to address some issues.

3-070-FallingInWater

            Drop a canister in column of water, collect data from video, model motion

3-075-RLCCircuits

            An introduction to RLC circuits is offered including definitions and modeling.

3-080-PendulumModeling

            Several different pendulum configurations are modeled and compared.

3-085-SimplePendulum

            Modeling pendulum motion and verification of period with data

3-090-OneMassSpring

            Data on a single mass spring system permits modeling of oscillator

3-090-Text-ChebyshevPolynomialSolution

            Small study of Chebyshev Equation for which there are polynomial solutions

3-101-SpringMassFirstTry

            Modeling a simple spring mass with no damping conjecturing solutions

3-102-SpringMassDamped

           Modeling a simple spring mass with damping conjecturing solutions

3-110-MilitarySpringMassApplication

    Modeling the shock absorber system for an Army vehicle.

3-140-TwoSpringOneMassFixedEnds

    Modeling two spring, single mass with fixed ends


Modeling Four - Second Order NonHomogeneous Differential Equation Models

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Narrative and Development

    Under development.

Activities

4-020-AnIEDBlast

     Modeling the effects of an Improvised Explosive Device

4-023-MysteryCircuit

Students assigned various input voltages to a circuit  to see what the circuit is.

4-035-ParEstSteadyState

    Input Output Analysis analyzing steady state to estimate parameters.

4-050-ResonanceBeats

    We study the notions of resonance and beats for undamped system

4-060-CircuitTuner

    Building the differential equation for a radio tuner.

 

Modeling Five - Linear Systems of Differential Equation Models

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Narrative and Development

    Under Development

Activities

5-005-Dialysis

    Modeling Dialysis Machine

5-007-ChemOpt

    Optimization for a Chemical Reaction

5-0012-LipoproteinMetabolism

    Medical study data to build and affirm build a model for low-density-lipoprotein.

5-014-TwoSpringMass

    Build Free Body Diagram and model for two spring configuration.

5-022- ColdPill

    Modeling flow of drug from gastrointestinal tract to bloodstream for peak

5-025-Eviction

    A model for eviction in the United States is built.

5-028-SaltCompartments

    Amount of salt in two water tanks is modeled in several ways

5-030-AirshedSulphur

    Analyzing a model of the production of sulphur compounds in a Montana airshed.

5-040-TunedMassDampers-Part I

    Applying a second mass to keep structure from experiencing resonance

5-040-TunedMassDampers-Part II

    Applying second mass to keep structure from large displacement

5-080-SpaceFlightRecolonize

    Modeling the recolonization of the human race on a distant planet.

 

Modeling Six - NonLinear Systems of Differential Equation Models

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Narrative and Development

    Under Development

Activities

6-001-Epidemic

    English Boarding School NonLethal Influenza

6-002-EulerCromerPendulum

    Using the study of nonlinear pendulum to implement numerical methods

6-004-VillageEpidemic

    Build a model of a mid seventeenth century English village epidemic

6-005-InsectColonyOpt

    Insect Colony Optimal Control

6-007-S-FunctionsAndDerivativesInSIRModels

    Relating functions and derivatives in SIR Models                    

6-008-PursuitModels

    Linearization and Support from Homogeneous System Analysis

6-012-RiverCrossing

    Building a model to help cross a river with current to land at specific spot

6-015- CombatingEbolaEpidemic

    Making policy recommendations from models of spread of Ebola

6-018-ExploringSIRModel

    Modeling rumor and disease spread.

6-020-AlgaePopulationSelf-Replenishment

    Investigate the massive algal blooms that struck Lake Chapala, Mexico

6-021-AcornsRodentsSnakes

    Building a three trophic level model of acorns, rodents, and snakes

6-023-DroneHeadingHome

    Moving against a headwind create model of drone flight to fixed delivery point

6-025-WhalesAndKrill

    Use Excel to observe qualitative behavior a predator-prey model

6-028-SaltCompartments

    Amount of salt in two water tanks is modeled when tank volumes are changing

6-030-SaltTorricelli

    Modeling complex salt levels in a falling column of water.

6-035-Shampoo

    Modeling the amount of shampoo in a bottle during a shower

6-040-StruggleForExistence

    Using historical data to model multiple species growth

6-070-BeerBubbles

    Modeling the rise and size of beer bubbles in a sitting glass of beer

 

Modeling Seven - Changing the Venue for Solution Success - Laplace Transforms

<Return to Table of Contents>

Narrative and Development

    Under Development

Activities

7-005-LaplaceTransformOverview

    Introduction and application of Laplace Transforms

7-008-MachineReplacement

    Laplace Transforms - Convolution Applications - Replacement Theory

7-010-MultipleDoses

    Modeling several multiple dose approaches for drugs is considered.

 

Modeling Eight - Representing Natural Phenomena with Sines and Cosines

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Narrative and Development

8-002-Text-TrigSumRepresentation

    Representing functions as sums of trigonometric functions.

Activities

    Under Development

 

Modeling Nine - Modeling with Differential Equations in Higher Dimensions

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Narrative and Development

9-001-Text-SkinBurnModelNumericalMethods

    Numerical methods for Heat Equation are introduced in context of skin burn issues.

Activities

9-012-PDEGuitarTuning

    Tuning a Stringed Instrument with the Wave Equation

9-020-HorizontalBeam

    Modeling a suspended beam and collecting data to justify the model

 

 

Modeling Ten – Techni que Narrative - General Modeling

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APPENDIX

Strategies for Analytic Solutions and Qualitative Analysis

First Order Differential Equations

1-001-Text-SepVariables

1-002-Text-IntegratingFactor

1-009-Text-Bifurcation

 

Second Order Differential Equations

3-090-Text-ChebyshevPolynomialSolution

 

Linear Systems of  Differential Equations

5-030-Text- LinNonHomoSystemSol

           

NonLinear Systems of  Differential Equations

    Under Development

 

Fourier Series Development

8-002-Text-TrigSumRepresentation

 

Partial Differential Equation Solution Strategies

Under Development