differential eqns_appplications.pdf

One SIMIODE Approach for Teaching Differential Equations

Modeling and Technology Approach

For Teaching Differential Equations

Modeling One

First Order Differential Equations

Narrative and Development

1. Modeling Death with M&M ’s and Simulation

2. Immigration Model with M&M’s

3. Modeling Change Discretely

4. Moving to Differential Equation Model

5. Introducing Mathematica’s DSolve (with modest analytic solutions in appendices as needed)

Activities

1. Modeling Immigration in a Petri Dish

2. Bank Investment Analysis

3. Bank Loan Analysis

4. Saving for Child’s College Education

5. Evaporation and Radioactive Decay Modeling

6. Modeling Sphere of Salt in Solution

7. Modeling the Spread of Oil Slick with Incomplete Data

8. Salt Mixing in Tank Modeling

9. Hang Time Modeling

10. Cooling/Dueling Coffee Modeling

11. Modeling the Emptying of a Column of Water

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

13. Tunnel Construction by Ants and Humans

14. Sublimation of Carbon Dioxide

Modeling OneA

Numerically Solving Differential Equations


1. Differential Equations Which Are Just Impossible!

2. Simple Step Iterating Method

3. Euler’s Method

4. Numerical Experiments

5. Improving on Euler’s Method

6. Runge-Kutta Method

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7. Comparison of Methods with Mathematica’s NDSolve approach

Activities

These are embedded in the development of the numerical methods.

Modeling Two

Limited Growth Models


1. Exponential Growth Population Model

2. Limited Growth Population Modeling

3. Logistic Model Development

4. Estimating Parameter Strategies for Logistic Growth Model

Activities

1. Parameter Estimation in Logistic Growth Models

2. Comparison with Transformed Data vs. Direct Optimization of Sum of Square Error

3. Simulation with M&M’s of Logistic Growth Model for Spread of Disease

4. Analysis of “Struggle for Existence” Paramecia Data from G. F. Gause, Soviet Ecologist

5. Sensitivity Analysis - Data Collection Experiment in Comparing Masses of Rocks

6. Modeling the Spread of Technological Innovation in the United States with Real Data

7. Maximum Sustainable Yield In Harvesting Models

8. Harvesting a Renewable Resources - Some Analysis

9. Running a Catfish Farm

Modeling Three

Second Order Homogeneous Differential Equation Models


1. Spring Mass Dashpot Modeling

2. Modeling with Newton’s Second Law of Motion

3. Free Body Diagram Modeling Tool

4. Using Initial Condition Information for Completion of Solution

5. General Forms of Solutions

6. Characteristic Equation and Roots and Their Significance

7. Complex Roots to Characteristic Equation and Their Meaning

8. Various Types of Damping - Under, Critical, and Over

9. Repeated Roots for Characteristic Equation - Self-Discovery

10. Bookkeeping and Rearrangements - Phase Angle Issues

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Activities

1. Estimating Resistance Parameter in Spring Mass Dashpot Modeling - Three Types of Damped

2. Design with First Passage Times

3. Modeling Parachuting

4. Modeling Falling Stack of Coffee Filters

5. Modeling Frequency of Spring Mass Dashpot Motion

6. Logarithmic Decrement

7. Buoyancy Force Modeling

8. Bad News - Positive Real Part of Characteristic Root

9. Rocket Thrust Modeling

10. Inverse Problems - Ascertaining Function Models and Parameters Given Observational Data

11. Keeping Costs Down - Manufacturing Cheaper Springs with High Performance

12. RLC Circuits - Basic Circuit Notions and Second Order Differential Equation Models

13. Swinging Along - Modeling Massless and Physical Pendula

14. Pendulum Motion and Logarithmic Decrement

Modeling Four

Second Order NonHomogeneous Differential Equation Models


1. Driver for Spring Mass Dashpot Modeling

2. Intelligent Conjecturing for NonHomogeneous Solution

3. Building General Solutions with Homogeneous and NonHomogeneous Parts

4. Transient and Steady State Portions of Solutions

5. Phase Angle for Solutions

6. Using Initial Condition Information for Completion of Solution

7. General Forms of Solutions

Activities

1. RLC Tuner Circuit - Working a Radio

2. Parameter Estimation Through Steady State Data

3. Forced Vibration and No Damping

4. Frequency Response - Maximum Steady State Amplitude

5. Building Swaying - Tuned Mass Dampers

6. Killer Speed Bumps

Modeling Five

Linear Systems of Differential Equation Models


1. Intelligent Conjecturing for Homogeneous Solution

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2. Eigenvalues - Voodoo Dispelled

3. Converting Second Order to System and Noting Meaning of Eigenvalues and Eigenvectors.

4. Building General Solutions with Homogeneous and NonHomogeneous Parts

5. Time Out for Small Motor Skills and Eye Hand Coordination Practice

6. Trial Run at Two Solution Strategies

7. First of Two Two Compartment Model Analyses

8. Second Two Compartment Model Analysis

Activities

1. Two Spring Configuration for Spring Mass Dashpot

2. Modeling Dialysis Machine

3. Mystery Circuit

4. Tuned Mass Dampers

5. Optimization for a Chemical Reaction

6. Stochastic Processes - An Infinite System of Differential Equations

7. Inverse Problems - Ascertaining Function Models and Parameters Given Observational Data

Modeling Six

NonLinear Systems of Differential Equation Models


1. Numerical Solution Strategy

2. Equilibria and Stability Analysis

3. Linearization and Support from Homogeneous System Analysis

4. Nonlinear Ecological Modeling

5. Predator-Prey Modeling

6. Competition Modeling

7. Higher Trophic Level Models

Activities

1. Competition in Ecology

2. Predator Prey Modeling with Satiation and Limited Growth

3. Predator Prey Modeling with Hiding

4. Mimicry in Nature

5. West Point Acorn, Rodent, Rattlesnake Populations

6. 1914 Influenza Epidemic

7. Epidemic Models

8. Flour Beetles Predation with Hiding

9. Predator Prey Modeling and Optimal Control

10. Insect Colony Optimal Control

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Modeling Seven

Changing the Venue for Solution Success

Laplace Transforms


1. Modeling with Spikes and Jumps

2. Transformations in General

3. General Laplace Transforms -Lots of Functions

4. Revisit Second Order Differential Equations and Linear Systems with Laplace Transform

5. Transfer Function Thinking

6. Solution Strategies with Laplace Transforms

7. Living in the Frequency Domain

8. Convolution Applications

Activities

1. RLC Filter Circuit and Laplace Transform ViewPoint

2. Convolution Applications - Replacement Theory

Modeling Eight

Representing Natural Phenomena with Sines and Cosines


1. Approximating Functions with sums of Sine Functions

2. General Fourier Series Modeling

Activities

1. Creating Complex Sounds from Simple Sounds

2. Analyzing Signals with Simple Functions

3. Orthonormal Families and Their Good Times

Modeling Nine

Modeling with Differential Equations in Higher Dimensions


1. Partial Differential Equations

2. Numerical Solutions

3. Modeling Spread of Heat in Limited Environment - Heat Equation

4. Applications of the Heat Equation - Formulating Conditions

5. Extensions of the Heat Equations

6. Making Sound with Waves - Wave Equation

7. Modes

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8. Analytic Solutions - Building for Success

9. Boundary Value Problems and Fourier Analysis - Bringing it All Together

Activities

1. Root Cellar Modeling with Heat Equation

2. Non invasive Analysis with the Heat Equation

3. Tuning a Stringed Instrument with the Wave Equation

4. Orthonormal Families and Their Good Times

APPENDICES

Strategies for Analytic Solutions

1. First Order Differential Equations

i) Separation of Variable

ii) Integrating Factors

iii) Conjecturing

2. Second Order Differential Equations

i) Conjecturing and EigenValues

ii) The Cases for EigenValues

iii) Homogeneous and NonHomogeneous Solutions and Building Final Solutions

3. Series Approach to Differential Equations

i) Conjecturing and Expectation

ii) Difference Equations

iii) Induction and Building Final Solutions

4. Qualitative Approach to Differential Equations

i) Phase Plane

ii) Equilibrium

iii) Stability

iv) Sensitivity

5. Linear Systems of Differential Equations

i) Equilibrium and Stability Issues

ii) Conjecturing, EigenVectors, and EigenValues

iii) Construction of Solutions - Homogeneous and NonHomogeneous Equations

6. NonLinear Systems of Differential Equations

i) Equilibrium and Stability Issues

ii) Linearization and Translation

iii) Some Special Situations - Orbits in Predator Prey Models

7. Fourier Series Development

i) General Formulae

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ii) Extensions - Odd and Even

iii) Orthonormal Family Theory

8. Partial Differential Equation Solution Strategies

i) Separation of Variables

ii) Boundary Value Problems

iii) Appearance of Fourier Series

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