Using MathUsing Mathtoto

Predict the FuturePredict the Future

Differential Equation ModelsDifferential Equation Models

OverviewOverview Observe systems, model relationships, Observe systems, model relationships,

predict future evolutionpredict future evolution One KEY tool: Differential EquationsOne KEY tool: Differential Equations Most significant application of calculusMost significant application of calculus Remarkably effectiveRemarkably effective Unbelievably effective in some Unbelievably effective in some

applicationsapplications Some systems inherently unpredictableSome systems inherently unpredictable WeatherWeather ChaosChaos

Mars Global Mars Global SurveyorSurveyor

Launched 11/7/96Launched 11/7/96 10 month, 435 million mile trip10 month, 435 million mile trip Final 22 minute rocket firingFinal 22 minute rocket firing Stable orbit around Mars Stable orbit around Mars

Mars Rover MissionsMars Rover Missions

7 month, 320 7 month, 320 million mile tripmillion mile trip

3 stage launch 3 stage launch programprogram

Exit Earth orbit at 23,000 mphExit Earth orbit at 23,000 mph 3 trajectory corrections en route3 trajectory corrections en route Final destination: soft landing on MarsFinal destination: soft landing on Mars

Interplanetary GolfInterplanetary Golf

Comparable shot in miniature golfComparable shot in miniature golf 14,000 miles to the pin14,000 miles to the pin

more than half way around the more than half way around the equatorequator

Uphill all the wayUphill all the way Hit a moving targetHit a moving target T off from a spinning merry-go-roundT off from a spinning merry-go-round

Course CorrectionsCourse Corrections

3 corrections in cruise phase3 corrections in cruise phase Location measurementsLocation measurements Radio Ranging to EarthRadio Ranging to Earth

Accurate to 30 feetAccurate to 30 feet Reference to sun and starsReference to sun and stars Position accurate to 1 part in 200 Position accurate to 1 part in 200

million -- 99.9999995% accuratemillion -- 99.9999995% accurate

How is this possible?How is this possible?

One word answer:One word answer:

Differential EquationsDifferential Equations

(OK, 2 words, so sue me)(OK, 2 words, so sue me)

ReductionismReductionism

Highly simplified crude Highly simplified crude approximationapproximation

Refine to microscopic scaleRefine to microscopic scale In the limit, answer is exactly rightIn the limit, answer is exactly right Right in a theoretical senseRight in a theoretical sense Practical Significance: highly Practical Significance: highly

effective means for constructing and effective means for constructing and refining mathematical modelsrefining mathematical models

Tank Model ExampleTank Model Example

100 gal water tank100 gal water tank Initial Condition: 5 pounds of salt Initial Condition: 5 pounds of salt

dissolved in waterdissolved in water Inflow: pure water 10 gal per minuteInflow: pure water 10 gal per minute Outflow: mixture, 10 gal per minuteOutflow: mixture, 10 gal per minute Problem: model the amount of salt in Problem: model the amount of salt in

the tank as a function of timethe tank as a function of time

In one minute …In one minute …

Start with 5 pounds of salt in the Start with 5 pounds of salt in the waterwater

10 gals of the mixture flows out10 gals of the mixture flows out That is 1/10 of the tankThat is 1/10 of the tank Lose 1/10 of the salt or .5 poundsLose 1/10 of the salt or .5 pounds Change in amount of salt is –(.1)5 Change in amount of salt is –(.1)5

poundspounds Summary: Summary: t = 1, t = 1, s = -(.1)(5)s = -(.1)(5)

CritiqueCritique Water flows in and out of the tank Water flows in and out of the tank

continuously, mixing in the processcontinuously, mixing in the process During the minute in question, the During the minute in question, the

amount of salt in the tank will varyamount of salt in the tank will vary Water flowing out at the end of the Water flowing out at the end of the

minute is less salty than water minute is less salty than water flowing out at the startflowing out at the start

Total amount of salt that is removed Total amount of salt that is removed will be less than .5 poundswill be less than .5 pounds

Improvement: ½ minuteImprovement: ½ minute

In .5 minutes, water flow is .5(10) = In .5 minutes, water flow is .5(10) = 5 gals5 gals

IOW: in .5 minutes replace .5(1/10) of IOW: in .5 minutes replace .5(1/10) of the tankthe tank

Lose .5(1/10)(5 pounds) of saltLose .5(1/10)(5 pounds) of salt Summary: Summary: t = .5, t = .5, s = -.5(.1)(5)s = -.5(.1)(5)

This is still approximate, but betterThis is still approximate, but better

Improvement: .01 minuteImprovement: .01 minute

In .01 minutes, water flow is .01(10) In .01 minutes, water flow is .01(10) = .01(1/10) of full tank= .01(1/10) of full tank

IOW: in .01 minutes replace .01(1/10) IOW: in .01 minutes replace .01(1/10) = .001 of the tank= .001 of the tank

Lose .01(1/10)(5 pounds) of saltLose .01(1/10)(5 pounds) of salt Summary: Summary: t = .01, t = .01, s = -.01(.1)(5)s = -.01(.1)(5)

This is still approximate, but even betterThis is still approximate, but even better

Summarize resultsSummarize resultst (minutes)t (minutes) s (pounds)s (pounds)

11 -1(.1)(5)-1(.1)(5)

.5.5 -.5(.1)(5)-.5(.1)(5)

.01.01 -.01(.1)(5)-.01(.1)(5)

Summarize resultsSummarize resultst (minutes)t (minutes) s (pounds)s (pounds)

11 -1(.1)(5)-1(.1)(5)

.5.5 -.5(.1)(5)-.5(.1)(5)

.01.01 -.01(.1)(5)-.01(.1)(5)

hh -h(.1)(5)-h(.1)(5)

)5(1.

t

s Limit )5(1.

dt

ds

Other TimesOther Times

So far, everything is at time 0So far, everything is at time 0 s = 5 pounds at that times = 5 pounds at that time What about another time?What about another time? Redo the analysis assuming 3 Redo the analysis assuming 3

pounds of salt in the tankpounds of salt in the tank Final conclusion:Final conclusion:

)3(1.dt

ds

So at any time…So at any time…

If the amount of salt is s,If the amount of salt is s,

)(1. sdt

ds

We still don’t know a formula for We still don’t know a formula for ss((tt))

But we do know that this unknown But we do know that this unknown function must be related to its own function must be related to its own derivative in a particular way.derivative in a particular way.

Differential EquationDifferential Equation

Function Function ss((tt) is unknown) is unknown It must satisfy It must satisfy s’ s’ ((tt) = -.1 ) = -.1 ss((tt) ) Also know Also know ss(0) = 5(0) = 5 That is enough information to That is enough information to

completely determine the function:completely determine the function:

ss((tt) = 5) = 5ee-.1-.1tt

DerivationDerivation Want an unknown function Want an unknown function ss((tt) with ) with

the property that the property that s’ s’ ((tt) = -.1 ) = -.1 ss((tt) ) Reformulation: Reformulation: s’ s’ ((tt) / ) / ss((tt) = -.1) = -.1 Remember that pattern – the Remember that pattern – the

derivative divided by the function?derivative divided by the function? (ln (ln ss(t))’ = -.1(t))’ = -.1 (ln (ln ss(t)) = -.1 (t)) = -.1 t + Ct + C s s ((tt) = ) = ee-.1-.1t + C t + C = = ee C C ee-.1-.1t t =Ae=Ae-.1-.1tt

s s ((00) ) =Ae=Ae0 0 =A=A Also know Also know ss(0) = 5. So (0) = 5. So s s ((tt) = 5) = 5ee-.1-.1t t

Relative Growth RateRelative Growth Rate In tank model, In tank model, s’ s’ ((tt) / ) / ss((tt) = -.1) = -.1 In general In general f’ f’ ((tt) / ) / ff((tt) is called the ) is called the

relative growth rate relative growth rate of of ff . . AKA the percent growth rate – gives AKA the percent growth rate – gives

rate of growth as a percentagerate of growth as a percentage In tank model, relative growth rate is In tank model, relative growth rate is

constantconstant Constant relative growth Constant relative growth rr always leads always leads

to an exponential function to an exponential function AeAertrt

In section 3.8, this is used to model In section 3.8, this is used to model population growthpopulation growth

Required KnowledgeRequired Knowledgeto set up and solve differential equationsto set up and solve differential equations

Basic concepts of derivative as Basic concepts of derivative as instantaneous rate of changeinstantaneous rate of change

Conceptual or physical model for how Conceptual or physical model for how something changes over timesomething changes over time

Detailed knowledge of patterns of Detailed knowledge of patterns of derivativesderivatives

Applications of Tank ModelApplications of Tank Model

Other substances than saltOther substances than salt Incorporate additions as well as Incorporate additions as well as

reductions of the substance over reductions of the substance over timetime

Pollutants in a lakePollutants in a lake Chemical reactionsChemical reactions Metabolization of medicationsMetabolization of medications Heat flowHeat flow

Miraculous!Miraculous!

Start with simple yet plausible modelStart with simple yet plausible model Refine through limit concept to an Refine through limit concept to an

exact equation about derivativeexact equation about derivative Obtain an exact prediction of the Obtain an exact prediction of the

function for all timefunction for all time This method has been found over This method has been found over

years of application to work years of application to work incredibly, impossibly wellincredibly, impossibly well

On the other hand…On the other hand… In some applications the method does not In some applications the method does not

seem to work at allseem to work at all We now know that the We now know that the formform of the of the

differential equation matters a great dealdifferential equation matters a great deal For certain forms of equation, theoretical For certain forms of equation, theoretical

models can never give accurate models can never give accurate predictions of realitypredictions of reality

The study of when this occurs and what The study of when this occurs and what (if anything) to do is part of the subject of (if anything) to do is part of the subject of CHAOS.CHAOS.