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.