integration with delta
TRANSCRIPT
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Delta
Functions:
Unit
Impulse
1.
Introduction
Inourdiscussionoftheunitstepfunctionu(t)wesawthatitwasanidealizedmodelofa quantitythatgoes from0 to1 veryquickly. Intheidealizationweassumeditjumpeddirectlyfrom0to1innotime.
Inthisnotewewillhaveanidealizedmodelofalargeinputthatactsoverashorttime. Wewillcallthismodelthedelta
function
orDirac
delta
functionorunitimpulse.
Afterconstructingthedeltafunctionwewilllookatitsproperties.Thefirstisthatitisnotreallyafunction. Thiswontbotherus,wewillsimply
callitageneralized
function. Thereasonitwontbotherusisthatthedeltafunctionisusefulandeasytoworkwith. Insideintegralsorasinputtodifferentialequationswewillseethatitismuchsimplerthanalmostanyotherfunction.
2.
Delta
Function
as
Idealized
Input
Supposethatradioactivematerialisdumpedinacontainer. Theequa-tiongoverningtheamountofmaterialinthetankis
.x+kx=q(t),
where,x(t)
istheamountofradioactivematerial(inkg),k
isthedecayrateofthematerial(in1/year), andq(t)
istherateatwhichmaterialisbeingaddedtothedump(inkg/year).
The input q(t) is in units of mass/time, say kg/year. So, the totalamountdumpedintothecontainerfromtime0totimetis
t
Q(t) = q(u)du.0
Equivalently.Q(t) =
q(t).
Tokeepthingssimplewewillassumethatq(t)
isonlynonzeroforashortamount of time and that the total amount of radioactivematerialdumpedoverthatperiodis1kg. Herearethegraphsoftwopossibilitiesforq(t)andQ(t).
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DeltaFunctions:UnitImpulse OCW18.03SC
t1/2
2
q(t) =
t1/2
1
Q(t)
t
1/8
8
q(t) =
t
1/8
1
Q(t)
Figure1:twopossiblegraphsofq(t)andQ(t),bothwithtotalinput=1.
ItiseasytoseethateachoftheboxesontheleftsideofFigure1hastotalareaequalto1. Thus,thegraphsforQ(t)riselinearlyto1andthenstayequalto1thereafter.Inotherwords,thetotalamountdumpedineachcaseis1.
Nowletqh(t)beaboxofwidthhandheight1/h.Ash 0,thewidthoftheboxbecomes0,thegraphlooksmoreandmorelikeaspike,yetitstillhasarea1(seeFigure2).
t1
1
h= 1
t1
2
2
h= 1/2
t1
16
16
h= 1/16
t
h 0, q(t) =(t)
Figure2:Boxfunctionsqh(t)becomingthedeltafunctionash 0.
Wedefinethedeltafunctiontobetheformallimit
(t) =
limqh(t).h 0
Graphically(t) isrepresentedasaspikeorharpoonatt = 0. It is aninfinitelytallspikeofinfinitesimalwidthenclosingatotalareaof1(seefigure2,rightmostgraph).
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DeltaFunctions:UnitImpulse OCW18.03SC
Asaninputfunction(t)representstheidealcasewhere1unitofma-
terialisdumpedinallatonceattimet
=
0.
3.
Properties
of(t)
Welistthepropertiesof (t)below.
1. Fromtheprevioussectionwehave
0 ift=0,(t) =
ift=0.
Thegraphisrepresentedasaspikeatt=0.(Seefigure2
2. Because(t)
isthelimitofgraphsofarea1,theareaunderitsgraphis1.Moreprecisely:
d
1 ifc
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DeltaFunctions:UnitImpulse OCW18.03SC
6. Wedefined(t)asalimitofasequenceofboxfunctions,allwithunit
areaandwhich,inthelimit,becomeainfinitespikeover t
=
0. Boxfunctionsaresimple,butnotspecial. Anysequenceof functionswiththesepropertieshas(t)asitslimit.
7. Inpracticalterms,youshouldthinkof(t)asanyfunctionofunitarea,concentratedveryneart
=0.
8. (t)isnotreallyafunction.Wecallitageneralizedfunction.
9. Inarrivingat thesepropertieswehave skippedover someimportanttechnicaldetailsintheanalysis.Generallyproperty(3)istakentobetheformaldefinitionof(t),fromwhichtheotherpropertiesfollow.
4.
Examples
of
integration
Properties(3)and(2)showthat (t) isveryeasytointegrate,asthefollowingexamplesshow:
5
Example
1. 7et
2cos(t)(t)dt=7.Allwehadtodowasevaluatethe
integrandatt
=5
0. 5
Example2. 7et2
cos(t)(t2)dt=7e4cos(2).Allwehadtodowas5
evaluatetheintegrandatt=2.
1Example
3.
7et2
cos(t)(t
2)
dt
=
0.Sincet
=
2isnotintheinterval5
ofintegrationtheintegrandis0ontheentireinterval.
Thevaluet
=0 representstheleftsideof0andt
=0+ istherightside.So,0isintheinterval[0,)andnotin[0+,).Thus
(t)dt=1 and (t)dt=0.0 0+
Infact,sincealltheareaunderthegraphisconcentratedat0,wecanevenwrite
0+
(t)dt=1.
0
5.
Generalized
Derivatives
Ourgoalinthissectionistoexplainproperty(5). Alookatthegraphoftheunitstepfunction u(t)showsthatithasslope0everywhereexcept
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DeltaFunctions:UnitImpulse OCW18.03SC
att=0andthatitsslopeisatt=0.
t
1u(t)
Thatis,itsderivativeis
u(t) =0 ift=0 ift=0.
Sinceu(t)hasajumpof1att = 0thisderivativematchesproperties(1)and(2)of(t)andweconcludethatu(t) =(t).
Nowthisderivativedoesnotexistinthecalculussense. Thefunctionu(t)isnotevendefinedat0.Sowecallthisderivativeageneralizedderiva-tive.
Wecanalsoexplainproperty (5)bylookingat theantiderivative of(t).Let
t
f(t) = ()d.
The fundamental theoremof calculusleadsus tosay that f(t) = (t).(Again,thisisonlyinageneralizedsensesincetechnicallythefundamentaltheoremofcalculusrequirestheintegrandtobecontinuous.)Property(3)
makesiteasytocompute
0 ift0.
Thatis,f(t) =u(t),sou(t)istheantiderivativeof(t).
Ingeneral,ajumpdiscontinuitycontributesadeltafunctiontothegen-eralizedderivative.
Example
4.
Supposef
(t)hasthefollowinggraph.
t
f(t) = t2
f(t) = 2
f(t) = 3t 7
2
-1
2
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Theformulaforeachpieceofthegraphisindicated.Forthesmoothparts
ofthegraphthederivativeisjusttheusualone. Eachjumpdiscontinuityaddsadeltafunctionscaledbythesizeofthejumptof(t).
2t ift
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18.03SC Differential Equations
Fall 2011
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