t to be equivalent in the sense of measurement if for all functions gt () with compact support, they integrate in the same way, i.e.

Transcription

1 Cocoure 8 Lecure #5 I oy lecure we begi o re he iuio of lier h orer ODE wih icoiuou /or oiffereible ipu The meho we ll evelop (Lplce Trform) will be pplicble o oher ype of ipu, bu i epecilly relev whe elig wih icoiuou ipu ipu efie oly by umericl The Mi Ie: Begiig wih lier h orer ODE wih iiil coiio ( iiil vlue problem), we ll rform hi io lgebric equio, olve hi equio, he rform bck i orer o prouce oluio o he iiil vlue problem We re oly cocere wih he oluio for > Big Ie #: Geerlize fucio, k fucio i oly goo how i i iegre - i priculr, el fucio ep fucio Big Ie #: We ll evie yemic wy of formlly olvig ODE wih uch ipu, he ue iegrio (covoluio) o prouce oluio o y give iiil vlue problem Suppoe h g () i fucio wih compc uppor, ie i vihe ouie ome cloe boue iervl We woul like o coier wo fucio f () f () o be equivle i he ee of meureme if for ll fucio g () wih compc uppor, hey iegre i he me wy, ie f() g() f() g() Si ifferely, [ f() f()] g() for ll fucio g () wih compc uppor I o hr o ee h for coiuou fucio hi me h ecerily f() f() for ll, bu we re relly ieree i wh hi me for icoiuou fucio fucio wih impule, ie el fucio Heviie fucio, box fucio, el fucio < The Heviie fucio [me for Oliver Heviie (85 95)] i u () For our purpoe i > relly oe mer how i i efie, becue i o relev whe iegrig hi fucio We c < lo efie rle Heviie fucio u() u ( ) Thee fucio c be cle > e o repree fucio correpoig o wichig o off For exmple, we c repree he < fucio f() 4 < < 5 4[ u() u5()] Thi i clle box fucio We c combie box fucio > 5 < ecery, eg () 4 5 g < < 4[ u() u5()] [ u5() u6()] 4 u() u5() u6() < < > 6 The Heviie fucio i co everywhere excep, becue i h jump icoiuiy here we uully ju y h i o iffereible However, we coul heuriiclly oberve h by coierig poi immeiely o he lef righ of he icoiuiy y coiuou pproximio o hi fucio woul hve o hve very lrge lope i he viciiy of We migh le ry o expre hi by yig < h u () δ (), he o-clle el fucio, bu hi oe relly mke much ee i erm of > riiol fucio We my, however, ill be ble o mke ee ou of hi if we ke he view h fucio i oly goo how i i iegre Similrly, u ( ) δ ( ), rle el fucio revie April 7, 9

2 Digreio Lier fuciol meureme Oe of he mo commo hig we o i vecor clculu i fiig he compoe or clr projecio of vecor i R i give irecio The ool ue o ccomplih hi k i he o prouc If u i ui vecor we hve h comp ( v u ) vu Thi i, i fc, lier fucio from R o R, ie v R vu R Thee re clle lier fuciol Iee, he r compoe of vecor v v, v, v i R re meure by oig h v v i, v v j, v If we le L ( v) comp ( v) vu, we ee h u v k uig he r ui vecor bi for L( c v + c v ) ( c v + c v ) u c v u+ c v u c L( v ) + c L( v ), o L i lier The Fourier coefficie re ju he meure of how much of give perioic fucio i ocie wih ech moe I relly o iffere h clculig he compoe of vecor i pecific irecio π π f () f ( )co π π π b ( )i π π π f π I hi ce, ech of he clculio of Fourier coefficie ke (perioic) fucio prouce rel π umber, eg L( f ) f ( )co π π Noe h ju w he ce wih vecor o prouc, L c lierly, ie π π π L( cf+ c f) π [ cf() + cf()]co c π f()co + c π f()co c L( f ) + c L( f ) R ( ) ( ) π π π So L i lo lier fuciol, hough i hi ce i ke fucio prouce rel umber There i, however, oe lier fuciol, rgubly he imple imgible oe, h we o uully hik of i erm of iegrio (hough mybe we houl), mely evluio Specificlly, if f() i fucio, we c, for y pecific vlue, coier L ( f) f( ) I quie imple o ee h L( cf + cf ) ( cf + cf )( ) cf ( ) + cf ( ) cl ( f) + cl ( f), o L i lier fuciol I i o efie i erm of iegrio, bu we will fi i ueful o o o oehele Geerlize fucio You c heuriiclly hik of he ep fucio u () y ice mooh fucio which i for < ε for > ε, where ε i poiive umber which i much mller h y ime cle we cre bou i he coex we re uyig he mome Similrly, he be wy for you o uer he el fucio (efie below) i o hik of i y mooh fucio which i zero excep i he immeie eighborhoo of which h iegrl A we ll ee, we c lo hik of he el fucio δ () δ ( ) he fucio you iegre gi i orer o evlue fucio repecively y Th i, f () () f () f () ( ) f ( ) How c we mke ee of hi? Mkig he mo of iegrio by pr I fir-yer Clculu we lere h for iffereible fucio u () v () he Prouc Rule pplie, ie [ uv () ()] uv () () + vu () () O y fiie iervl [ b, ] we c iegre boh ie of he Prouc Rule hi pply he Fumel b b b Though Theorem of Clculu o ge h ubvb ( ) ( ) uv ( ) ( ) [ uv () ()] uv () () + vu () () revie April 7, 9

3 we ofe hik of Iegrio by Pr efie formlly by uv uv vu, he e reul i relly wh b b b hi me, ie we c y h u () v () [ uv ] v () u () If oe of hee fucio h compc uppor, ie if i vihe ouie of ome cloe, boue iervl, he we c exe he reul o he eire rel lie implify he eme coierbly (ice he vlue of he prouc of he fucio will vih ouie ome iervl Specificlly, if g () h compc uppor if f() + + i y fucio, we c y h f () g() f () g () We c cully ue hi o efie erivive f () i geerlize wy, ie geerlize erivive I re o he oio h fucio c be ueroo by how hey re iegre o ju by how hey re evlue Thee geerlize fucio re lo kow iribuio Perhp he mo impor illurio of hi i he geerlize erivive of he Heviie fucio u () We formlly clle hi he el fucio u () δ () eve hough i i relly mke ee he poi of icoiuiy for he Heviie fucio However, we c y h if u () δ () he for y fucio g () wih compc uppor: () g() u () g() u() g () g () [ g()] g() Th i, if we iegre fucio gi he el fucio, hi i imply evluio of h fucio I i hi relly hi propery h efie he el fucio geerlize fucio Similrly, we c o he me for he rle Heviie fucio u () u ( ) o coclue h i geerlize erivive δ () δ ( ) i uch h for y fucio g () wih compc uppor: + + g() () g() ( ) g( ) You c lo ke equeil pproch o mke ee of hi i erm of limi, ie if you ucceively pproxime he el fucio by equece of coiuou fucio fk () where he uppor (omi where i ozero) ge rrower [ ε, + ε ] he vlue grow reciproclly i uch wy h ech ep he k k + + εk k ε k k (we cll uch fucio probbiliy eiie), he you c how iegrl i lwy f () f () + h lim g() fk () g() k Noe: The Fumel Theorem of Clculu well ll he uul rule of iffereiio lo pply o geerlize erivive, o we cully hve geerlize clculu for elig wih hee geerlize fucio or iribuio (hough i my ke while geig ue o i) Biclly, we exe he uul rule of iffereiio o geerlize fucio ogeher wih he fc h u ( ) δ ( ) A fucio f() i regulr or piecewie mooh if i c be broke io piece ech hvig ll higher ( erivive uch h ech brekpoi ) ( f ( ) f ) ( + ) exi A igulriy fucio i lier combiio of hife el fucio A geerlize fucio f() i um f() fr() + f() of regulr fucio igulriy fucio Ay regulr fucio f() h geerlize erivive f (), wih regulr pr f r () he regulr erivive of f() wherever i exi, igulr pr f () give by um of erm ( f( + ) f( )) δ ( ) ru over he icoiuiie of f revie April 7, 9

4 Now, o ge bck o he Mi Ie, how c we olve lier iffereil equio [ pd ( )] x () q () by rformig i io lgebric equio, olvig h lgebric equio, he rformig bck o prouce oluio o iiil vlue problem? A we will oly be cocere wih forwr ime, we ll preume h q () ifie q () for < The Lplce Trform Defiiio: The Lplce rform of fucio f() i efie by L [ f()] F( ) e f() where he ew (complex) vrible i uch h i rel pr Re( ) (he iegrl woul oherwie o coverge) Noe h he lower limi of he iegrl iice h i iclue i iee o re poeil icoiuiie el fucio We will liberlly mke ue of he coveio h fucio of will be repreee by lower ce me i Lplce rform by he correpoig upper ce me, eg L [ x ( )] X( ) Lieriy Becue he Lplce rform i efie iegrl, i ey o ee h L[ f () + bg()] L[ f ()] + bl [ g()] F( ) + bg() Specificlly, L[ f () + bg()] e [ f () + bg()] e f () + b e g() L[ f ()] + bl[ g()] F() + bg() Thi will permi u o rform iffereil equio erm-by-erm ( rform bck well) Ivere rform: F () eeilly eermie f() for Thi will geerlly llow u o prouce oluio o give Iiil Vlue Problem by imply recogizig, erm by erm, oluio by ieifyig which fucio gve rie o ech erm of he rforme iffereil equio Some Clculio ) For our purpoe, ice we re oly cocere wih, he co fucio f() he Heviie < fucio u () re iiiguihble Thu > e [] [ u( )] e L L + Here we ue he fc h for >, lim e Iee, hi i ill he ce eve if we permi o be complex wih poiive rel pr, ie Re( ) > ) If f() ) If f() e, we clcule F( ) L [ ] e e [] + + L, we clcule Thi, ogeher wih lieriy, eble u o clcule he Lplce rform of y polyomil fucio 4 e F() L [ ] e + e + [] L 4) -erivive rule: L [ f( )] F ( ) We c eblih hi by oig h if he L, o [ f( )] F ( ) F () e f() e f() [ f()] From hi we ee h L[ ] L[ ] L [] ;! 4! L 4 L[ ] L[ ] L [ ] 4 5 ; o o Geerlly, + F( ) L [ f()] e f(),! L[ ] L[ ] L [ ] ; 4! L [ ] revie April 7, 9

5 5) If f() e i expoeil fucio (relly f() ue () ice we re oly cocere wih ), ( ) ( ) e L [ e ] e e e ( ), o r 6) -hif rule: L [ e f ( )] F ( r) To eblih hi, we clcule r r ( r) [ e f()] e e f() e f() F( r) L [ e ] L imply by oig he ubiuio 7) Trformig erivive: For y geerlize fucio, L [ f ( )] F( ) f ( ) where f ( ) repree he iiil vlue of f() The uuul oio i here becue we will be elig wih icoiuou geerlize fucio where we my ee o iiguih lef-h from righ-h limi We c eblih hi -erivive rule by oig h L [ f ()] e f () If we ue Iegrio by Pr wih u e v f (), we ge u e v f(), o [ ()] () () L f e f e f + e f () [] + F () F () f ( ) For eco erivive, oe h f () f (), o we c pply he bove reul o ge h L [ f( )] L[ f( )] f( ) F ( ( ) f( )) f( ) F ( ) f( ) f( ) L [ f ( )] F( ) f( ) f( ), o Coiuig, we ge h L [ f ( )] F( ) f( ) f( ) f ( ), o o Geerlly, L f F f f f ( ) ( ) [ ( )] ( ) ( ) ( ) ( ) 8) Trformig he el fucio: Oe of our mo fumel rform i L [ δ ( )] Thi i relively ey o ee oce you re comforble wih he iegrl formlim cocerig he el fucio how hey rele o evluio Specificlly, L [ ()] e () e ice hi i relly ju evluio of he fucio e 9) Trformig ie coie: L [co( )] L [i( )] + + We c erive ech of hee iepeely, bu if we ue Euler Formul lieriy we hve h: i L[ e ] L[co( ) + ii( )] L[co( )] + il [i( )], i i i [ e + ] + i L + i + i Tkig rel imgiry pr eprely we ge h L [co( )] L [i( )] + + We ll o hi li we go he ee rie Exmple: Solve he Iiil Vlue Problem x+ x + x e, x(), x () Ol Fihful Soluio: The homogeeou equio x+ x + x i ey o olve I chrceriic polyomil i p ( ) + + ( + )( + ) which yiel he wo roo Thi give he wo iepee oluio e e, ll homogeeou oluio re of he form xh () ce + ce Noe h boh of hee homogeeou oluio re rie i he ee h hey ecy expoeilly icree 5 revie April 7, 9

6 Nex, we ee o fi priculr oluio x () h ifie he ihomogeeou iffereil equio Oe p look he righ-h-ie we ee h he Expoeil Repoe Formul (ERF) wo work here i reoce We c, however, ue he Reo Repoe Formul o ge he priculr oluio e e xp () e, o he geerl oluio i x() xh() + xp() ce + ce + e I erivive p ( ) i x x() c+ c () ce ce e + e Subiuig he (re) iiil coiio give x() c c +, hee c be olve o give c, c, o he oluio i x e e + e () Solvig irecly by Lplce rform: We clcule he followig Lplce rform: () L k ( e ) k wih regio of covergece Re( ) > k, o L ( e ) + () If he Lplce rform of x () i X(), he he Lplce rform of i erivive re L( x ( )) X ( ) x( ) L( x( )) X( ) x( ) x ( ) I he ce of re iiil coiio x( ) x ( ), hee re grely implifie, i fc L ( pdx ( )) px () () Specificlly, L ( x+ x+ ) x X() + X() + X() ( + + ) X() p() X() If we ow rform he eire iffereil equio, we ge ( + + ) X( ) + A B C We he olve for X() + + ( + )( + + ) ( + )( + ) + + ( + ) There re my goo wy o fi he ukow A, B, C For exmple, if we muliply hrough by he commo eomior o cler frcio, we ge A ( + ) + B ( + )( + ) + C ( + ) Pluggig i he pecific vlue quickly yiel h A C Pluggig i, for exmple, uig he vlue for A C he yiel B So X() ( + ) + + ( + ) Coulig our ble of commo Lplce rform, we ee h ( + ) L ( e ), o rformig bck (uig lieriy) give + L ( e ), L ( e ), + x() e e + e 6 revie April 7, 9

7 Properie of he Lplce rform Defiiio: L [ f()] F() e f() for Re( ) Lieriy: L[ f () + bg()] L[ f ()] + bl [ g()] F() + bg() Ivere rform: F () eeilly eermie f() r -hif rule: L [ e f ( )] F ( r) 4 -hif rule: L [ f ( )] e F ( ) if f() for < Thi my lo be expree L [ f( )] e F ( ) where 5 -erivive rule: L [ f( )] F ( ) 6 -erivive rule: L [ f ( )] F( ) f ( ) L [ f ( )] F( ) f ( ) f ( ) L ( ) ( ) [ f ( )] F ( ) f( ) f( ) f ( ) 7 Covoluio rule: L [ f() g ()] FG ( ) ( ), f( ) if > f ( ) u( ) f( ) if < ( f g)( ) f( ) g( ) 8 Weigh fucio: L [ w ( )] W( ), w () he ui impule repoe If q () i regre he ipu igl i pdx ( ) q (), W() p () Formul for he Lplce rform L [] L [ δ ( )] L[ δ( )] L [ δ ( )] e e L[ u ( )] L [ u ( )] L [ e ] L []! L [ ] + ( ) L [ f( )] ( ) F ( ) L [ u( ) f( )] e F( ) L[ u( ) f( )] e L [ f( + )] L [co( )] + L [i( )] + L [ co( )] ( + ) L [ i( )] ( + ) [ co( )] L e z z ( z) + z L [ e i( )] ( z) + Noe by Rober Wier 7 revie April 7, 9