An Introduction to Asymptotic Expansions

Transcription

1 A Itroductio to Asmptotic Expasios R. Shaar Subramaia Dpartmt o Chmical ad Biomolcular Egirig Clarso Uivrsit Asmptotic xpasios ar usd i aalsis to dscrib th bhavior o a uctio i a limitig situatio. Wh a uctio ( x, ε dpds o a small paramtr ε, ad th solutio o th govrig quatio or this uctio is ow wh ε =, a prturbatio mthod ma prov usul i obtaiig a solutio or small valus o ε. Such a approach is particularl attractiv wh th govrig quatio is oliar ad o gral tchiqus ar availabl or xact solutio. I ε appars as a multiplicativ actor i a trm i th govrig quatio, th stadard approach is to tr a powr sris solutio o th ollowig orm: ( ε ε ε x, = x + x + x +... ( whr th smbol stads or highr ordr trms. Th sris is isrtd ito th govrig quatio ad boudar coditios, ad coicits o li powrs o ε ar th groupd to obtai a sris o quatios or th coicit uctios j ( x, which ar th solvd i a squtial mar. Th rsultig sris d ot covrg or a valu o ε ; vrthlss, th x, ε wh ε is small. solutio ca b usul i approximatig th uctio Covrgt ad Asmptotic Sris Computatioall, a covrgt sris a j is ot alwas usul, bcaus covrgc is a j= cocpt rlatig to th bhavior o th trms i th sris at th tail d, that is, as j. That a sris covrgs sas othig about how rapidl th trms will dcras i magitud. O th othr had, i a asmptotic sris, th trms will usuall dcras rapidl with j at irst or suicitl small ε. Somtims, th ma bgi to icras with icrasig j at som poit atr dcrasig iitiall. Wh th trms ar dcrasig rapidl, i w sum just th irst w trms ad w ow that th rror icurrd is o th ordr o th xt trm, w ca gt a good stimat o th sum. This is wh asmptotic sris, v wh divrgt, ar practicall usul. Th mai problm with asmptotic sris is that o vr ows how accurat th aswr is. Th rsults must b validatd b compariso with som othr rprstatio o th xpctd aswr. Nvrthlss, asmptotic sris ma b th ol mas o obtaiig a aaltical solutio o a diicult problm, ad ar usd commol or this purpos. To illustrat th idas rgardig computatioal utilit, writ a computr program to sum th irst trms i th Talor sris or si θ giv blow, irst or θ =.5, ad th or θ = 4. + θ si θ = ( ( ( +! =

2 4 Th sris is ow to b uiorml covrgt or all valus o θ. Th rsult or θ = rlcts th act that th prcisio o machi computatio is iit. Errors itroducd b th limitd prcisio lad to a absurd rsult wh th sum is calculatd. You ca s a dmostratio o this b pritig ach trm ad th sum as succssiv trms ar addd. Now, cosidr th ollowig sris or th complmtar rror uctio. x rc( x~ + x π = 5 ( ( ( x This sris divrgs or all valus o x. It is a asmptotic sris that rprsts th uctio i th limit as x. I spit o its divrgc, it is usul or computig th complmtar rror uctio or larg valus o x, bcaus th trms i th sris dcras rapidl with icrasig or small valus o ad th rror icurrd b trucatig th sris at a crtai trm is o th ordr o magitud o th xt trm, which is much smallr tha th trm rtaid as log as x is larg ad w us ol a small umbr o trms. Tr calculatig rsults rom this sris or x = 5,, ad ad chc th sum atr addig ach trm agaist th xact rsult. Also, s i ou ca dmostrat to oursl that this is a divrgt sris. Som Basic Cocpts Som basic cocpts i usig asmptotic sris ar dscribd xt. Two smbols ar commol usd to dscrib th bhavior o a uctio ( ε i th limit as ε. Th ar " O " ad " o" ad ar trmd big oh ad littl oh. I w hav two dirt uctios o ε, aml ( ε ad g ( ε, w sa that ( ε = O( g( ε i lim ε ε g ( ε < (4 g ε. I words, this is statd as ollows: ( ε is o th ordr o I th limit is zro, th th smbol o is usd. ( ε o( g( ε ( ε ( ε = i lim = (5 ε g I th abov, th uctio g ( ε is trmd a gaug uctio. A commo st o gaug uctios is th st o powrs o psilo (, εε,,.... Ths powrs ar ot usd to dscrib th bhavior o som othr uctio o psilo. For xampl, w ma writ siε ε (6

3 which should b rad as si psilo is asmptoticall qual to psilo. Ev though th phras as ε approachs zro is omittd, it is implid. O cours, ta ε ε (7 at ladig ordr, so that w s that dirt uctios ca hav idtical asmptotic rprstatios. Powr sris ar just o tp o asmptotic sris. A mor gral asmptotic sris or a x, ε is o th orm uctio N (, ε ( ε ( ε x = (8 = Not that w hav trmiatd th sris at a iit valu o th idx. Thror, covrgc is ot a issu hr. Th uctios ( ε must satis + ( ε lim =, =,,,... (9 ε ε This mas that ach mmbr o th st o uctios approachs zro mor rapidl tha th prvious mmbr as ε. W call th st o uctios { ( ε } a asmptotic squc i th mmbrs satis th coditio giv i Equatio (9. Not that th st o powrs o psilo is idd a asmptotic squc. Th coicit uctios ( x ca b dtrmid uiqul rom th proprt o th mmbrs o a asmptotic squc otd abov. First, b dividig both sids o Equatio (8 b ( ε ad taig th limit as ε, w obtai th ollowig rsult or th ladig ordr coicit x. ( x ( ε ( ε = lim x, ( ε Now, subtract ( ε ( x rom both sids o Equatio (8, divid b as ε. This ilds ( x ε (, ε ( ε ( ε ( ε ε, ad ta th limit = lim x ( Usig this procdur, it is straightorward to show that th coicit uctio asmptotic sris ca b writt as j x i th

4 j j (, ε ( ε ( ε x x = j = = lim,,,,... ε j ( ε ( Th coicits i th asmptotic sris or a giv uctio dpd o th choic o th squc; oc th squc is did, th coicits ar uiqul dtrmid b Equatios ( ad (. I a giv problm, w usuall do ot ow th dpdc o ( x, ε o ε so that th rsults i ths quatios should b rgardd ol as ormal diitios o th coicit uctios. Nxt, w dmostrat how ths coicits ar dtrmid i xampl cass. A Itgral Cosidr th itgral I ( ε did as show blow. t I ( ε = dt ( + εt Procdig to itgrat b parts, w obtai t t ε I ( ε = { } { } dt + εt ( + εt t = ε dt ( + εt (4 { t } { t } ε = ε dt + εt ( + εt t = (! ε + (! ε dt + εt Cotiuig to itgrat b parts i this mar, w ca show that As a iiit sris, I = (! + (!... + ( ([ ]! ε ε ε ε ( (! + I ε t ( + εt + ( ε = ( (! dt (5 ε (6 = is divrgt or all valus o ε. But, or rlativl small valus o ε, th sris i Equatio (6, trucatd atr a small umbr o trms, provids a good approximatio o th itgral. 4

5 A Dirtial Equatio Cosidr th dirtial quatio alog with th iitial coditio W ow that th solutio is + ε = (7 = (8 = ε x (9 Lt us s how a asmptotic xpasio ca b dvlopd or ( x, ε. Writ ε x = = ( Substitut this xpasio ito th govrig quatio (7, ildig ( + ε + ε = = = Rarrag this quatio to writ it as ε ( + = ( = with th covtio that =. W ca s that b taig th limit ε, w obtai = ( ad b subtractig this rsult rom Equatio (, dividig both sids b ε, ad taig th limit ε agai, w gt = (4 Rpatig th procss as ma tims as dd lads to =, =,,,... (5 W could also hav writt Equatio (5 b ormall sttig th coicit o ε to zro i Equatio ( or ach valu o. B isrtig th asmptotic xpasio giv i Equatio ( ito th iitial coditio, w obtai ε = (6 which ilds = δ = (7 whr δ ij = wh i = j ad othrwis. It is ow as th Krocr dlta. Th solutio o = alog with = is =. Usig this, w ca solv 5

6 th quatio or, which is cotiuig th procss, w id ( x, ε ca b writt as = =, alog with x = x /!, (, ε x ε = x /! = to ild = x. B =, ad so o. Th solutio or x (8 =! ε which is th Talor sris or th xpotial uctio = x. This sris happs to covrg uiorml or all valus o ε ad x. I this xampl, our attmpt to id a powr sris xpasio i ε has ld to a covrgt sris, v though w caot xpct th sam i othr problms. Cocludig Rmars W hav s how a usul approximatio to th solutio o problms ivolvig a small paramtr ca b obtaid b xpadig i a asmptotic sris i that paramtr. This mthod is ow as prturbatio. It ca b show that th simpl tchiqu illustratd hr ails i th small paramtr multiplis th highst ordr drivativ i a dirtial quatio. This is bcaus th ordr o th dirtial quatio is rducd wh th small paramtr is st qual to zro. This lads to qualitativ dircs i th solutio, ad i boudar valu problms, th iabilit to satis th complt st o boudar coditios o th problm. Also, a simpl prturbatio mthod ca ail v wh th small paramtr ol multiplis a low ordr drivativ i th domai is uboudd, as ca occur i idalizd mathmatical problms. Ths problms ar hadld b usig sigular prturbatio tchiqus. You ca lar mor about prturbatio mthods rom a o th ollowig rrcs. Rrcs. A.H. Nah, Prturbatio Mthods, Joh Wil & Sos, Nw Yor (97.. M. Va D, Prturbatio Mthods i Fluid Mchaics, Parabolic Prss, Staord, Calioria (975.. J. Kvoria ad J.D. Col, Prturbatio Mthods i Applid Mathmatics, Sprigr-Vrlag, Nw Yor ( E.J. Hich, Prturbatio Mthods, Cambridg Uivrsit Prss, Cambridg, UK (99. 6