Numerical Solution of Differential and Integral Equations

Transcription

1 5 Numerical Soluio o Diereial ad Iegral Equaios The aspec o he calculus o Newo ad Leibiz ha allowed he mahemaical descripio o he phsical world is he abili o icorporae derivaives ad iegrals io equaios ha relae various properies o he world o oe aoher. Thus, much o he heor ha describes he world i which we live is coaied i wha are kow as diereial ad iegral equaios. Such equaios appear o ol i he phsical scieces, bu i biolog, sociolog, ad all scieiic disciplies ha aemp o udersad he world i which we live. Iumerable books ad eire courses o sud are devoed o he sud o he soluio o such equaios ad mos college maors i sciece ad egieerig require a leas oe such course o heir sudes. These courses geerall cover he aalic closed orm soluio o such equaios. Bu ma o he equaios ha gover he phsical world have o soluio i closed orm. Thereore, o id he aswer o quesios abou he world i which we live, we mus resor o solvig hese equaios umericall. gai, he lieraure o his subec is volumious, so we ca ol hope o provide a brie iroducio o some o he basic mehods widel emploed i idig hese soluios. lso, he subec is b o meas closed so he sude should be o he lookou or ew echiques ha prove icreasigl eicie ad accurae.

2 Numerical Mehods ad Daa alsis 5. The Numerical Iegraio o Diereial Equaios Whe we speak o a diereial equaio, we simpl mea a equaio where he depede variable appears as well as oe or more o is derivaives. The highes derivaive ha is prese deermies he order o he diereial equaio while he highes power o he depede variable or is derivaive appearig i he equaio ses is degree. Theories which emplo diereial equaios usuall will o be limied o sigle equaios, bu ma iclude ses o simulaeous equaios represeig he pheomea he describe. Thus, we mus sa somehig abou he soluios o ses o such equaios. Ideed, chagig a high order diereial equaio io a ssem o irs order diereial equaios is a sadard approach o idig he soluio o such equaios. Basicall, oe simpl replaces he higher order erms wih ew variables ad icludes he equaios ha deie he ew variables o orm a se o irs order simulaeous diereial equaios ha replace he origial equaio. Thus a hird order diereial equaio ha had he orm '''(x α"(x β'(x γ(x g(x, (5.. could be replaced wih a ssem o irs order diereial equaios ha looked like '(x αz'(x β '(x γ (x g(x z'(x (x. (5.. '(x z(x This simpliicaio meas ha we ca limi our discussio o he soluio o ses o irs order diereial equaios wih o loss o geerali. Oe remembers rom begiig calculus ha he derivaive o a cosa is zero. This meas ha i is alwas possible o add a cosa o he geeral soluio o a irs order diereial equaio uless some addiioal cosrai is imposed o he problem. These are geerall called he cosas o iegraio. These cosas will be prese eve i he equaios are ihomogeeous ad i his respec diereial equaios dier sigiical rom ucioal algebraic equaios. Thus, or a problem ivolvig diereial equaios o be ull speciied, he cosas correspodig o he derivaive prese mus be give i advace. The aure o he cosas (i.e. he ac ha heir derivaives are zero implies ha here is some value o he idepede variable or which he depede variable has he value o he cosa. Thus, cosas o iegraio o ol have a value, bu he have a "place" where he soluio has ha value. I all he cosas o iegraio are speciied a he same place, he are called iiial values ad he problem o idig a soluio is called a iiial value problem. I addiio, o id a umerical soluio, he rage o he idepede variable or which he soluio is desired mus also be speciied. This rage mus coai he iiial value o he idepede variable (i.e. ha value o he idepede variable correspodig o he locaio where he cosas o iegraio are speciied. O occasio, he cosas o iegraio are speciied a diere locaios. Such problems are kow as boudar value problems ad, as we shall see, hese require a special approach. So le us begi our discussio o he umerical soluio o ordiar diereial equaios b cosiderig he soluio o irs order iiial value diereial equaios. The geeral approach o idig a soluio o a diereial equaio (or a se o diereial equaios is o begi he soluio a he value o he idepede variable or which he soluio is equal o he iiial values. Oe he proceeds i a sep b sep maer o chage he idepede variable ad move

3 5 - Diereial ad Iegral Equaios across he required rage. Mos mehods or doig his rel o he local polomial approximaio o he soluio ad all he sabili problems ha were a cocer or ierpolaio will be a cocer or he umerical soluio o diereial equaios. However, ulike ierpolaio, we are o limied i our choice o he values o he idepede variable o where we ca evaluae he depede variable ad is derivaives. Thus, he spacig bewee soluio pois will be a ree parameer. We shall use his variable o corol he process o idig he soluio ad esimaig his error. Sice he soluio is o be locall approximaed b a polomial, we will have cosraied he soluio ad he values o he coeicies o he approximaig polomial. This would seem o impl ha beore we ca ake a ew sep i idig he soluio, we mus have prior iormaio abou he soluio i order o provide hose cosrais. This "chicke or egg" aspec o solvig diereial equaios would be removed i we could id a mehod ha ol depeded o he soluio a he previous sep. The we could sar wih he iiial value(s ad geerae he soluio a as ma addiioal values o he idepede variable as we eeded. Thereore le us begi b cosiderig oe-sep mehods. a. Oe Sep Mehods o he Numerical Soluio o Diereial Equaios Probabl he mos cocepuall simple mehod o umericall iegraig diereial equaios is Picard's mehod. Cosider he irs order diereial equaio '(x g(x,. (5.. Le us direcl iegrae his over he small bu iie rage h so ha which becomes x h d g(x, dx, (5..4 x x h (x g(x, dx, (5..5 x Now o evaluae he iegral ad obai he soluio, oe mus kow he aswer o evaluae g(x,. This ca be doe ieraivel b urig eq (5..5 io a ixed-poi ieraio ormula so ha x h (k (k (x h g[x, (x]dx x. (5..6 (k (k (x (x h more ispired choice o he ieraive value or ( k- (x migh be (k (x (k [ (x h]. (5..7 However, a eve beer approach would be o admi ha he bes polomial i o he soluio ha ca be achieved or wo pois is a sraigh lie, which ca be wrie as (k (x a(x x {[ (x h](x x [ (x ](x h x]}/ h. (5..8 While he righ had side o equaio (5..8 ca be used as he basis or a ixed poi ieraio scheme, he ieraio process ca be compleel avoided b akig advaage o he ucioal orm o g(x,. The liear

4 Numerical Mehods ad Daa alsis orm o ca be subsiued direcl io g(x, o id he bes value o a. The equaio ha cosrais a is he simpl x h ah g[x,(ax ] dx. (5..9 x This value o a ma he be subsiued direcl io he ceer erm o equaio (5..8 which i ur is evaluaed a x x h. Eve should i be impossible o evaluae he righ had side o equaio (5..9 i closed orm a o he quadraure ormulae o chaper 4 ca be used o direcl obai a value or a. However, oe should use a ormula wih a degree o precisio cosise wih he liear approximaio o. To see how hese various orms o Picard's mehod acuall work, cosider he diereial equaio '(x x, (5.. subec o he iiial codiios Direc iegraio ields he closed orm soluio (. (5.. x / e. (5.. The rapidl varig aure o his soluio will provide a ormidable es o a iegraio scheme paricularl i he sep size is large. Bu his is exacl wha we wa i we are o es he relaive accurac o diere mehods. I geeral, we ca cas Picard's mehod as z (x z(z dz, (5.. where equaios ( (5..8 represe various mehods o speciig he behavior o (z or purposes o evaluaig he iegrad. For purposes o demosraio, le us choose h which we kow is ureasoabl large. However, such a large choice will serve o demosrae he relaive accurac o our various choices quie clearl. Furher, le us obai he soluio a x, ad. The aive choice o equaio (5..6 ields a ieraio ormula o he orm x h (k (k h z (x hdz [h(x h / ] (x h x (x. (5..4 This ma be ieraed direcl o ield he resuls i colum (a o able 5., bu he ixed poi ca be oud direcl b simpl solvig equaio (5..4 or ( (x h o ge ( (x h ( hx h /. (5..5 For he irs sep whe x, he limiig value or he soluio is. However, as he soluio proceeds, he ieraio scheme clearl becomes usable. 4

5 5 - Diereial ad Iegral Equaios Table 5. Resuls or Picard's Mehod ( (B (C (D i ( ( ( c ( / 7/ i ( ( ( c ( Esimaig he appropriae value o (x b averagig he values a he limis o he iegral as idicaed b equaio (5..7 eds o sabilize he procedure ieldig he ieraio ormula x h (k (k (k (x h z[(x (x hdz [h(x h / ][(x (x h]/, x (5..6 he applicaio o which is coaied i colum (b o Table 5.. The limiig value o his ieraio ormula ca also be oud aalicall o be [h(x h/(x ]/ ( (x h (5..7 [ h(x h//], which clearl demosraes he sabilizig iluece o he averagig process or his rapidl icreasig soluio. Fiall, we ca ivesigae he impac o a liear approximaio or (x as give b equaio (5..8. Le us assume ha he soluio behaves liearl as suggesed b he ceer erm o equaio (

6 Numerical Mehods ad Daa alsis This ca be subsiued direcl io he explici orm or he soluio give b equaio (5.. ad he value or he slope, a, obaied as i equaio (5..9. This process ields a (x (x h//[-(x h/-(h /], (5..8 which wih he liear orm or he soluio gives he soluio wihou ieraio. The resuls are lised i able 5. i colum (c. I is empig o hik ha a combiaio o he righ had side o equaio (5..7 iegraed i closed orm i equaio (5.. would give a more exac aswer ha ha obaied wih he help o equaio (5..8, bu such is o he case. ieraio ormula developed i such a maer ca be ieraed aalicall as was doe wih equaios (5..5 ad (5..7 o ield exacl he resuls i colum (c o able 5.. Thus he bes oe ca hope or wih a liear Picard's mehod is give b equaio (5..8 wih he slope, a, speciied b equaio (5..9. However, here is aoher approach o idig oe-sep mehods. The diereial equaio (5.. has a ull amil o soluios depedig o he iiial value (i.e. he soluio a he begiig o he sep. Tha amil o soluios is resriced b he aure o g(x,. The behavior o ha amil i he eighborhood o x x h ca shed some ligh o he aure o he soluio a x x h. This is he udameal basis or oe o he more successul ad widel used oe-sep mehods kow as he Ruge-Kua mehod. The Ruge-Kua mehod is also oe o he ew mehods i umerical aalsis ha does o rel direcl o polomial approximaio or, while i is cerail correc or polomials, he basic mehod assumes ha he soluio ca be represeed b a Talor series. So le us begi our discussio o Ruge-Kua ormulae b assumig ha he soluio ca be represeed b a iie alor series o he orm k (k h' (h /!" L (h / k!. (5..9 Now assume ha he soluio ca also be represeed b a ucio o he orm h{α g(x, α g[(x µ h,( b h] α g[(x µ h,( b h] L α k g[(x µ k h,( b k h]}. (5.. This raher covolued expressio, while appearig o deped ol o he value o a he iiial sep (i.e. ivolves evaluaig he ucio g(x, all abou he soluio poi x, (see Figure 5.. B seig equaios (5..9 ad (5.. equal o each oher, we see ha we ca wrie he soluio i he rom α α L α k k, (5.. where he i s ca be expressed recursivel b hg(x, M k hg[(x hg[(x hg[(x µ h,( µ µ k h,( M h,( λ λ λ,, k, ] λ λ, k, ] L λ k,k k ]. (5.. Now we mus deermie k values o α, k values o µ ad k (k/ values o λ i,. Bu we ol have k 6

7 5 - Diereial ad Iegral Equaios erms o he Talor series o ac as cosrais. Thus, he problem is hopelessl uder-deermied. Thus ideermiec will give rise o eire amilies o Ruge-Kua ormulae or a order k. I addiio, he algebra o elimiae as ma o he ukows as possible is quie ormidable ad o uique due o he udeermied aure o he problem. Thus we will coe ourselves wih dealig ol wih low order ormulae which demosrae he basic approach ad aure o he problem. Le us cosider he lowes order ha provides some isigh io he geeral aspecs o he Ruge-Kua mehod. Tha is k. Wih k equaios (5.. ad (5.. become α α hg(x. (5.. hg[(x µ h,( λ ] Here we have dropped he subscrip o λ as here will ol be oe o hem. However, here are sill our ree parameers ad we reall ol have hree equaios o cosrai. Figure 5. show he soluio space or he diereial equaio ' g(x,. Sice he iiial value is diere or diere soluios, he space surroudig he soluio o choice ca be viewed as beig ull o alerae soluios. The wo dimesioal Talor expasio o he Ruge- Kua mehod explores his soluio space o obai a higher order value or he speciic soluio i us oe sep. 7

8 Numerical Mehods ad Daa alsis I we expad g(x, abou x,, i a wo dimesioal alor series, we ca wrie g(x, g(x, g[(x µ h,( λ ] g(x, µ h λ µ x λ g(x, µλ g(x, x L Makig use o he hird o equaios (5.., we ca explicil wrie as g(x, g(x, hg(x, h µ λg(x, x g(x, g(x, h µ λ g (x, x Direc subsiuio io he irs o equaios (5.. gives µλg(x, h g(x g(x, x x,. (5..4. (5..5 g(x, g(x, h( α αg(x, h µ λg(x, x. (5..6 g(x, g(x, g(x, h α µ λ g (x, µλg(x, x x We ca also expad ' i a wo dimesioal alor series makig use o he origial diereial equaio (5.. o ge ' g(x, g(x, g(x, g(x, g(x, " ' g(x, x x " " g(x, g(x, g(x, g(x, ''' ' g(x,. (5..7 x x x x g(x, g(x, g(x, g(x, g(x, g(x, x Subsiuig his io he sadard orm o he Talor series as give b equaio (5..9 ields g(x, g(x, h g(x, g(x, hg(x, h λg(x, g (x, x 6 x. g(x, g(x, g(x, g(x, g(x, g(x, x x (5..8 Now b comparig his erm b erm wih he expasio show i equaio (5..6 we ca coclude ha he ree parameers α, α, µ, ad λ mus be cosraied b 8

9 5 - Diereial ad Iegral Equaios ( α α αµ. (5..9 α λ s we suggesed earlier, he ormula is uder-deermied b oe cosrai. However, we ma use he cosrai equaios as represeed b equaio (5..9 o express he ree parameers i erms o a sigle cosa c. Thus he parameers are α c α c µ λ c. (5.. ad he approximaio ormula becomes g(x, g(x, h g(x, g(x, hg(x, h λg(x, g (x, x 8c x. g(x, g(x, x (5.. We ca mach he irs wo erms o he Talor series wih a choice o c. The error erm will ha be o order O(h ad speciicall has he orm h ''' g(x, " R [ 4c]. (5.. 4c Clearl he mos eecive choice o c will deped o he soluio so ha here is o geeral "bes" choice. However, a umber o auhors recommed c ½ as a geeral purpose value. I we icrease he umber o erms i he series, he uder-deermiaio o he cosas ges rapidl worse. More ad more parameers mus be chose arbiraril. Whe hese ormulae are give, he arbirariess has oe bee removed b ia. Thus oe ma id various Ruge-Kua ormulae o he same order. For example, a commo such ourh order ormula is ( / 6 hg(x, hg[(x h,( ] hg[(x h,( ] hg[(x h,( ]. (5.. Here he "bes" choice or he uder-deermied parameers has alread bee made largel o he basis o experiece. I we appl hese ormulae o our es diereial equaio (5.., we eed irs speci which Ruge-Kua ormula we pla o use. Le us r he secod order (i.e. exac or quadraic polomials ormula give b equaio (5.. wih he choice o cosas give b equaio (5..9 whe c ½. The ormula he becomes 9

10 Numerical Mehods ad Daa alsis hg(x,. (5..4 hg[(x h,( ] So ha we ma readil compare o he irs order Picard ormula, we will ake h ad (. The akig g(x, rom equaio (5.. we ge or he irs sep ha hx ((( h(x h( (( (. (5..5 (x h ( ( ( ( ( The secod sep ields hx ((( h(x h( (( ( 5. ( (x h ( ( ( ( (5 4 Table 5. Sample Ruge-Kua Soluios Secod Order Soluio Fourh Order Soluio Sep h h/ c h i i i i. [, 9/].. [/4, 45/64] δ.7 h'.85* Sep i i i i.5 [.886,.984] [.89, 5.96] δ.845 h'.65 * This value assumes ha δ.

11 5 - Diereial ad Iegral Equaios The Ruge-Kua ormula eds o uder-esimae he soluio i a ssemaic ashio. I we reduce he sep size o h ½ he agreeme is much beer as he error erm i his ormula is o O(h. The resuls or h ½ are give i able 5. alog wih he resuls or h. I addiio we have abulaed he resuls or he ourh order ormula give b equaio (5... For our example, he irs sep would require ha equaio (5.. ake he orm (x hx h(x h(x h(x h ((( h( h( h( (( ( [( ( (( (( [ ( ( ( [ ( 5 8 ( ( ( ] ] 8 ]/ (5..7 The error erm or his ormula is o O(h 5 so we would expec i o be superior o he secod order ormula or h ½ ad ideed i is. These resuls demosrae ha usuall i is preerable o icrease he accurac o a soluio b icreasig he accurac o he iegraio ormula raher ha decreasig he sep size. The calculaios leadig o Table 5. were largel carried ou usig racioal arihmeic so as o elimiae he roud-o error. The eecs o roud-o error are usuall such ha he are more serious or a dimiished sep size ha or a iegraio ormula ieldig suiabl icreased accurac o mach he decreased sep size. This simpl acceuaes he ecessi o improve soluio accurac b improvig he approximaio accurac o he iegraio ormula. The Ruge-Kua pe schemes eo grea populari as heir applicaio is quie sraigh orward ad he ed o be quie sable. Their greaes appeal comes rom he ac ha he are oe-sep mehods. Ol he iormaio abou he ucio a he previous sep is ecessar o predic he soluio a he ex sep. Thus he are exremel useul i iiiaig a soluio sarig wih he iiial value a he boudar o he rage. The greaes drawback o he mehods is heir relaive eiciec. For example, he orh order scheme requires our evaluaios o he ucio a each sep. We shall see ha here are oher mehods ha require ar ewer evaluaios o he ucio a each sep ad e have a higher order. b. Error Esimae ad Sep Size Corol umerical soluio o a diereial equaio is o lile use i here is o esimae o is accurac. However, as is clear rom equaio (5.., he ormal esimae o he rucaio error is oe more diicul ha idig he soluio. Uoruael, he rucaio error or mos problems ivolvig diereial equaios eds o mimic he soluio. Tha is, should he soluio be moooicall icreasig, he he absolue rucaio error will also icrease. Eve moooicall decreasig soluios will ed o have rucaio errors ha keep he same sig ad accumulae as he soluio progresses. The commo eec o rucaio errors o oscillaor soluios is o iroduce a "phase shi" i he soluio. Sice he eec o rucaio error eds o be ssemaic, here mus be some mehod or esimaig is magiude. lhough he ormal expressio o he rucaio error [sa equaio (5..] is usuall raher ormidable, such expressios alwas deped o he sep size. Thus we ma use he sep size h isel o

12 Numerical Mehods ad Daa alsis esimae he magiude o he error. We ca he use his esimae ad a a priori value o he larges accepable error o adus he sep size. Viruall all geeral algorihms or he soluio o diereial equaios coai a secio or he esimae o he rucaio error ad he subseque adusme o he sep size h so ha predeermied oleraces ca be me. Uoruael, hese mehods o error esimae will rel o he variaio o he sep size a each sep. This will geerall riple he amou o ime required o eec he soluio. However, he icrease i ime spe makig a sigle sep ma be ose b beig able o use much larger seps resulig i a over all savigs i ime. The geeral accurac cao be arbiraril icreased b decreasig he sep size. While his will reduce he rucaio error, i will icrease he eecs o roud-o error due o he icreased amou o calculaio required o cover he same rage. Thus oe does o wa o se he a priori error olerace o low or he roud-o error ma desro he validi o he soluio. Ideall, he, we would like our soluio o proceed wih raher large sep sizes (i.e. values o h whe he soluio is slowl varig ad auomaicall decrease he sep size whe he soluio begis o chage rapidl. Wih his i mid, le us see how we ma corol he sep size rom oleraces se o he rucaio error. Give eiher he oe sep mehods discussed above or he muli-sep mehods ha ollow, assume ha we have deermied he soluio a some poi x. We are abou o ake he ex sep i he soluio o x b a amou h ad wish o esimae he rucaio error i. Calculae his value o he soluio wo was. Firs, arrivig a x b akig a sigle sep h, he repea he calculaio akig wo seps o (h/. Le us call he irs soluio, ad he secod,. Now he exac soluio (eglecig earlier accumulaed error a x could be wrie i each case as k e αh L,, (5..8 k α( e h L, where k is he order o he approximaio scheme. Now α ca be regarded as a cosa hroughou he ierval h sice i is us he coeicie o he Talor series i or he (kh erm. Now le us deie δ as a measure o he error so ha k k δ( αh /( δ. (5..9,, Clearl, k δ ( h, (5..4 so ha he sep size h ca be adused a each sep i order ha he rucaio error remais uiorm b k h h δ( / δ(. (5..4 Iiiall, oe mus se he olerace a some pre-assiged level ε so ha δ ε. (5..4 I we use his procedure o ivesigae he sep sizes used i our es o he Ruge-Kua mehod, we see ha we cerail chose he sep size o be oo large. We ca veri his wih he secod order soluio or we carried ou he calculaio or sep sizes o h ad h½. Followig he prescripio o equaio (5..9 ad (5..4 we have, ha or he resuls speciied i Table 5.,

13 5 - Diereial ad Iegral Equaios δ,, δ. (5..4 h h ((./.7.85 δ Here we have acil assumed a iiial olerace o δ.. While his is arbirar ad raher large or a olerace o a soluio, i is illusraive ad cosise wih he spiri o he soluio. We see ha o maiai he accurac o he soluio wihi. we should decrease he sep size slighl or he iiial sep. The error a he ed o he irs sep is.6 or h, while i is ol abou.4 or h ½. B comparig he umerical aswers wih he aalic aswer, c, we see ha acor o wo chage i he sep size reduces he error b abou a acor o our. Our saed olerace o. requires ol a reducio i he error o abou % which implies a reducio o abou 6% i he sep size or a ew sep size h '.84h. This is amazigl close o he recommeded chage, which was deermied wihou kowledge o he aalic soluio. The amou o he sep size adusme a he secod sep is made o maiai he accurac ha exiss a he ed o he irs sep. Thus, δ,, δ h h. (5..44 ((.7 / δ Normall hese adusmes would be made cumulaivel i order o maiai he iiial olerace. However, he coveie values or he sep sizes were useul or he earlier comparisos o iegraio mehods. The rapid icrease o he soluio aer x causes he Ruge-Kua mehod o have a icreasigl diicul ime maiaiig accurac. This is abudal clear i he drasic reducio i he sep size suggesed a he ed o he secod sep. he ed o he irs sep, he relaive errors where 9% ad % or he h ad h½ sep size soluios respecivel. he ed o he secod sep hose errors, resulig rom compariso wih he aalic soluio, had umped o 55% ad % respecivel (see able 5.. While a acor o wo-chage i he sep size sill produces abou a acor o our chage i he soluio, o arrive a a relaive error o 9%, we will eed more like a acor o 6 chage i he soluio. This would sugges a chage i he sep size o a abou a acor o hree, bu he recommeded chage is more like a acor o 6. This dierece ca be udersood b oicig ha equaio (5..4 aemps o maiai he absolue error less ha δ. For our problem his is abou. a he ed o sep oe. To keep he error wihi hose oleraces, he accurac a sep wo would have o be wihi abou.5% o he correc aswer. To ge here rom 55% meas a reducio i he error o a acor o 6, which correspods o a reducio i he sep size o a acor o abou 8, is close o ha give b he esimae. Thus we see ha he equaio (5..4 is desiged o maiai a absolue accurac i he soluio b adusig he sep size. Should oe wish o adus he sep size so as o maiai a relaive or perceage accurac, he oe could adus he sep size accordig o (k h h {[ δ( ] [ δ( ]. (5..45 While hese procedures var he sep size so as o maiai cosa rucaio error, a sigiica price i he amou o compuig mus be paid a each sep. However, he amou o exra eor eed o be used ol o esimae he error ad hereb corol i. Oe ca solve equaios (5..8 (eglecig erms o order greaer ha k o provide a improved esimae o. Speciicall

14 Numerical Mehods ad Daa alsis e k, δ( (. (5..46 However, sice oe cao simulaeousl iclude his improveme direcl i he error esimae, i is advisable ha i be regarded as a "sae acor" ad proceeds wih he error esimae as i he improveme had o bee made. While his ma seem udul coservaive, i he umerical soluio o diereial equaios coservaism is a virue. c. Muli-Sep ad Predicor-Correcor Mehods The high order oe sep mehods achieve heir accurac b explorig he soluio space i he eighborhood o he speciic soluio. I priciple, we could use prior iormaio abou he soluio o cosrai our exrapolaio o he ex sep. Sice his iormaio is he direc resul o prior calculaio, ar greaer levels o eiciec ca be achieved ha b mehods such as Ruge-Kua ha explore he soluio space i he vicii o he required soluio. B usig he soluio a pois we could, i priciple, i a (- degree polomial o he soluio a hose pois ad use i o obai he soluio a he (s poi. Such mehods are called muli-sep mehods. However, oe should remember he caveas a he ed o chaper where we poied ou ha polomial exrapolaio is exremel usable. Thus such a procedure b isel will geerall o provide a suiable mehod or he soluio o diereial equaios. Bu whe combied wih algorihms ha compesae or he isabili such schemes ca provide ver sable soluio algorihms. lgorihms o his pe are called predicor-correcor mehods ad here are umerous orms o hem. So raher ha aemp o cover hem all, we shall sa a ew higs abou he geeral heor o such schemes ad give some examples. predicor-correcor algorihm, as he ame implies, cosiss o basicall wo pars. The predicor exrapolaes he soluio over some iie rage h based o he iormaio a prior pois ad is iherel usable. The correcor allows or his local isabili ad makes a correcio o he soluio a he ed o he ierval also based o prior iormaio as well as he exrapolaed soluio. Cocepuall, he oio o a predicor is quie simple. I is simples orm, such a scheme is he oe-sep predicor where h'. (5..47 B usig he value o he derivaive a x he scheme will ssemaicall uder esimae he proper value required or exrapolaio o a moooicall icreasig soluio (see igure 5.. The error will build up cumulaivel ad hece i is usable. beer sraeg would be o use he value o he derivaive midwa bewee he wo soluio pois, or aleraivel o use he iormaio rom he prior wo pois o predic. Thus a wo poi predicor could ake he orm h'. (5..48 lhough his is a wo-poi scheme, he exrapolaig polomial is sill a sraigh lie. We could have used he value o direcl o i a parabola hrough he wo pois, bu we did' due o he isabiliies o be associaed wih a higher degree polomial exrapolaio. This deliberae reecio o he some o he iormaioal cosrais i avor o icreased sabili is wha makes predicor-correcor schemes o-rivial ad eecive. I he geeral case, we have grea reedom o use he iormaio we have regardig i ad ' i. I we were o iclude all he available iormaio, a geeral predicor would have he 4

15 orm a h bi ' i i i i i R 5 - Diereial ad Iegral Equaios, (5..49 where he a i s ad b i s are chose b imposig he appropriae cosrais a he pois x i ad R is a error erm. Whe we have decided o he orm o he predicor, we mus impleme some sor o correcor scheme o reduce he rucaio error iroduced b he predicor. s wih he predicor, le us ake a simple case o a correcor as a example. Havig produced a soluio a x we ca calculae he value o he derivaive ' a x. This represes ew iormaio ad ca be used o modi he resuls o he predicio. For example, we could wrie a correcor as (k (k h[' ' ]. (5..5 Thereore, i we were o wrie a geeral expressio or a correcor based o he available iormaio we would ge Figure 5. shows he isabili o a simple predicor scheme ha ssemaicall uderesimaes he soluio leadig o a cumulaive build up o rucaio error. (k α ii h βi ' i hβ i i '. (5..5 Equaios (5..5 ad (5..5 boh are wrie i he orm o ieraio ormulae, bu i is o a all clear ha (k 5

16 Numerical Mehods ad Daa alsis he ixed-poi or hese ormulae is a beer represeaio o he soluio ha sigle ieraio. So i order o miimize he compuaioal demads o he mehod, correcors are geerall applied ol oce. Le us ow cosider cerai speciic pes o predicor correcor schemes ha have bee oud o be successul. Hammig gives a umber o popular predicor-correcor schemes, he bes kow o which is he dams-bashorh-moulo Predicor-Correcor. Predicor schemes o he dams-bashorh pe emphasize he iormaio coaied i prior values o he derivaive as opposed o he ucio isel. This is presumabl because he derivaive is usuall a more slowl varig ucio ha he soluio ad so ca be more accurael exrapolaed. This philosoph is carried over o he dams-moulo Correcor. classical ourh-order ormula o his pe is ( ' ' ' ' 5 h( / 4 O(h. (5..5 ' ' ' 5 h(9 9 5 / 4 O(h Legh sud o predicor-correcor schemes has evolved some special orms such as his oe ' ' ' ' z ( / h( / 75 u z 77(z c / 75. (5..5 c ( / h(5u' 9' 4' 9' / 7 6 c 4(z c / 75 O(h where he exrapolaio ormula has bee expressed i erms o some recursive parameers u i ad c i. The derivaive o hese iermediae parameers are obaied b usig he origial diereial equaio so ha u ' g(x, u. (5..54 B good chace, his ormula [equaio (5..5] has a error erm ha varies as O(h 6 ad so is a ih-order ormula. Fiall a classical predicor-correcor scheme which combies dams-bashorh ad Mile predicors ad is quie sable is paramericall ( i.e. Hammig p6 ' ' ' ' z ( h( / 48 u z 6(z c /7. (5..55 c ( h(7u' 5' ' ' / 48 6 c 9(z c /7 O(h Press e al are o he opiio ha predicor-correcor schemes have see heir da ad are made obsolee b he Bulirsch-Soer mehod which he discuss a some legh. The quie properl poi ou ha he predicor-correcor schemes are somewha ilexible whe i comes o varig he sep size. The sep size ca be reduced b ierpolaig he ecessar missig iormaio rom earlier seps ad i ca be expaded i iegral muliples b skippig earlier pois ad akig he required iormaio rom eve earlier i he soluio. However, he Bulirsch-Soer mehod, as described b Press e. al. uilizes a predicor scheme wih some special properies. I ma be parameerized as 6

17 5 - Diereial ad Iegral Equaios z (x z z hz' z k z k hz' k k,,, L,. (5..56 ( 5 (z z hz' O(h z' g(z, x I is a odd characerisic o he hird o equaios (5..56 ha he error erm ol coais eve powers o he sep size. Thus, we ma use he same rick ha was used i equaio (5..46 o uilizig he iormaio geeraed i esimaig he error erm o improve he approximaio order. Bu sice ol eve powers o h appear i he error erm, his sigle sep will gai us wo powers o h resulig i a predicor o order seve. ( ( 7 h {4 (x h / [x ( / (h]}/ O(h. (5..57 This ields a predicor ha requires somehig o he order o ½ evaluaios o he ucio per sep compared o our or a Ruge-Kua ormula o ierior order. Now we come o he aspec o he Bulirsch-Soer mehod ha begis o diereiae i rom classical predicor-correcors. predicor ha operaes over some iie ierval ca use a successivel icreasig umber o seps i order o make is predicio. Presumabl he predicio will ge beer ad beer as he sep size decreases so ha he umber o seps o make he oe predicio icreases. O course pracical aspecs o he problem such as roudo error ad iie compuig resources preve us rom usig arbiraril small sep sizes, bu we ca approximae wha would happe i a ideal world wihou roud-o error ad uilizig ulimied compuers. Simpl cosider he predicio a he ed o he iie ierval H where H αh. (5..58 Thus α (xh ca be ake o be a ucio o he sep size h so ha, α (xh (xαh (h. (5..59 Now we ca phrase our problem o esimae he value o ha ucio i he limi Lim (h Y (x H. (5..6 h α We ca accomplish his b carrig ou he calculaio or successivel smaller ad smaller values o h ad, o he basis o hese values, exrapolaig he resul o h. I spie o he admoiios raised i chaper regardig exrapolaio, he rage here is small. Bu o produce a rul powerul umerical iegraio algorihm, Bulirsch ad Soer carr ou he exrapolaio usig raioal ucios i he maer described i secio. [equaio (..65]. The superiori o raioal ucios o polomials i represeig mos aalic ucios meas ha he sep size ca be quie large ideed ad he coveioal meaig o he 'order' o he approximaio is irreleva i describig he accurac o he mehod. 7

18 Numerical Mehods ad Daa alsis I a case, remember ha accurac ad order are o somous! Should he soluio be described b a slowl varig ucio ad he umerical iegraio scheme operae b iig high order polomials o prior iormaio or he purposes o exrapolaio, he high-order ormula ca give ver iaccurae resuls. This simpl sas ha he iegraio scheme ca be usable eve or well behaved soluios. Press e. al. 4 sugges ha all oe eeds o solve ordiar diereial equaios is eiher a Ruge- Kua or Bulirsch-Soer mehod ad i would seem ha or mos problems ha ma well be he case. However, here are a large umber o commercial diereial equaio solvig algorihms ad he maori o hem uilize predicor-correcor schemes. These schemes are geerall ver as ad he more sophisicaed oes carr ou ver ivolved error checkig algorihms. The are geerall quie sable ad ca ivolve a ver high order whe required. I a eve, he user should kow how he work ad be war o he resuls. I is ar oo eas o simpl ake he resuls o such programs a ace value wihou ever quesioig he accurac o he resuls. Cerail oe should alwas ask he quesio "re hese resuls reasoable?" a he ed o a umerical iegraio. I oe is geuiel skepical, i is o a bad idea o ake he ial value o he calculaio as a iiial value ad iegrae back over he rage. Should oe recover he origial iiial value wihi he accepable oleraces, oe ca be reasoabl coide ha he resuls are accurae. I o, he dierece bewee he begiig iiial value ad wha is calculaed b he reverse iegraio over he rage ca be used o place limis o he accurac o he iiial iegraio. d. Ssems o Diereial Equaios ad Boudar Value Problems ll he mehods we have developed or he soluio o sigle irs order diereial equaios ma be applied o he case where we have a coupled ssem o diereial equaios. We saw earlier ha such ssems arose wheever we deal wih ordiar diereial equaios o order greaer ha oe. However, here are ma scieiic problems which are irisicall described b coupled ssems o diereial equaios ad so we should sa somehig abou heir soluio. The simples wa o see he applicabili o he sigle equaio algorihms o a ssem o diereial equaios is o wrie a ssem like ' g(x,,, L ' g (x,,, L M M ' g (x,,, L, (5..6 as a vecor where each eleme represes oe o he depede variables or ukows o he ssem. The he ssem becomes r r r ' g(x,, (5..6 which looks us like equaio (5.. so ha everhig applicable o ha equaio will appl o he ssem o equaios. O course some care mus be ake wih he ermiolog. For example, equaio (5..4 would have o be udersood as sadig or a eire ssem o equaios ivolvig ar more complicaed iegrals, bu i priciple, he ideas carr over. Some care mus also be exeded o he error aalsis i ha he error 8

19 5 - Diereial ad Iegral Equaios erm is also a vecor R r (x. I geeral, oe should worr abou he magiude o he error vecor, bu i pracice, i is usuall he larges eleme ha is ake as characerizig he accurac o he soluio. To geerae a umerical iegraio mehod or a speciic algorihm, oe simpl applies i o each o he equaios ha make up he ssem. B wa o a speciic example, le's cosider a orh order Ruge- Kua algorihm as give b equaio (5.. ad appl i o a ssem o wo equaios. We ge u u,, hg [(x hg [(x hg [(x hg (x u hg [(x hg (x hg [(x u hg [(x,, ( (u, h,(,,,, u, h,( h,(, h,( h,( h,(,,,,, u,(,,,( u / 6,(,(, u,(,(, / 6,,,, u ] ] u u u u ] ] ] ]. (5..6 We ca geeralize equaio (5..6 o a arbirar ssem o equaios b wriig i i vecor orm as r r r (. (5..64 r r The vecor ( cosiss o elemes which are ucios o depede variables i, ad x, bu which all have he same geeral orm varig ol wih g i (x, r. Sice a h order diereial equaio ca alwas be reduced o a ssem o irs order diereial equaios, a expressio o he orm o equaio (5..6 could be used o solve a secod order diereial equaio. The exisece o coupled ssems o diereial equaios admis he ieresig possibili ha he cosas o iegraio required o uiquel speci a soluio are o all give a he same locaio. Thus we do o have a ull complime o i, 's wih which o begi he iegraio. Such problems are called boudar value problems. comprehesive discussio o boudar value problems is well beod he scope o his book, bu we will examie he simpler problem o liear wo poi boudar value problems. This subclass o boudar value problems is quie commo i sciece ad exremel well sudied. I cosiss o a ssem o liear diereial equaios (i.e. diereial equaios o he irs degree ol where par o he iegraio cosas are speciied a oe locaio x ad he remaider are speciied a some oher value o he idepede variable x. These pois are kow as he boudaries o he problem ad we seek a soluio o he problem wihi hese boudaries. Clearl he soluio ca be exeded beod he boudaries as he soluio a he boudaries ca serve as iiial values or a sadard umerical iegraio. The geeral approach o such problems is o ake advaage o he lieari o he equaios, which 9

20 Numerical Mehods ad Daa alsis guaraees ha a soluio o he ssem ca be expressed as a liear combiaio o a se o basis soluios. se o basis soluios is simpl a se o soluios, which are liearl idepede. Le us cosider a se o m liear irs order diereial equaios where k values o he depede variables are speciied a x ad (m-k values correspodig o he remaiig depede variables are speciied a x. We could solve (m-k iiial value problems sarig a x ad speciig (m-k idepede, ses o missig iiial values so ha he iiial value problems are uiquel deermied. Le us deoe he missig se o iiial values a x b r ( (x which we kow ca be deermied rom iiial ses o liearl idepede rial iiial values r ( (x b r ( ( (x (x, (5..65 The colums o ( r ( (x are us he idividual vecors (x. Clearl he marix will have o be r ( diagoal o alwas produce (x. Sice he rial iiial values are arbirar, we will choose he elemes o he (m-k ses o be so ha he missig iiial values will be i (x δi, (5..66 r ( (x. (5..67 r Iegraig across he ierval wih hese iiial values will ield (m-k soluio ( (x a he oher boudar. Sice he equaios are liear each rial soluio will be relaed o he kow boudar r values ( (x b r ( r ( (x [ (x ], (5..68 so ha or he complee se o rial soluios we ma wrie r ( (x ( (x, (5..69 where b aalog o equaio (5..65, he colum vecors o ( r (x are ( (x. We ma solve hese equaios or he ukow rasormaio marix so ha he missig iiial values are r ( r ( (x - (x. (5..7 I oe emplos a oe sep mehod such as Ruge-Kua, i is possible o collapse his eire operaio o he poi where oe ca represe he complee boudar codiios a oe boudar i erms o he values a he oher boudar r a ssem o liear algebraic equaios such as r r (x B(x The marix B will deped ol o he deails o he iegraio scheme ad he ucioal orm o he equaios hemselves, o o he boudar values. Thereore i ma be calculaed or a se o boudar values ad used repeaedl or problems dierig ol i he values a he boudar (see Da ad Collis 5. 4

21 5 - Diereial ad Iegral Equaios To demosrae mehods o soluio or ssems o diereial equaios or boudar value problems, we shall eed more ha he irs order equaio (5.. ha we used or earlier examples. However, ha equaio was quie illusraive as i had a rapidl icreasig soluio ha emphasized he shorcomigs o he various umerical mehods. Thus we shall keep he soluio, bu chage he equaio. Simpl diereiae equaio (5.. so ha x Y" ( x e ( x. (5..7 Le us keep he same iiial codiio give b equaio (5.. ad add a codiio o he derivaive a x so ha (. (5..7 '( e This isures ha he closed orm soluio is he same as equaio (5.. so ha we will be able o compare he resuls o solvig his problem wih earlier mehods. We should o expec he soluio o be as accurae or we have made he problem more diicul b icreasig he order o he diereial equaio i addiio o separaig he locaio o he cosas o iegraio. This is o loger a iiial value problem sice he soluio value is give a x, while he oher cosrai o he derivaive is speciied a x. This is pical o he classical wo-poi boudar value problem. We ma also use his example o idicae he mehod or solvig higher order diereial equaios give a he sar o his chaper b equaios (5.. ad (5... Wih hose equaios i mid, le us replace equaio (5..7 b ssem o irs order equaios ' (x (x, (5..74 ' (x ( x (x which we ca wrie i vecor orm as r r ' (x, (5..75 where ( x. (5..76 ( x The compoes o he soluio vecor r are us he soluio we seek (i.e. ad is derivaive. However, he orm o equaio (5..75 emphasizes is liear orm ad were i a scalar equaio, we should kow how o proceed. For purposes o illusraio, le us appl he ourh order Ruge-Kua scheme give b equaio (5..6. Here we ca ake speciic advaage o he liear aure o our problem ad he ac ha he depedece o he idepede variable acors ou o he righ had side. To illusrae he uili o his ac, le g (x, [ (x], (5..77 i equaio (

22 Numerical Mehods ad Daa alsis 4 The we ca wrie he ourh order Ruge-Kua parameers as h h h (h ( h h h (h ( h h (h h ( h ( h h. (5..78 where h (x h (x (x, (5..79 so ha he ormula becomes 4 4 h ( h ( 6 h 4 ( 6 h (. (5..8 Here we see ha he lieari o he diereial equaio allows he soluio a sep o be acored ou o he ormula so ha he soluio a sep appears explicil i he ormula. Ideed, equaio (5..8 represes a power series i h or he soluio a sep ( i erms o he soluio a sep. Sice we have bee careul abou he order i which he ucios i muliplied each oher, we ma appl equaio (5..8 direcl o equaio (5..75 ad obai a similar ormula or ssems o liear irs order diereial equaios ha has he orm 4 4 h ( h 4 ( 6 h 4 ( 6 h r r. (5..8 Here he meaig o i is he same as i i ha he subscrip idicaes he value o he idepede variable x or which he marix is o be evaluaed. I we ake h, he marices or our speciic problem become 4. (5..8 Keepig i mid ha he order o marix muliplicaio is impora, he producs appearig i he secod order erm are

23 5 - Diereial ad Iegral Equaios 4 6. (5..8 The wo producs appearig i he hird order erm ca be easil geeraed rom equaios (5..8 ad (5..8 ad are. ( Fiall he sigle marix o he irs order erm ca be obai b successive muliplicaio usig equaios(5..8 ad (5..84 ieldig 8 9. (5..85 Like equaio (5..8, we ca regard equaio (5..8 as a series soluio i h ha ields a ssem o liear equaios or he soluio a sep i erms o he soluio a sep. I is worh oig ha he coeicies o he various erms o order h k are similar o hose developed or equal ierval quadraure ormulae i chaper 4. For example he lead erm beig he ui marix geeraes he coeicies o he rapezoid rule while he h(, 4, /6 coeicies o he secod erm are he amiliar progressio characerisic o Simpso's rule. The higher order erms i he ormula are less recogizable sice he deped o he parameers chose i he uder-deermied Ruge-Kua ormula. I we deie a marix P(h k so ha k k (h r r P P, (5..86 he series aure o equaio (5..8 ca be explicil represeed i erms o he various values o k P.

24 Numerical Mehods ad Daa alsis For our problem he are: 4 P P 6 7 P 6 7 P P (5..87 The boudar value problem ow is reduced o solvig he liear ssem o equaios speciied b equaio (5..86 where he kow values a he respecive boudaries are speciied. Usig he values give i equaio (5..7 he liear equaios or he missig boudar values become k k P ( P ( (5..88 k k ( P( P ( The irs o hese ields he missig soluio value a x [i.e. (]. Wih ha value he remaiig value ca be obaied rom he secod equaio. The resuls o hese soluios icludig addiioal erms o order h k are give i able 5.. We have ake h o be ui, which is ureasoabl large, bu i serves o demosrae he relaive accurac o icludig higher order erms ad simpliies he arihmeic. The resuls or he missig values ( ad ( (i.e. he ceer wo rows coverge slowl, ad o uiorml, oward heir aalic values give i he colum labeled k. Had we chose he sep size h o be smaller so ha a umber o seps were required o cross he ierval, he each sep would have produced a marix k ip ad he soluio a each sep would have bee relaed o he soluio a he ollowig sep b equaio ( Repeaed applicaio o ha equaio would ield he soluio a oe boudar i erms o he soluio a he oher so ha 44

25 r 5 - Diereial ad Iegral Equaios r k k k k ( P P PL P Q. (5..89 r Table 5. Soluios o a Sample Boudar Value Problem or Various Orders o pproximaio \ K 4 ( ( ( ( e Thus oe arrives a a similar se o liear equaios o hose implied b equaio (5..86 ad explicil give i equaio (5..88 relaig he soluio a oe boudar i erms o he soluio a he oher boudar. These ca be solved or he missig boudar values i he same maer as our example. Clearl he decrease i he sep size will improve he accurac as dramaicall as icreasig he order k o he approximaio ormula. Ideed he sep size ca be variable a each sep allowig or he use o he error correcig procedures described i secio 5.b. Table 5.4 Soluios o a Sample Boudar Value Problem \ K 4 ( ( ( ( se o boudar values could have bee used wih equaios (5..8 o ield he soluio elsewhere. Thus, we could rea our sample problem as a iiial value problem or compariso. I we ake he aalic values or ( ad ( ad solve he resulig liear equaios, we ge he resuls give i Table 5.4. Here he ial soluio is more accurae ad exhibis a covergece sequece more like we would expec rom Ruge-Kua. Namel, he soluio ssemaicall lies below he rapidl icreasig aalic soluio. For he boudar value problem, he reverse was rue ad he ial resul less accurae. This is o a ucommo resul or wo-poi boudar value problems sice he error o he approximaio scheme is direcl releced i he deermiaio o he missig boudar values. I a iiial value problem, here is assumed o be o error i he iiial values. 45

26 Numerical Mehods ad Daa alsis This simple example is o mea o provide a deiiive discussio o eve he resriced subse o liear wo-poi boudar value problems, bu simpl o idicae a wa o proceed wih heir soluio. oe wishig o pursue he subec o wo-poi boudar value problems urher should begi wih he veerable ex b Fox 6. e. Parial Diereial Equaios The subec o parial diereial equaios has a lieraure a leas as large as ha or ordiar diereial equaios. I is beod he scope o his book o provide a discussio o parial diereial equaios eve a he level chose or ordiar diereial equaios. Ideed, ma iroducor books o umerical aalsis do o rea hem a all. Thus we will ol skech a geeral approach o problems ivolvig such equaios. Parial diereial equaios orm he basis or so ma problems i sciece, ha o limi he choice o examples. Mos o he udameal laws o phsical sciece are wrie i erms o parial diereial equaios. Thus oe ids hem prese i compuer modelig rom he hdrodamic calculaios eeded or airplae desig, weaher orecasig, ad he low o luids i he huma bod o he damical ieracios o he elemes ha make up a model ecoom. parial derivaive simpl reers o he rae o chage o a ucio o ma variables, wih respec o us oe o hose variables. I erms o he amiliar limiig process or deiig diereials we would wrie F(x, x, L, x F(x, x, L, x, Lx F(x, x, L, x x, Lx Lim. (5..9 x x x Parial diereial equaios usuall relae derivaives o some ucio wih respec o oe variable o derivaives o he same ucio wih respec o aoher. The oio o order ad degree are he same as wih ordiar diereial equaios. lhough a parial diereial equaio ma be expressed i muliple dimesios, he smalles umber or illusraio is wo, oe o which ma be ime. Ma o hese equaios, which describe so ma aspecs o he phsical world, have he orm z(x, z(x, z(x, z z a(x, b(x, c(x, F x,,z,. x x x (5..9 ad as such ca be classiied io hree disic groups b he discrimiae so ha [b (x, a(x, c(x, ] < Ellipic [b (x, a(x, c(x, ] Parabolic [b (x, a(x, c(x, ] > Hperbolic. (

27 5 - Diereial ad Iegral Equaios Should he equaio o ieres all io oe o hese hree caegories, oe should search or soluio algorihms desiged o be eecive or ha class. Some mehods ha will be eecive a solvig equaios o oe class will ail miserabl or aoher. While here are ma diere echiques or dealig wih parial diereial equaios, he mos wide-spread mehod is o replace he diereial operaor b a iie dierece operaor hereb urig he diereial equaio io a iie dierece equaio i a leas wo variables. Jus as a umerical iegraio scheme ids he soluio o a diereial equaio a discree pois x i alog he real lie, so a wo dimesioal iegraio scheme will speci he soluio a a se o discree pois x i,. These pois ca be viewed as he iersecios o a grid. Thus he soluio i he x- space is represeed b he soluio o a iie grid. The locaio o he grid pois will be speciied b he iie dierece operaors or he wo coordiaes. Ulike problems ivolvig ordiar diereial equaios, he iiial values or parial diereial equaios are o simpl cosas. Speciig he parial derivaive o a ucio a some paricular value o oe o he idepede variables sill allows i o be a ucio o he remaiig idepede variables o he problem. Thus he ucioal behavior o he soluio is oe speciied a some boudar ad he soluio proceeds rom here. Usuall he iie dierece scheme will ake advaage o a smmer ha ma resul or he choice o he boudar. For example, as was poied ou i secio. here are hiree orhogoal coordiae ssems i which Laplace's equaio is separable. Should he boudaries o a problem mach oe o hose coordiae ssems, he he iie dierece scheme would be oall separable i he idepede variables greal simpliig he umerical soluio. I geeral, oe picks a coordiae ssem ha will mach he local boudaries ad ha will deermie he geomer o he grid. The soluio ca he proceed rom he iiial values a a paricular boudar ad move across he grid uil he eire space has bee covered. O course he soluio should be idepede o he pah ake i illig he grid ad ha ca be used o esimae he accurac o he iie dierece scheme ha is beig used. The deails o seig up various pes o schemes are beod he scope o his book ad could serve as he subec o a book b hemselves. For a urher iroducio o he soluio o parial diereial equaios he reader is reerred o Sokoliko ad Redheer 7 ad or he umerical implemeaio o some mehods he sude should cosul Press e.al. 8. Le us ow ur o he umerical soluio o iegral equaios. 5. The Numerical Soluio o Iegral Equaios For reasos ha I have ever ull udersood, he mahemaical lieraure is crowded wih books, aricles, ad papers o he subec o diereial equaios. Mos uiversiies have several courses o sud i he subec, bu lile aeio is paid o he subec o iegral equaios. The diereial operaor is liear ad so is he iegral operaor. Ideed, oe is us he iverse o he oher. Equaios ca be wrie where he depede variable appears uder a iegral as well as aloe. Such equaios are he aalogue o he diereial equaios ad are called iegral equaios. I is oe possible o ur a diereial equaio io a iegral equaio which ma make he problem easier o umericall solve. Ideed ma phsical pheomea led hemselves o descripio b iegral equaios. So oe would hik ha he migh orm as large a area or aalsis are do he diereial equaios. Such is o he case. Ideed, we will o be able o devoe as much ime o he discussio o hese ieresig equaios as we should, bu we shall sped eough ime so ha he sude is a leas amiliar wih some o heir basic properies. O ecessi, we will resric our discussio o hose iegral equaios where he ukow appears liearl. Such liear equaios are more racable ad e describe much ha is o ieres i sciece. 47

28 Numerical Mehods ad Daa alsis a. Tpes o Liear Iegral Equaios We will ollow he sadard classiicaio scheme or iegral equaios which, while o exhausive, does iclude mos o he commo pes. There are basicall wo mai classes kow as Fredholm ad Volerra aer he mahemaicias who irs sudied hem i deail. Fredholm equaios ivolve deiie iegrals, while Volerra equaios have he idepede variable as oe o he limis. Each o hese caegories ca be urher subdivided as o wheher or o he depede variable appears ouside he iegral sig as well as uder i. Thus he wo pes o Fredholm equaios or he ukow φ are b F(x K(x, φ(d Fredholm Tpe I a b, (5.. φ(x F(x λ K(x, φ(d Fredholm Tpe II a while he correspodig wo pes o Volerra equaios or φ ake he orm x F(x K(x, φ(d Volerra Tpe I a x. (5.. φ(x F(x λ K(x, φ(d Volerra Tpe II a The parameer K(x, appearig i he iegrad is kow as he kerel o he iegral equaio. Is orm is crucial i deermiig he aure o he soluio. Cerail oe ca have homogeeous or ihomogeeous iegral equaios depedig o wheher or o F(x is zero. O he wo classes, he Fredholm are geerall easier o solve. b. The Numerical Soluio o Fredholm Equaios Iegral equaios are oe easier o solve ha a correspodig diereial equaio. Oe o he reasos is ha he rucaio errors o he soluio ed o be averaged ou b he process o quadraure while he ed o accumulae durig he process o umerical iegraio emploed i he soluio o diereial equaios. The mos sraigh-orward approach is o simpl replace he iegral wih a quadraure sum. I he case o Fredholm equaios o pe oe, his resuls i a ucioal equaio or he ukow φ(x a a discree se o pois used b he quadraure scheme. Speciicall F(x Σ K(x, φ( W R (x. (5.. Sice equaio (5.. mus hold or all values o x, i mus hold or values o x equal o hose chose or he quadraure pois so ha x,,, L,. (5..4 B pickig hose paricular pois we ca geerae a liear ssem o equaios rom he ucioal equaio (5.. ad, eglecig he quadraure error erm, he are F(x i Σ K(x i, φ( W Σ i φ(x i,,, L,, (5..5 which ca be solved b a o he mehods discussed i Chaper ieldig 48

29 k k 5 - Diereial ad Iegral Equaios φ( x F(x,,, L,. (5..6 The soluio will be obaied a he quadraure pois x so ha oe migh wish o be careul i he selecio o a quadraure scheme ad pick oe ha coaied he pois o ieres. However, oe ca use he soluio se φ(x o ierpolae or missig pois ad maiai he same degree o precessio ha geeraed he soluio se. For Fredholm equaios o pe, oe ca perorm he same rick o replacig he iegral wih a quadraure scheme. Thus k φ( x F(x λ K(x, φ( W R (x. (5..7 Here we mus be a lile careul as he ukow φ(x appears ouside he iegral. Thus equaio (5..7 is a ucioal equaio or φ(x isel. However, b evaluaig his ucioal equaio as we did or Fredholm equaios o pe we ge φ( x F(x λ K(x, φ( W, (5..8 which, aer a lile algebra, ca be pu i he sadard orm or liear equaios ha have a soluio i i F(x [ δ λk(x, W ] φ( B φ(x i,,, L,, (5..9 i i i k k i i φ( x B F(x,,, L,. (5.. k Here he soluio se φ(x ca be subsiued io equaio (5..7 o direcl obai a ierpolaio ormula or φ(x which will have he same degree o precisio as he quadraure scheme ad is valid or all values o x. Such equaios ca be solved eiciel b usig he appropriae Gaussia quadraure scheme ha is required b he limis. I addiio, he orm o he kerel K(x, ma iluece he choice o he quadraure scheme ad i is useul o iclude as much o he behavior o he kerel i he quadraure weigh ucios as possible. We could also choose o break he ierval a b i several pieces depedig o he aure o he kerel ad wha ca be guessed abou he soluio isel. The subseque quadraure schemes or he subiervals will o he deped o he coiui o polomials rom oe sub-ierval o aoher ad ma allow or more accurae approximaio i he sub-ierval. For a speciic example o he soluio o Fredholm equaios, le us cosider a simple equaio o he secod pe amel (x x d. (5.. Comparig his o equaio (5..7, we see ha F(x, ad ha he kerel is separable which leads us immediael o a aalic soluio. Sice he iegral is a deiie iegral, i ma be regarded as some cosa α ad he soluio will be liear o he orm (x αx ( αd x( α. (5.. 49

30 Numerical Mehods ad Daa alsis This leads o a value or α o α /4. (5.. However, had he equaio required a umerical soluio, he we would have proceeded b replacig he iegral b a quadraure sum ad evaluaig he resulig ucioal equaio a he pois o he quadraure. Kowig ha he soluio is liear, le us choose he quadraure o be Simpso's rule which has a degree o precisio high eough o provide a exac aswer. The liear equaios or he soluio become ( ([(( 4( ( (]/ 6 ( ( [(( 4( ( ([(( 4( ( ( (]/ 6 (]/ 6 ( ( / 6 ( / / ( / 6, (5..4 which have he immediae soluio ( (. ( ( 4 Clearl his soluio is i exac agreeme wih he aalic orm correspodig o α/4, (x x/4. (5..6 While here are variaios o a heme or he soluio o hese pe o equaios, he basic approach is icel illusraed b his approach. Now le us ur o he geerall more ormidable Volerra equaios. c. The Numerical Soluio o Volerra Equaios We ma approach Volerra equaios i much he same wa as we did Fredholm equaios, bu here is he problem ha he upper limi o he iegral is he idepede variable o he equaio. Thus we mus choose a quadraure scheme ha uilizes he edpois o he ierval; oherwise we will o be able o evaluae he ucioal equaio a he releva quadraure pois. Oe could adop he view ha Volerra equaios are, i geeral, us special cases o Fredholm equaios where he kerel is K(x,, > x. (5..7 bu his would usuall require he kerel o be o-aalic However, i we choose such a quadraure ormula he, or Volerra equaios o pe, we ca wrie F(x i K(x i, x φ(x W x k a kh i i,,, L,. (5..8 No ol mus he quadraure scheme ivolve he edpois, i mus be a equal ierval ormula so ha 5

31 5 - Diereial ad Iegral Equaios successive evaluaios o he ucioal equaio ivolve he pois where he ucio has bee previousl evaluaed. However, b doig ha we obai a ssem o liear equaios i ( ukows. The value o φ(a is o clearl speciied b he equaio ad mus be obaied rom he ucioal behavior o F(x. Oe cosrai ha supplies he missig value o φ(x is df(x φ (a K(a, a. (5..9 dx The value o φ(a reduces equaios (5..8 o a riagular ssem ha ca be solved quickl b successive subsiuio (see secio.. The same mehod ca be used or Volerra equaios o pe ieldig i F(x i φ(x i K(x i, x φ(x W i,,, L, x a kh k. (5.. Here he diicul wih φ(a is removed sice i he limi as x a φ(a F(a. (5.. Thus i would appear ha pe equaios are more well behaved ha pe equaios. To he exe ha his is rue, we ma replace a pe equaio wih a pe equaio o he orm x K(x, F '(x K(x, x φ(x φ(d. a x (5.. Uoruael we mus sill obai F'(x which ma have o be accomplished umericall. Cosider how hese direc soluio mehods ca be applied i pracice. Le us choose equaio (5.., which served so well as a es case or diereial equaios. I seig ha equaio up or Picard's mehod, we ured i io a pe Volerra iegral equaio o he orm x x a (x x d. (5.. I we pu his i he orm suggesed b equaio (5..7 where he kerel vaishes or > x, we could wrie x d (x i x i (x x ( W, W, > i. (5..4 Here we have isured ha he kerel vaishes or >x b choosig he quadraure weighs o be zero whe ha codiio is saisied. The resulig liear equaios or he soluio become ( [(( 4(( ( [(( 4( ( [(( 4( ( ( ((]/ 6 (, ((]/ 6 ( (]/ 6 ( i /, i / 5( / 6, i. (5..5 The mehod o usig equal ierval quadraure ormulae o varig degrees o precisio as x icreases is expresses b equaio (5..8, which or our example akes he orm x i d (x i (x x ( W. (5..6 5

32 Numerical Mehods ad Daa alsis This resuls i liear equaios or he soluio ha are ( ( ( [(( ( ( ]/ ( / 4,. (5..7 ( [(( 4( ( (]/ 6 ( / 5( / 6 The soluios o he wo ses o liear equaios (5..5 ad (5..7 ha represe hese wo diere approaches are give i able 5.5 Table 5.5 Sample Soluios or a Tpe Volerra Equaio Fredholm Sol. Triagular Sol. alic Sol. (... % Error.%.% (½ % Error 6.8%.8% ( % Error -.8% -6.% s wih he oher examples, we have ake a large sep size so as o emphasize he relaive accurac. Wih he sep size agai beig ui, we ge a raher poor resul or he rapidl icreasig soluio. While boh mehod give aswers ha are slighl larger ha he correc aswer a x ½, he rapidl all behid he expoeiall icreasig soluio b x. s was suggesed, he riagular soluio is over all slighl beer ha he Fredholm soluio wih he discoiuous kerel. Whe applig quadraure schemes direcl o Volerra equaios, we geerae a soluio wih variable accurac. The quadraure scheme ca iiiall have a degree o precisio o greaer ha oe. While his improves as oe crosses he ierval he rucaio error icurred i he irs several pois accumulaes i he soluio. This was o a problem wih Fredholm equaios as he rucaio error was spread across he ierval perhaps weighed o some degree b he behavior o he kerel. I addiio, here is o opporui o use he highl eicie Gaussia schemes direcl as he pois o he quadraure mus be equall spaced. Thus we will cosider a idirec applicaio o quadraure schemes o he soluio o boh pes o iegral equaios. B usig a quadraure scheme, we are acil assumig ha he iegrad is well approximaed b a polomial. Le us isead assume ha he soluio isel ca be approximaed b a polomial o he orm φ(x i Σ α ξ (x. (5..8 Subsiuio o his polomial io he iegral o eiher Fredholm or Volerra equaios ields K(x, φ(d α K(x, ξ (d R α H (x R

33 5 - Diereial ad Iegral Equaios Now he eire iegrad o he iegral is kow ad ma be evaluaed o geerae he ucios H (x. I should be oed ha he ucio H (x will deped o he limis or boh classes o equaios, bu is evaluaio poses a separae problem rom he soluio o he iegral equaio. I some cases i ma be evaluaed aalicall ad i ohers i will have o be compued umericall or a chose value o x. However, oce ha is doe, pe oe equaios o boh classes ca be wrie as F(x i Σ α H (x i i,,, L,, (5.. which cosiue a liear ssem o ( algebraic equaios i he ( ukows α. These, ad equaio (5..8 suppl he desired soluio φ(x. Soluio or he pe equaios is ol slighl more complicaed as equaio (5..8 mus be direcl isered io he iegral equaio a evaluaed a xx i. However, he resulig liear equaios ca sill be pu io sadard orm so ha he α s ca be solved or o geerae he soluio φ(x. We have said ohig abou he ucios ξ (x ha appear i he approximaio equaio (5..8. For omial polomial approximaio hese migh be x, bu or large such a choice eds o develop isabiliies. Thus he same sor o care ha was used i developig ierpolaio ormulae should be emploed here. Oe migh eve wish o emplo a raioal ucio approach o approximaig φ(x as was doe i secio.. Such care is usiied as we have iroduced a addiioal source o rucaio error wih his approach. No ol will here be rucaio error resulig rom he quadraure approximaio or he eire iegral, bu here will be rucaio error rom he approximaio o he soluio isel [i.e. equaio (5..8]. While each o hese rucaio errors is separael subec o corol, heir combied eec is less predicable. Fiall, we should cosider he easibili o ieraive approaches i coucio wih quadraure schemes or idig soluios o hese equaios. The pe equaios immediael sugges a ieraive ucio o he orm φ b (k (k ( x F(x λ K(x, φ (d. (5.. a Rememberig ha i is φ(x ha we are aer, we ca use equaio (.. ad he lieari o he iegral equaios wih respec o φ(x o esablish ha he ieraive ucio will coverge o a ixed poi as log as λ b K(x, d <. (5.. a Equaio (5..7 shows us ha a Volerra equaio is more likel o coverge b ieraio ha a Fredholm equaio wih a similar kerel. I λ is small, he o ol is he ieraive sequece likel o coverge, bu a iiial guess o φ ( (x F(x. (5.. suggess isel. I all cases iegraio required or he ieraio ca be accomplished b a desireable quadraure scheme as he prelimiar value or he soluio φ (k- (x is kow. 5

34 Numerical Mehods ad Daa alsis d. The Iluece o he Kerel o he Soluio lhough he lieari o he iegral operaor ad is iverse relaioship o he diereial operaor eds o make oe hik ha iegral equaios are o more diicul ha diereial equaios, here are some suble diereces. For example, oe would ever aemp a umerical soluio o a diereial equaio ha could be show o have o soluio, bu ha ca happe wih iegral equaios i oe is o careul. The presece o he kerel uder he operaor makes he behavior o hese equaios less raspare ha diereial equaios. Cosider he apparel beig kerel K(x, cos(x cos(, (5..4 ad a associaed Fredholm equaio o he irs pe a a F(x cos(xcos(φ(d cos(xz(a. (5..5 Clearl his equaio has a soluio i ad ol i F(x has he orm give b he righ had side. Ideed, a kerel ha is separable i he idepede variables so as o have he orm K(x, P(xQ(, (5..6 places cosrais o he orm o F(x or which he equaio has a soluio. Neverheless, i is coceivable ha someoe could r o solve equaio (5..5 or ucioal orms o F(x oher ha he hose which allow or a value o φ(x o exis. Udoubedl he umerical mehod would provide some sor o aswer. This probabl promped Baker 9, as repored b Craig ad Brow, o remark 'wihou care we ma well id ourselves compuig approximae soluios o problems ha have o rue soluios'. Clearl he orm o he kerel is crucial o aure o he soluio, ideed, o is ver exisece. Should eve he codiios imposed o F(x b equaio (5..5 be me, a soluio o he orm φ(x φ(x ζ(x, (5..7 where φ(x is he iiial soluio ad ζ(x is a ai-smmeric ucio will also sais he equaio. No ol are we o guaraeed exisece, we are o eve guaraeed uiqueess whe exisece ca be show. Foruael, hese are oe us mahemaical cocers ad equaios ha arise rom scieiic argumes will geerall have uique soluios i he are properl ormulaed. However, here is alwas he risk ha he ormulaio will iser he problem i a class wih ma soluios ol oe o which is phsical. The ivesigaor is he aced wih he added problem o idig all he soluios ad decidig which oes are phsical. Tha ma prove o be more diicul ha he umerical problem o idig he soluios. There are oher was i which he kerel ca iluece he soluio. Craig ad Brow devoe mos o heir book o ivesigaig he soluio o a class o iegral equaios which represe iversio problems i asroom. The show repeaedl ha he presece o a iappropriae kerel ca cause he umerical mehods or he soluio o become wildl usable. Mos o heir aeio is direced o he eecs o radom error i F(x o he subseque soluio. However, he rucaio error i equaio (5.. ca combie wih F(x o simulae such errors. The implicaios are devasaig. I Fredholm equaios o Tpe, i λ is large ad he kerel a weak ucio o, he he soluio is liable o be exremel usable. The reaso or his ca be see i he role o he kerel i deermiig he soluio φ(x. K(x, behaves like a 54

35 5 - Diereial ad Iegral Equaios iler o he coribuio o he soluio a all pois o he local value o he soluio. I K(x, is large ol or x he he coribuio o he res o he iegral is reduced ad φ(x is largel deermied b he local value o x [i.e. F(x]. I he Kerel is broad he disa values o φ( pla a maor role i deermiig he local value o φ(x. I λ is large, he he role o F(x is reduced ad he equaio becomes more earl homogeeous. Uder hese codiios φ(x will be poorl deermied ad he eec o he rucaio error o F(x will be disproporioael large. Thus oe should hope or o-separable Kerels ha are srogl peaked a x. Wha happes a he oher exreme whe he kerel is so srogl peaked a x ha i exhibis a sigulari. Uder ma codiios his ca be accommodaed wihi he quadraure approaches we have alread developed. Cosider he ulimael peaked kerel K(x, δ(x-, (5..8 where δ(x is he Dirac dela ucio. This reduces all o he iegral equaios discussed here o have soluios φ(x F(x pe. (5..9 φ(x F(x( λ pe Thus, eve hough he Dirac dela ucio is "udeied" or zero argume, he iegrals are well deied ad he subseque soluios simple. For kerels ha have sigulariies a x, bu are deied elsewhere we ca remove he sigulari b he simple expedie o addig ad subracig he aswer rom he iegrad so ha φ (k b ( x F(x λ K(x, [ φ( φ(x]d λφ(x K(x, d. (5..4 a We ma use he sadard quadraure echiques o his equaio i he ollowig codiios are me: b K(x, d <, x a. (5..4 Lim{K(x, [ φ( φ(x]} x The irs o hese is a reasoable cosrai o he kerel. I ha is o me i is ulikel ha he soluio ca be iie. The secod codiio will be me i he kerel does o approach he sigulari aser ha liearl ad he soluio saisies a Lipshiz codiio. Sice his is rue o all coiuous ucios, i is likel o be rue or a equaio ha arises rom modelig he real world. I his codiio is me he he coribuio o he quadraure sum rom he erms where (i ca be omied (or assiged weigh zero. Wih ha sligh modiicaio all he previousl described schemes ca be uilized o solve he resulig equaio. lhough some addiioal algebra is required, he resulig liear algebraic equaios ca be pu io sadard orm ad solved usig he ormalisms rom Chaper. I his chaper we have cosidered he soluio o diereial ad iegral equaios ha arise so oe i he world o sciece. Wha we have doe is bu a brie surve. Oe could devoe his or her lie o he sud o hese subecs. However, hese echiques will serve he sude o sciece who wishes simpl o use hem as ools o arrive a a aswer. s problems become more diicul, algorihms ma eed o become more sophisicaed, bu hese udameals alwas provide a good begiig. a b 55

36 Numerical Mehods ad Daa alsis Chaper 5 Exercises. Fid he soluio o he ollowig diereial equaio ', i he rage. Le he iiial value be (. Use he ollowig mehods or our soluio: a. a secod order Ruge-Kua b. a -poi predicor-correcor. c. Picard's mehod wih seps. d. Compare our aswer o he aalic soluio.. Fid he soluio or he diereial equaio x " x' (x -6, i he rage wih iiial values o '((. Use a mehod ou like,bu explai wh i was chose.. Fid he umerical soluio o he iegral equaio (x ((x x 5 5 d, x. Comme o he accurac o our soluio ad he reaso or usig he umerical mehod ou chose. 4. Fid a closed orm soluio o he equaio i problem o he orm (x ax bx c, ad speciicall obai he values or a,b, ad c. 5. How would ou have umericall obaied he values or a, b, ad c o problem 4 had ou ol kow he umerical soluio o problem? How would he compare o he values obaied rom he closed orm soluio? 56

37 6. We wish o id a approximae soluio o he ollowig iegral equaio: 5 - Diereial ad Iegral Equaios (x x x ( d. a. Firs assume we shall use a quadraure ormula wih a degree o precisio o wo where he pois o evaluaio are speciied o be x.5, x.5, ad x.75. Use Lagrage ierpolaio o id he weighs or he quadraure ormula ad use he resuls o id a ssem o liear algebraic equaios ha represe he soluio a he quadraure pois. b. Solve he resulig liear equaios b meas o Gauss-Jorda elimiaio ad use he resuls o id a ierpolaive soluio or he iegral equaio. Comme o he accurac o he resulig soluio over he rage. 7. Solve he ollowig iegral equaio: B(x / B(E -x d, where E (x e -x d/. a. Firs solve he equaio b reaig he iegral as a Gaussia sum. Noe ha Lim E x, x b. Solve he equaio b expadig B( i a Talor series abou x ad hereb chagig he iegral equaio io a h order liear diereial equaio. Cover his equaio io a ssem o irs order liear diereial equaios ad solve he ssem subec o he boudar codiios ( B( B, B' ( B"( B (. Noe ha he iegral equaio is a homogeeous equaio. Discuss how ha ilueced our approach o he problem. 57

38 Numerical Mehods ad Daa alsis Chaper 5 Reereces ad Supplemeal Readig. Hammig, R.W., "Numerical Mehods or Scieiss ad Egieers" (96 McGraw-Hill Book Co., Ic., New York, Sa Fracisco, Toroo, Lodo, pp Press, W.H., Flaer, B.P., Teukolsk, S.., ad Veerlig, W.T., "Numerical Recipies he r o Scieiic Compuig" (986, Cambridge Uiversi Press, Cambridge, pp Press, W.H., Flaer, B.P., Teukolsk, S.., ad Veerlig, W.T., "Numerical Recipies he r o Scieiic Compuig" (986, Cambridge Uiversi Press, Cambridge, pp Press, W.H., Flaer, B.P., Teukolsk, S.., ad Veerlig, W.T., "Numerical Recipies he r o Scieiic Compuig" (986, Cambridge Uiversi Press, Cambridge, pp Da, J.T., ad Collis, G.W.,II, "O he Numerical Soluio o Boudar Value Problems or Liear Ordiar Diereial Equaios", (964, Comm..C.M. 7, pp Fox, L., "The Numerical Soluio o Two-poi Boudar Value Problems i Ordiar Diereial Equaios", (957, Oxord Uiversi Press, Oxord. 7. Sokoliko, I.S., ad Redheer, R.M., "Mahemaics o Phsics ad Moder Egieerig" (958 McGraw-Hill Book Co., Ic. New York, Toroo, Lodo, pp Press, W.H., Flaer, B.P., Teukolsk, S.., ad Veerlig, W.T., "Numerical Recipies he ar o scieiic compuig" (986, Cambridge Uiversi Press, Cambridge, pp Baker, C.T.N., "The Numerical Treame o Iegral Equaios", (977, Oxord Uiversi Press, Oxord.. Craig, I.J.D., ad Brow, J.C., (986, "Iverse Problems i sroom - Guide o Iversio Sraegies or Remoel Sesed Daa", dam Hilger Ld. Brisol ad Boso, pp. 5. Craig, I.J.D., ad Brow, J.C., (986, "Iverse Problems i sroom - Guide o Iversio Sraegies or Remoel Sesed Daa", dam Hilger Ld. Brisol ad Boso. 58