CH08 - 44‘? Irlll'rF'Ilnl 't_I' ll 'lr'l'Ii-J' l' '5'...

Info iconThis preview shows pages 1–21. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 44‘? Irlll'rF'Ilnl 't_I' ll 'lr'l'Ii-J' l' '5' .irrhnsrin, L tat, Hiess. R. [1. and Arnold. .l. T.. transduction to LittertrAt'gebrn {5th etil [Readinghfltt “ix “"4” 5‘" “"3 hflmuge'“w“5 System. canes“; "'l 8 {a} Determine a fundamental ma '1 __ '.- Eq. it}. Enter to Problem 25 of Section 16. fi _ I _ I {h} It ttrt :— e‘”. determine the solution of the system {i} that also satisfies il,l:.._::I-.I_IE:‘ I: conditions sttl} = I]. _ -- . I I . . _ _.t-_‘L - and 15 verify that the given 1vector ts the general sol'ufiflnlfifi ' t. I. 1 'r 'a IF-‘FJ' tem. and then solve the nonhomogenenus system mm“:- ln each of Problems [4 3. —2 —31’ J” 1 l rl+f 2 r: 15.ts':(__ a. 3+ I...t__l ' l ‘H 3 L1 to. Let s : dirt“ he the genera! solution of s.’ 2 Pulse + gut, and [at x =vfl) ha iflmé ol' the same system. By considering the dtfterettce g6“) -- titlt-Shtifh; particular solution I where tltt'} is the general solution of the homogeneousf'sjrg‘tgm that em} : utt} + Ho. '1. : Pitta. I? t'fonsitler the initial talne problem '.'t'. = As. + girl. stt'tt = a". tat tst rel'errine to Problem lite} in Section Ilshoat that 1f s- = at t is“ + ] (Pt! — rigor} as. UP“) [hi5 Pflifltwe have discussed methods for solving differential equations be nsme a analytical techniques such as integration or series expansions Lisuattt'. the emphhast: seas on finding an exact expression for the solution. Unfortunately. there are main. 3 “pi—J“ H“ + GEN-A” - “Igm d3. giggighprgblems in engineering and science. especially nonlinear ones. to n hieh I” _ ‘ o 5 either do not apply or are very complicated to use. in this chapter - we discuss an alternative approach. the use of numerical approsimation methods to obtain an accurate approximation to the solution of an initial value Dtt't'f‘lti‘m- “'s' _ _ _ F P _ _m present these methods in the simplest possible contest. namely. a sinele scalar first _ "—""— order equation. However. the}: can readily he extended to st'slcmahiti tirst order equations. and this is outlined briefly in Section an The prt'tecijtares described her.- can he executed easii}.r on personal computers as melt as on some pocket calculator. t"'J Tahoe. also that t rsalo. it ith those of Problem 3? in Section 3.7. .If' '1 and linear algebra is at'ailahte in an}; introductory.r hook on the subject. a. rtratriees .r r-.--oru.-~ent.'itire sample. {one s_t_' . Eh rrtt'nrari- Lint-antittehrn thth ed.) [New York: Wileyflflflfll. iii. - 1L'li'_"~1- 'IIIIE; intuit. l'l .tltt'l l 3.1 The Euler or Tangent Line Method To discuss the development and use -. .Jttlthson—‘oesles. ltJtll I Holman. 3.. Lit-mettritrt Litrenr .Jtt'_t:t"i'tt'tt ll'ill'l ed.l {Upper Saddle River. NJ: PrflflliflE~HfliL HIE]. . '__ ;.. ' ._ .. "i noon-tees! i rt--r.'-..to--_ s e " .- E'L. -l". Leon. '5. 1.. Littear.--tl'gehnt nah Application:- thth ch (Hen. York: Macmillan. Zflflll. strane. t1. Linearatteehrn and to Annotations ch ethhIen' ‘t’ork: Academic Press. 1993). mainly on the first order initial t‘alu. trtt-i'tiettt Cttl'tt-llhiitT-T‘ u: re. it; Qt h i-_- t... L L__ . .t't —.- I I t!.t , iii and the initial condition :_lllllr- 1h- eetaneie It'l -'.‘-t--' plan. We assume that the functionsf and)“. are continuous on some r containing the point Hearst. "Hien. hs- Theorem 2.4.2. there esists a anion-e stem at t' = no} of the given problem in some intersal about r... it Ea ti l I'st‘tt.‘~l1ittte.t‘.lit..r‘= 441 Euler's niethed eensists ef repeatedly evaluating Eq. (3) 05(4): each step In eseeute the next step. In this way you flhffllfl'ifl-Eflq that appreximate the values at the snlutraa-e‘tfiahn painted eat that a eemputer Pregl'flm ffll' alias-'2" -r - f{|.l'|1r2.. a; J"... . . . In Seelinn 2.7 we I t the structure given helnw. The specific instruenens can be written” tn'any;3_'_ pit-grammng language. '4' -.' 'I The Euler Merlin-rt fiitep 1. define ftt,_vl Step 2. input initial values til and yfl Step 3. input step size it and number ef steps a Step ti. nulpul it} and }’U bitep 5. far i frtirn 1 It] it [It]: iHep ti. kl I {like} 1- :_t'+lfl*kl l = i' + h Htep 7. ntttput t' and y _. Step 8. end Ini- Milt-l: it Hume evnniples nf Euler's methnd appear in Sectien 2.7. As anethery .. '. 'III' .1 II cunsider the initial value prnhlem __‘ .- ,i'rzl—i'd'ilr’t ytll'} = l. Equatinn (S) is a first nrder linear equatinn. and it is easily Vitrified that the. '- l' satisfying the initial etmtlilinn (6) is 3.3 I _ _ 1 3 iv 4t If .; _‘Ir' — till” a— 1! H 'I'fi TEE . :;:'.=-. . Since the exact seluliun is knewn. we the net need numerical methods ':_ “mil” “"1"? Pml‘tlfl‘m (5). {6}. On the ether hand. the availability of the-e.:tact-‘"-”==i ntakes it easy tn determine the accuracy at any numerical procedure ' ' en this prnhlem. We will use this prehlem thrnughnut the chapter te illes. - Iii-.53 1-57"; .- eempare different numerical metheds. The sniutiens nt' Eq. (5-) diverge rath-i-ii'i'i‘i' ; [rem eaeh ether. se we slieuld expect that it will be fairly difficult teaan '~- ' Tbbegin tu MmMmhmMfi-ul mummies and alt-n It'- suggest ways tn ennstm mare mute ntgnnthm II. is helpha te met-trim um alternative ways In leak at the Euler method. First, let us write the differential equation it i It the pan: 1 = a. in the a -rrn 1.; - 3. a} Chapter'flt Nam-tart. . ' r to write. the problem as an integral I. ‘ “3””: mm” (1* ‘2)! “1" a“. . Another way to proceed is I _ _v = t;th is a solution of the mine 1,”, we obtain it.-|.-I I The Backward Eh'lhr-Ft'lrmttlht At 'variati a .. . . . . PPWWHI-mg- the derivative In fig. (3) by the backward difference quotient fatal Wm!“ = [ fihmnldh on on the-Euler formula can he ohtainedhy f r. ' '1 little) “ tetra—31.)]? it"lnsteadof [the fm-w- d [ff-I - -- r r m this way WE flhtaifl . .. . _ at t erence quotient used Ill Eq. (9). In first term“: = rill.le +f flfrtf’lfl] {ff- {filo} _ hallo-l) E hfttmyn}. t. ‘ or The meant! in Est- {101 is represented seemstfisflllr as the arse under the a I is = ls.-. + afresh). II' we approximate the integral 53’ Figure 3.1.1 between: = r... andr = in“. i g I .- t'tr. elrtl or its value fir... about] at r z: r... then we are approximating the act-1131511 hr the area of the shaded rectangle. In this way we obtain . fl. . suns-.1 E em +.f1r...etslltrt+. e s.) fl. = dillnl + h__t'[r,..err,.t|. H Hall}. to obtain an approximation rut. for tt: i r.,+1 t. we make a second appmflmmgg- br replacing out”?! by its approximate value _v., in Eq. t l 1}. This gtves the Euler it. | = 1.. + Kittens}: I. .r-‘t more accurate algorithm can he obtained hgappmxtmflsmog:- the integral more accurater This is discussed in Section 8.3. _- Stopping- the index up from H. to tr + 1, we obtain the backward Euler formula yn+l = yo + hf“n+h,}'n+l}- AEEHElIl‘lg that y.” is known and y“; is to be calculated.observe that Eq. (13) does not prov: e an explicit formula for v.1“. Rather, it is an equation that implicitly defines yum and must he solved to determine the value of gen“ . How difficult this is depends entirer on the nature of the function f. ' Use the backward Euler formula (13) and step sites ft = 0.05. [1025. [ill]. and 01111 to find E h a M P L E approstmate values of the solution of the initial value problem (5}.{6} on the interval t] 5 r 5 2. 2 For this problem the backward Euler formula (13} becomes .r'r 1! PM] = yo +_rn+1+ 4_lrrt+l y' 2 fit. ottll We 1|rtrill show the first two steps in detail so that it will be clear how the method works At the first step we have yr=_va+ltt1—r. «1- 4st,} = l + teaser — {1-05 + 4y. 1. Solving this equation for 'v] . we obtain r... 1‘ tr. e. 1.0475ft}.s : tenures. FIGLJRE 3.1.1 integral derivation of the Euler method. IFri—l Observe that hocause the differential equation is linear. the implicit equation for it I~ also linear and therefore easy to solve. Nest. v; :3“ +htl — :1 +411: : lfithlfiilfi +1 tll.ll:~lt. — t‘I-l + Jr‘s. at third approach is to assume that the solution j.“ = on] has a Tavlor series-about the point t... Then ' which leads to if} = LES-«ljffifllfi = it‘llrilhi'i-‘i -1:- _ _ ,, fl: . _ . I . . _ _ . , ., 11-bit” + h} = ¢Ilfn| + ¢u [in it: + qt; {Int— + - - - ‘ Conttnmng the eomputatronsonacomputcr. at: certain the-results shown in Tahlt -s-r._ in. 2! - values given by the backward Euler method are uniformlr too large for this problem. v. here-as or the values obtained from the Fuler method were too small. in this problem the errors are he somewhat larger for the backward Euler method than for the Enter method. atria-ugh Int (12-) small values of h the differences are insignificant. Since the hacltward Euler method appears etrttttt = eta.) +f'[r“.¢rr,,i]h + edit”)?! +. _ E to he no more accurate than the Euler method. and is somewhat more complicated. a naturar H [ha Hales i5 terminalfld after the firm 1WD [Ermi and ¢lfrr+t l and If? [tel affl'ffiplmfl question is whv it should even he mentioned. The answer is that it is the simplest esamplc of a bi“ [hair appmflmflte l"film—‘5 _l*a+i and ya. we again obtain the Euler formula It class of methods known as backward differentiation formulas that are very usetul tor certain more terms in the series are retained. a more accurate formula is obtained. Futillflr types of differential equations. We will return to this issue later In this chapter. ' It 1'. 1'- E I I ‘ fy = 1 1. cm arisen 11f Results fer the Numertcal 51:1] utten n I '- I II I 335331513 -- T _ 5:11.115 11 = 11.1125 11 = 11-01 a .. 11 11111 11d F 1.111111111111111 1.111111111111111 1.111111112131113 1mg 11 1 1.5111555 1.5414315 1.52.111 2.509531 111 1.1111 1151111 2.1121 1.111111 2.5411355 3.83963” 111' 141145.111 111111111555 5.9255121 5.8131232 11:1 1111111155111 5.5111115511 5.5955353 3.74??fim 111 1111111111155 t1.111151111111312 11151111113 .4 164 11—1 1111115111 511.41.111.51 111.451.1115 55. 1119 11 11511111311 1111 1.1111151 542.1245: 455.05525 111 5111111511111 5455121111 111151115 355111415 —I——- ___ -...1 'ulr'u Errors 1‘11 Numerical Approximations. The use of a numerical pTDEEdUIE.ISIJCh3.511153%?- 1'11r111ula. 111 salve 1111 initial value prehlem raises anumher 11f questinns that answered hetere the apprnsimate numerical scluttnn can he accented a5 sattsfanggfig tine 111' these is the questicn 11f canvergence. That 15. as the step Stze h tends 11111111: values ml the numerical selutinn 11.13. . . . ._v,,. . . . appreach the enrrespufljw 5‘11} 1.1111111 111- 11111 actual 511111ti11n'j’ If we assume that. the answer 15 affirmative; 1'1-11111'1115 1l1e i1111111115111 pr1‘1ctie1-1I question at h1.1w rapidly the numerical appr‘flximfifififl 1:1 1111 1 11-1-11 111 the s11lu111111. In 11thcr w111cls.ht1wsmall a step s1ze 15 needed tn 1111151111111. 511.11'111'1le1* 11 elven level 111'11eeuraev'i' We want to use a step size that is small 5111111333.- 111 11-1111 1111.- 1-1-1111ire1'J aeeuraev. but 11111 11111 small. A11 unnecessarily small 111.111 11 the ealeulatiens. makes them mere expensive, and 1n seme came-mar? 1'1 1:11:11; .1 1111511! accuracy. 11.1.1. :1 111.. 1-1111d11n1e11111l 511urce5 11f error in sclving an initial value 111111115111: -1-_.1|!1.. L111 115 111-11 assume that 11111 cemputer i5 such that we can carry 131111151] “'1‘. '.'-1"~. 51:1-11151111111, 1.1.111 emnplele accuracy; that is. we can retaln an 1nfin1te numhemf ._1__ _._11-1;.E 111111-11 F111- difference E1. between the salutinn _1- = 111(1) cf the initial 11.111115 1 l 1 131.1111! its numerical apprcsimatlnn 15 g1ven h}; E 11 : [fi' H H l _‘ ,‘1’11 111111 '11 human as 1l1e glnhal truncatinn errnr. It arises frem two causes: First. 111111511 1111-11 we use 1111 1111p111sin1ate f11rmula 11:1 determine (11,1111; 5613131111111 the input (131.31%! 1111111 11111-11 are 1'111lv appresimatelv cerrect since in general 11111") is net equal'tn 1.1-1- 11111111111: 111111; ’1“ = 11111111. then the «only errer in geing cue step is due 111 the 1115-11 an 111111111sia1ate [111mula. This e1111r is kncwn as the Inca! truncation err-111' 1'11-_ _ The sec11111l fundamental scurce 11f errnr i5 that we carr},r cut the cemputatmnstn arithmetic with nnlv a finite number 111' digits. This leads 11.1 a mend-eff 1111111931 delined hv ' RH : ,‘1’11 _ an where 1”“ is the v11er 1111111111131111111111111111511l frem the given numerical methud. The ahselute value 111' the tntal errer in ccmputing 11111”) is given by 11111.11 ~— 1’..1 211111.11 — _11, + _1-.. — 1’11. 116) primarily tn the local _h'uneatien errer. which is somewhat simpler. The round-eff errer 15 clearly mare randem in nature. It depends 1111 the type at mmpfllfir m the sequence 111 which the camputatinns are carried cut. the methed 11f munditm cfl. and 51:1 forth. An analysis at reund-ett errer is heyend the scope at this book: but 1t 15 pessible 111 51131 mere abeut it than ene might at first expect (see. fer example. Henrici). Same cf the dangers {rem reund-nff errer are discussed in Prnhlems 2:"- thmugh 27 and in Sectihn 3.5. Local Trancctinn Error fur the Euler Method. Let us assume that the selutinn ‘11- = @111 cf the tntttal value prehlcm {1). (2) has a cuntinueus seccnd derivative in the interval at 1ntere-st. Tc ensure this. we can assume that f._f,. and 1‘; are centinueus 0511M that if f has these nrnperlies and if 111 is a selutien cf the initial value preblem 1 I 1.121. than 11'111=_f[1.1111r1]. and by the chain rule, 11”1'1125111111111+_f..11.1111111111'111 =f1lf1¢lfll+f1-[t.1111111f[t.1a111]. 11111 Since the rivht side nf this e uatian is centinucus. 111" is alsn centinueus. 111 ‘1} Then. makine use at a Tavlnr clvnemial with a remainder 111 11-11 .1111i 1:..- ah11ut 1r . 1:1 - P - P we nhtain e1111+h1:11111n1+11111..1hw£1611i_11*1-. tl'r'tl where F" is some paint in the intervai 1.; is 1- 1,; +11. Subtracting; E11. 141 from E1]. (19}. and noting that 11111,I +111: 11111.11:111n1l1:- 1111 2: 151 .1111. 11.1.11:- 11111i 111.11 ¢lr”+]l Filth-41' = — y“; 'l" "" 'i 1:" l. 'l1._ -'r- :15}. ll l.'l.:'- |_h" ll - T131 ccmpute the Inca! truneatien errer 111' 5111111} E11. 1 21-1 '11 11.11 ~~1-'_1.:11111 . .1 that is. we take 11.111111“ 11111.11. Then 11e1n1n1etliatelj. see 1111111E1 1:111 tic-.1. 111.. 111:1: truncation errrn‘ 11,111 is t'f1urt ___:il’ll'fj+:IIL—l—-1LI._ : 1IIII “HIS ll'lt: It'ttliil t'rnneatien fitTfll‘ liftl ll‘tt.‘ Eul1t' fila‘lih‘wl 1.1 :‘I1'1"1.1‘Il"=.11 11'1111; 13.11.}; (If the step 11111111111111 the ptflpdt‘ltnnnlltt {1111:1111 1*1 111:1‘11‘15 1111 H11. 551111111 1:' 111 1111 cf the snlutiun 1,11. The espressien given h}. an. 1?111l1-111..1l~1 1111 11 -11111 1.1 11.1 .1 different lt‘tl‘ each step. 17‘». unit'crm [1111111111. 1.1li1l 1111 .111 Intelsat 11.1 H. 1~1 3311 1.11 1111. H1... --.-I1 .1 {I_ '1 4 I 111,]' .11” .. ._. "1. —. ' :11. “21111.; J wherfl M 15 “IL. minimum [1f 15111111 1111 the interval [11.31]. 5111111 F111 1.._. i I' " ' 1' ' E' 1' I1 1111 a cansideratinn cf the wurst pess1hle easefl-thaI 11-. Ha lar11st 111151111111 1.11111 1 Bq.(22)flhave If N ~ . . . - “ii LT _|I- I" - .- w... "E"! ' I ' I' ' - -s-;~:'_!..-,: .. Ho'wmr' “1: mm {m “mm by I mum“ ' '- - .- _ _ .. - errer is prawn-inane! tn I12. Thus-if}: ts reduced h!- E _ _ - I _ 1h: flflanges fer estimating E" is mere difficult than that far em- I: ._ the Inca] truncatien error, we can make an intuitive estimated-11113531”. agate -. errur at a fixed T :r a, as Enilnws Suppnse that we take a steps a? ,I i" = m + ah. In each step the errer is at mest Mill/2; thus the must "th2. Netting that It = [T —- tel/h. we find that the l2519-]-liars};r_.I fur the Euler methnd in geing frem ta In T is beuaded by .3: ':':. Iii—2"— =tT"‘tfllT. I}; I .I‘ ' This argument is net cuntpletc since it dues nut take inte aeeeunt_.itiitiel L. I.“ ,F “l mu: step will have in succeeding steps. Nevertheless, it can he shfi'f 131:; ---.'.; 9' - _ Elm-Hm _ Ileph II but! el- ulniml truncatiun error in using the Euler methed en a finite interval ‘ - Iiiun .I t'uttsiutti times It: see Prnblem 23 fer mere details. The Euler fiesta“; --. lee m d“! H .i lust nrtJer tttcthnd because its glehal truncatien errer is 'ifij~:_si_—.. _fleedfi_,s_tepmwm_1wm4=zu‘mfldmmflt In :‘1:r Them-etl-uflifmmuepfieMinflermmem-fldmm :r'nm'e'l' it] the step Silvie. . . I . - - ' PEA. [herein-me it Is mun: accessiblewe Witt hereafter use the lecal truncatten- ' arse-:4! .-- I t'ut'ttit ir-ut measure iii the accuracy {It a numerical methed and fercempaeingen ::! n ustlu ttla II we ha tie H prttlrt intermatien aheut the selutien Elf the give.in 1m Ii‘ticttt we can use the result ('21 ) tu nhtain mere precise infermatien aheilfi Incui truncut it In err: 1r varies with I. As an tsaan‘tlztlel censider the illustrative :‘jjr-fjéit. n resultsiaateremlfithfimmmmmwmuepmfligm danger ef nuances-labile Inland-GE emu Anetherappmch‘uteteqnhewmm We“ threughetttthe interntbyMMIhesfii-psiflur m lathe example preblemwemuttlneed tnreduceh by: Mere-taint! seems-gm =Utflf=lAMMIhltmtflri'mmiflthtfltpmuflm#fl- y' = l - t + 4y. 39(0) = 1 .3: " i All modem mmpmer eludes fi‘tl'sul'irg differential equate-ms base the {mt-titty e-t " ' adjusting the step size as needed We as: mm: In the; m en in the he“ mum tilt the interval U :j t :J 2. Let y = 9M!) he the snlutien fit the initial 3:35“ :' {35}. The“. as nuled previuusly. e ._ fl _ a -——r —--— --——-—- -~*—~—-— — - - - a - ~— — W} — {4r 3 + 1% “16 PROBLEMS beach at mm 1 throngs-s and was” man. A the use“ en! the pm me and lhcrct'ure —-—a———r—-..—_- “In: PM I“ r .= r; Luigi.” n a em) = we“. (a) Use the Euler method was s = ate .. . (1'!) Use the Euler mind lath h I: fill-— htluattnn {2 I.) then states that (c) U“ the ME“: w in h I “(5‘ testes? _ (a) Use immanmmwu a one En+l=T. tn-fihrfltn't'h. ‘2’ Ly=3+tfiyh Jim-L] ‘2 lrtfl—iw'i'. “than: ‘2 3‘f_3!_3h mu] ‘2. tr-h+e‘“. We! The appearance at the [actnr 19 and the rapid grewth ef e“Ir explain Wt!!!" I H'I this seetien with h = {LBS were net very accurate. tn earth tit-Pm WINE a”?! . I I _ I. I Fl. : ' _r {a} Use...ttie EulermelhaflWEHt-tt: '- - ' {in .r If {b} He: the Euler methed with hi='tl.fl‘1?§-l 0025- 1 _ .. {e} Use the haekward Euler method = b 25 {d} Use the backward Euler method with h = I}, 1 7. v’=t}.5—t+2y. ytfll= I3 3-)”=-5"3fs s. r m, ytfl}:3 filey=at+e . J 14 C em plate the ealeulatiens leading tn the entries in eeluttms three-antl-ftittjti-."?§;;'.J Ii Using three terms in the Tayler Sfififlfi seen in Eq. (12.) and taking-il- =19} titer-statute..- i I appresimate values ef the setutien et' the illustrative example it Elm; ital I _ n I 1nd {1 :t Campfire the results with these using the Euler'methed-anflt _— . r. -—- “- iijtiii tent-3.1:"; 3.;_. values. _ fiz- .I - .J'J' _. II I a. L- Hirtr; If_v = fiat-L what ts _t '3' 1' estimate the least truneatien errer fer the Eaters-l7“? -. . .Ittsttnfl teat- ;ifig.i;_.; 3 ' I ' 1- . in: _I_:. --II _ a sir: ' in each ttf Prehlems If} and l? terms at the seiutien t- : stir}. Obtain a heund fer a“, in terms ef t and Mt} ._ I __ I - . - i t s - ' htain amere attire-rite- ‘1 m t Inwreai ti 1. r 5 l. E} usinga termula [erthe selutien,e ‘ ‘_ I _1 __ _Z _x 1-”: r. Fm h = til campute a beund ter es and eempare it with the aetual erttiht Iti'E-tt_':-,1_-____ .- _ .r £1: _ I I n I I I. .- .atse sentputv at haunt‘l it" “1*:- ETTUT E4 in “13 {With “all ,- t .- I. . 3‘?“ te fit} about t = est. in i' : "v - 1 vttii = l 17- J” = — t +2)“. Jim): 1- it; s: theEutermetthutretainingenlvthreedigitsthreughem ' ' h' ' ' J“ " ~ ' “1E amt-Putnam determine aphtesistste values at“ the selutien at l' = 0.1. a2, 0.}, me In etteh et Prehlems th' threes-h El ehtain a fermula fer the teen! truneati'en errer :_. t 3 nnlhed in terms et t and the selutien git. .- - - flat 'l‘er saith-e} the—feltevrt'ngiuitial value emblems (a) Y'=.1-_t+4.a suits: .. 1H tlrfr T-t'l. _‘i'iiii=i _lr'l=5f-‘3fis _‘gtt ~,~' -.—. first e _r, _vttt = 3 21— Jr" = 2! +5”. rtfl} =1 (e) y’ =2Jt—3t, ytfl)=t Camparethe matte—with these ehtained in Esample t and in Prehtems t and 3. The small. differs-ems-hetwfln seine at these results reunded in three digits and the present results are due in rennet-eff errer. The reuse—eff errer vt-eeld hemme important if the -, eem utatien e u' a. in] Determine the selutien ,v = [hit] and draw a graph UfJ" = ‘5'“) fflrfl 5' 5' L55.- - 26 Th- 1; II II' I "E Dian? SEE-PS- ‘ ' [hi Determine apprevimate values el tilt” at r = 0.10.4. and 0.6 using the ti'i‘iiitttil 7 ' ' ' e D wiqu pmbmm fllmmlfi a mgr that {FEMS Mi?“ “f muni‘i‘im firm “hm - -**‘e-'-'--'- nearly equal hustlers are subtracted and the dtfierenee is then multiplied by. a large with it : til. Draw a hreken-line graph fer the appresimate selutien-and-ee'mp” ".sitti3g1i'itii fis- numbfl Evaluate rte-quantity the graph e!" the esaet selutien. - _ . - ~ te} Repeat the eemputatien ef part {b} fer [i 5 t 5 0.4, but take ti = 0.1. {d} Shetv hv eemputing the least truneatien errer that neither ef these steer--= 'I eientlj.r small. Determine a value et it te ensure that the teen] truneatien-ert'ett-tth "r litiz'i threegheut the interval 0 5 t 5 1. That sueh a small value ef h is requif; '- +1.- trem the fact that mas W'tt}! is large. '3. s; its] 22. fittest-tier the initial value prehlem -- 3." = ees 5st. ytfl} = 1. iii; I1-".I H . ' taste tseti mm'I'Itttt‘t-t titl'lfli in the fettewing ways: (a) First reund each entry.t inthe determinant te twe digits. (1ft) First reurtd each entry in the determinant in three digits. (e) Retain all feur'digits. Centpare this value with the results in pins tat and {hi ._ t-l - ,- [#:5: 23. In this prehlem we discuss the glehal truneatien errerasseeiated with the fer the initial value prehlem y“ = flay). _vtrfl} = y“. Assuming that the Hit“ _ _.':._ .: I" an: flaminuflufi in a clflgfld’ haunded [Egifln R “f the [y'planfl that innmdgs'ii= i .i ' 27. The distributive law all! --e} = ah -- er dees net held. in generf'l. it the predeets are “ne’ul- it can be ShflWfl that 1'th E fixifilfi fl mflfilfinl L Suflh that Why) ‘ffltJ’l ‘5: iii?-'1':'--"‘.‘i -' _ IrtJt'tttdEtt'l eff tea smaller number ef-digits Te shew this in a specific ease. take a = nae. where [L y} and it, ji} are any [we peints in R with the same I eeurditlatfi- [553 1" h = 3.191. and e = 11?. After-eneh mnifiplieatien. reund eff the last digit- et Seetien 2.3]. Further. we assume that Jt‘} is eentinueus, se the selutien tilt hearsay. i.l:'i..falJJ-"“ ' .-."| __._. 8.2 Improvements on the Euler Methnd ff t-I 1|- .1_|I ' ch . . . rev e fish I - . .-|.. 1. .. . .. ‘__.. I a- . u l I I I ‘ltlt'l'l'tflllh’ reblemsthe Eu I I fillih‘ftcientlv accErate results. much effert has been deveted tn the develcm__ _ .3 re efficient ntethcids 1n the next three sectiune We Will drecuss stame- mu ‘ r - u _ I niethcds. Censider the initial value prcblem 'l 1- }" =fll'._l'l. THU} =—‘ F1} l t r em} denute its selutiun. Recall frcirn Eq. (1U) Di Sectien 3.1 thatfié if _ = .- and I I _ I ‘ inteuratine the eiven differential equaticn trem r” tc t,.+1 . we ebtatn if”_l - [flirIl—r—l} : l a. l.. I the Euler turmula _1'l'l'l,-i = yr: + “flirt-Fill} as ill‘l'tllle‘d hr replacinallt. tit-till in Eq. {I} bv its apprcsimate value fumyfl} at “m It u'llLiPt"lll[ til the interval cil' integratmn. it: Erwfuijd gut” Farm” at better appreisimatc t'crmula can he ehtaincd it‘ll-1e erhttl in. [fl is apprt‘uimnted tttcitc HCCUTHWIF- Om: will? 1'0 fit} this ts ill H ind m “h. Jam—3g: at its values at the twe endpctnts, namely, I: _I1 f. i; __:_f__mll,_a 1H 1 This is equivalent te apprcstmatrng the area has - :h Fiuure all] hetueen t = l“ and t = Isa] b3“ “1‘3 3W“ Elf “1'3 Shades fills {.UMHIT. m: Irwin-E {emit and retinal} he their respective appresimnte .. hi this has we uhtain. truth EL]. {2}. . l‘lrl- li'le' !i"|i~."__'T' I F._'|.'”:l I iii]... i11f+l .- _rii:_ l'_I,I I II' 7' 1}: --t = 'l' Hun..- the titti;h~.'it.'.lt l_.__5 appears as uric cf the arguments at f on the right side-ref Eu. i-lII- this aquatic-It delines the. implicitly rather than explicitly. Dependiflgfill the nature ul the funetiun t"- it may be fairly difficult tcl salve Eq. {4) her jun“. This fir-tri FIGURE 8.2.] Dcrivatiun cf the imprcved Euler methcd. er method requires a ve r)! small step SEE-tflspa I I if -_ EXAMPLE __.. _ filmy") +fll'n + ht)!" + hfflmynllh yfl+l "" n+ 2 "+ H i n n . =yn+f f“ +2hy +hflh' (5) where In“ has been replaced by a. + h. Equetten (5). gives an explicit terntula fer cumputing ya“, the apprmtimate value cf tetra“), In terms cf the data at In. This tenants is knewn as the impreved Euler fermnla er the Helm fermula. The imprcved Euler terrnula is an example of a twin-stage methcd; that is, we first calculate y" + lrfi. from the Euler [ennula and then use this result he calculate ts.“ frcrn Eq. (5). The impreved Euler fermula (S) decis represent an imprevement ever the Euler fermula (3) because the lflfifll truncatien errcr in using Eq. (5) is prcpcrtienal te lfiwhile fer the Euler methed it is Prflpflrttflnhl to it}. This errcr estimate fer the irnprcived Euler fermula is established in Preblem 14. It can else be shcwn that far a finite interval the glchal truncation errcr fer the impreved Euler fermula is heunded by a constant times hi. so this methccl is a seccnd crder methnd. Nate that this greater accuran is achieved at the expense cf mere ccmputatienai werk. since it is new necessary te evaluate firth twice in erder te ge frem I” In a,“ r If f (t. 5*} depends cal}! en t and net en y, then sclving the differential equatien Jv’ = f (t, y} reduces tn integrating ftt}. In this case the impreved Euler fermula t5) hccnrncs it he *J'i. = glfttnl +fun +hJ]. (6} which is just the trapezcid rulc fer numerical integratiun. Use the impruved Euler ferntula {5} tr: calculate apprcsiniate values cf the sclutiun cf the initial value prtiblem _v' = l -- l +4v. trill} = l I“! Te make clear exactly: what ccmputaticns are required. we shew a ccuple of steps tu detail Fer this prchlem fttnvt = l — l’ + 41v; hence Jill : I ‘— 'r - I" and filH-l-tll'tni-tllj} :l-t'1rlitl- Us It J Further, I” = U. in. :2 Land fit 2 I - 111+ it}. in is. it h : t}.|_’_-‘_‘i. [l'luli fut, + Fay“ + hilt} = t — ittflfi + 4[1 + tuttflfiujul : it Then. frcm Eq. (5]. t'; = l -i- [Gilli + 5.4?5H1H335l = [lit-“$.35. lair At the secend step we must calculate ft = 1 .- enas + titisassrsi = stasis. v. + afi = 1.130935 + artisans-tests} = Heathen a 'I - r .tI — I - -. and tits v. + tint = t — 1.1.05 + 4tI.2titl-tttti25'i = 61123625. Then. fret“ ELI- [5]” l ltnet‘ti + it} sitstasts + tihflhlfittltflflfit = 1.2?49671375. _r: : . . - - .. . s nhtained by using the impreved Euler methed with h. that 2.1. Te cempare the results el the imprpvetl Euler vi ith these ml the Euler methed. [little that the lllll'lrLWCd Eltlfi{hnii::::d evaluflfl3 n” m mill_ih Sign while the Euler methed requiresenly inumin I[ms-:ttee mufl Ul- [hg computing time in each step is spent in eta ua tugfi. ‘ f '31 _ ~ -' ~~ ' t hle wav te estimate the tetal ctttlllll-llme “NEIL m3“- flr a 31%“ Step 15 Li ham" 1 i e as manv evaluattens elf as the Euler methed. ' ‘ = - ‘ ires twic IITl titted hultr nttlhttLl rt-tl'J 1 -_ a . I H I PHI-W n illicit the imprinted Euler methed l'er step stir: it WEI UIWS 1h“ him“: number Uf evaluflflllllt' ‘ - ,_ I .1 til i as the Euler ntethed with step are. it. .__ Further results [m it 1'; r :3 and t't = ll.tll are given in Table ti. T 'tfll E *t 1 I it t‘ernparistin e-I' Results Using the Euler and lmpreved Euler methudgfir the lltitlitl ‘t-ltlttt: l’I'ttl‘llr.‘t'tl '-I = I J I + h ' m '_ _ EMU-___- ? littprevetl Euler .- it; ll [ll—m _ ——J't“=tl.lllll a = ates h = uttl Exact. . I __ I _ _ _ ___ __ . __. —- .—— -—---——-—--————-a—-———_.._.,._, 1] —— t—tit ti lt[J-| ttt |_tll_ll h'll'lltll l.lll'lllllllllll littltltltfl] ii I t Fitsjettt Itttttnjt‘stl lhllTLH'fi: l-fil-lllfifigfi 1160905518 ‘ ‘* tit—Ma‘s“: 2.51 ll l 15” Ifitlllh‘tlti lilhl't'lil? 3.5mm H. ‘- F-‘ttltl-i—lf; trtjtt" t fill 3-523435: 3.323914fi J__. l. |.|5_‘.T|_"'tl :1 Tri-tstfi stresses flatten s e nTTtlrsUj H.tit~l-l'-lll.ttl stems? stream ' II 5 l -_ —-t a“. “4 _tejfifist e-l.-l‘rl?*~l3l [14.83ll72'3 _.r -. .. saturates erasure trusts-as 47925919 ~ --i _t454 truth-t Hunt-Jill Biizfim'sl 35432001 ltltle a; t _ 't. nu can see that the impreved Euler methed with h = -"=t'..'~_t1 91th." t'tsttlts than the Euler methed with h : Il.lll. Nete that te reacht = Ewltlt'jlymr sI-._l" its. innit-tired Euler methed requires let] evaluatiens el_f.while the Euler I Elm. T‘ilere neteunrthv is that the impreved Euler methed with it = 11.325 eIlLLl‘lllt rite-re accuratg than the Euler methud with it : lilillll [200i] evaluatiuns tiff}. ant-tie v. uh settiething like tine-twelfth at the cemputing effert. the impreved Euler mellttfl “EM? “5“”“5 1-”? “ll-“t Efltltlem that are cemparahle te. er a bit better than. these generatedllgt the Euler methed- this illustrates that. compared In the Euler methed. the impreved Euler [lletl'ltttl is clearh mere ellicient. yielding substantially better results er requiring runaway. ti'ttal cernputing el't'ert. er heth. I _- The percentage errers at r = 2 fer the intpreved Euler methed are 1.23- % ler it = “-jlflt'r l-U'l- III = '"t'l!‘t'_' re ii; iii rt. s A cemputer pregram fer the Euler methed can be readily medified te hulllij'llal“ the impreved Euler methed instead. All that is required is te replace Stepshtttztltfl algerithm in Sectien H.l hy the fellewing: Chapter 3. . Variatinn aFStag Sim - _. . size as a calculatten p _ er ' ' I ' I - I sfl;:s:i:p:sttatfit level. The geal ts te use an mere steps than necessary and. at the Win dfiscrii‘eeh eep same-neutral ever the accuracy at the apprexhnatien. Here we ew this can he dene. Suppese that after in steps we have reached the i I . s Pl? HI (trays). We cheese a step size it and calculate ya“. Next we need In estimate 1 e errer We have made in calculatin [we c ' ' - ' ' ‘ ‘ ‘ alculated values Is an estimate ejfll ef the errer in using the angina] methed. If the estimated errer is different frem the errer telerance s. then we adjust the step size and repeat the calculatten. The key In making this adjustment efficiently is knewtng hew the lecal truncatien errer en.” depends en the step size it. Fer the Euler methed the lecal truncatten errer is prepertienal te It: . se te bring the estimated errer dewn (er up) In the telerance levels. we must multiply the eriginal step size by the Te illustrate this precedure.censider the example prehlem {7"}. fit" = l — l + 4y, it’ll-[ll I l- Yeu can verify that after ene step with h = [1.1 we ebtain the values t.:'t and 1.505 frettt the Euler methed and the impreved Euler methed. respectively. Thus the estimated errer in using the Euler methed is [High If we have ehesen an errer telerance of 0.05. fer instance, then we need in ElLlJLlSl the step size dewnward by the faster \/l}.{]:'i;'{l.tlllfi E 0.73. Reunding dewnward in he eenservatit e. let us cheese the adjusted step size ft = [till Then. frem the Euler lerrrtula. tie ehtain Using the impreved Euler tttethnrl. we Herein 3;. = l Betti-:1. s-t the 1"slili'ltt'e'tlCt'l’t'tl’ in using the Euler ferrnula islltt-lefifi. which is slightly less than the syn-titled inlet-an- . The actual errer. hased en a eetttpartsun with the st ~lutltin itself- is stint-sis hat greater. namely. 0.05 l 22. We can fellew the same precedure at each step el the calculi-then therehi Laying the lecal truncatien errer appresimately censtant threugheut the entire nunieiti. ll precess. Medern adaptive cedes fer selving differential equatiens adieu the step size as they preceed in very much this way. altheugh they usually use mere -teeurate fermulas than the Euler and impreved Euler lermulas. Censequently. the} are .ihle te achieve heth efficiency and accuracy by using very small steps enly wlte re the} a re really needed. m -_.‘ - ' flawlmfl" Wuhannudu h ' .— _ 1 I .Tr‘rfi _:l.'J . i -_ _:..'f._ .'_"-"l._.--'T Earl—1.... h _ II I . é. :- * '- fl‘} ‘—"-' will." a! .“"-=‘*':.-iT-‘”.':=:-. ' ’l 1 fifmfifl‘ HIM-Iflfl ‘3' f'miflhr'l _: I. ' ' hr: -_ Hie}, In mt. "I runaan 7 Ihnmh [2 find awmu willie-oi Ihmq‘fi “In: pmbkm i“ l =- H E III. I 5. and 2.". H fa] “III III: imp-'mcd hukr mlhml mlhh 1:110:15. r . E5” {h} Nu: lhc unprmcfl' Euifl Int-“mild *“h h " fl'mli ' * I r m H Ea Hm I '17: 3' I"! “5' " Jfi' - - _ _ s. . g. - - . u h a if L “I” _ ., ’91, w, y a: Barr-fr, no) ‘1‘. at? It if -_-I._n_'!‘__m Iii-1 " -‘ 2- - v -—- * - ' '-'~m..umm a...“ *flf' um 1- hill-III . ‘ ' [nut III-- ' Ill .- ' half-1‘ '_ " = g -. r ..,.|l.!ur.- tln; I._.ili.'u|.-1HrifnItdlllnfllll'lhc Hunt" 3" “alum ‘ _‘ I I A I ,I I !h+l fl 1... 1! ll Whit-n. m rilahhhh lhul lhc Imal trum'nliun emu fur III: $ 9' a1 I... f, ‘ "I" ' : WWII“ IHHI r fr. mu 1 --2 ii I. H.- + .‘IHEH 4' I'I'I, HH] :HS fifiitfi H III I‘+!.l :H'rl'. t. Inul-uHu-n.-1I In II! H W: Hi'iuml: “13' "W * 35L 1. . fir... p . hm: dcnwtiwu that! ill": Cflflllnufllfl lhllfl'llh m -.- L. .1“! [mill-1| dtrlwtllvfl]. Ihrn ll [{1th that 3;: 1a ,_ 2.x “l'll‘r :1“! . w"!!! 1 Ffi} I gun, ih1"+' «pun-1r g3 Huh + + a!“ k. wlwn‘ r. r, - I + h. flux-mm: that y. .-= tfih'fll. Fl 1.” "1lmw Hm, In: h“. m “wen by lzq. {5}. .ll-n i 1" 5 E i i I I E E 1",...1. Il‘fil‘flttl rail #5“th W. + M. thingy.” dummlh + d’M'f ._ . . if- N...” #3! {hi Making um "I 1h: Inn.- I,th 1V“! - LILMHI + f,|1.¢u}]¢'uj m A: - I .I I " upplmimnnnn wilh n remainder I'm I funclhm Finn at m “111th I fi-fimrafmmflu ' ' .' '1... l.- ..' Fla 4» hit + In at Pimh: + fitmhih + fitmhik z! ..'.'.F 313': . '1 1+" l=rlfr+fiIfTH1llII --:'_ H ; Chapter 3. ' . -'-_un' Tr '=- 1. _—l- r _ , + - r since itis not I __ _ _ t .II n .I Ths local truncation errors II I I I I I a II I III I lLr153ECli:t3lt’. The Euler and impffl‘tstd Emmi mathflds baking “3' What ISJmEmfllfll-lafighe an "3 nettle algorithmlgor meshed outlined .. t. i I oi 113' class of methods Z‘ EC 10!! .- . chewing-finest ' . ' I' , _ - LdllLLl the Range K11 II IIIIIIIIIIIIIIIIIII dIIIIIElIIIIIIId by RIIIIIEIII IIIIIIIIIRIIIIIII by the fflnufingI InSIePEnttheEuleralgrmlhmmustberqII-md for these methods are proportion'al'm-heéIfit e discuss the metho iethod is now called the classic fourth order four-stage Runge-Kutta magma; SIIIIIII III III =IIIIIIIII III a referred to simply as the Range—Katie method. and we will followihg II 2=m+05rh F'l'fljthtkl . I I This method has a local truncation error that is prqpmfififia-l rdcrs of magnitude more accurate than the impmafid k3 2f“ III IIIIII III III III IIIS III II III IIIEI method and three orders of magnitude better than the Euler method. It is ralafiggly k4 =f“ 4.th + h III B} o use and is sufficiently accurate to handle many PfflblEMS effitl‘ieutly. y = y+ (Mb) 1: {k1 +2 a k2 + 2 a B + k4 is espeeially true of adaptive Runge—Kutta methods. in which provision is “1&ng II = II IIIII l vary the step size as needed. We return to this issue at the end of the sectmn Rungesliutta formula involves a weighted average of values of f (r, y} mam but it is ofte practice for brevity. to hi". Thus it is two o simple t Note that if 1’ does not depend on y. then e , - i s' area] ,3 e: r E i, .11 is iven by H “all I'll pttll'llh ll'l ll'lt. IlllL i _ 1+1 E- III kn! "' flit")... kn; = kin-5 =fflfl + IIIHZL k“ = fun + In . 1 a s . _ II : II II II sat + 2k". + Jam + as) I (a) and Eq. (2) reduces- to _ l'l-r-l , H 6 __ . h IIIIIIIIIW .lr’n+i “Jr’s = Elfin-t} +4ftt. + hfll refitfl + in]. [5} k III : If-{IIII III” Equation {5) can be identified as Simpson's3 rule for the approximate evaluation of III II = III III “I II + lIIIIIIIII II the integral of y’ = fit}. 'Ihe fact that Simpson‘s rule has an error proportional to Jr" a- _ a 2 ._ I b is consistent with the local truncation error in the Runge-Kutta formula- kn} : first + %h-yrl "l— §hkfl2ji II Etna :flin “Final!” +hkrt3il- _ , _ Use the Range-Knits method to calculate approsirnate values of the solution I'- : out at the 'E ii, ~E.i:il tit-“I -'r- 2k”:- + 2k”; + entitle can be interpreted as an average slope. Nata EEAMPLE “filial Willie Prflfilsm tits: _' I. is the slope at the left end of the interval. k”: is the slope at the ulidllflifli 1 _ii' = l wten-4y. yilli = 1. [hi .-. the l' alt-r itiriiiula to go from t” to t” + tit/3s ifas i5 HISECflnd appmflmmififl‘lfl Taking h = 0.2.we have :t.._ .:t the tiiidpoint. and as is the slope at t” + h using the Euler formula-arid . _ Eli._- -.n urn; ti.:,. lti git th‘ill‘l f” it“! I” + ft. km :‘flfl'h = :1 “Li” : “1' ' s.a=fttl+t'l.l.l +ll.5l=bu‘. llkr; = l 3-H. vii. h- iltlit rtigliin principle it is not difficult to show that Eq. (2) differs from theTaletjr kit} Iflll—i- + “Will : fit-h; iii. - : l ‘1'- I '-.l‘ltlllislt in at the solution a: hy terms that are proportional to hi. the algebra isra‘tilet 1t. agility} 'l'htis a-e isill simply accept the fact that the local truncation errorin IIIII =II.II] II “I” II I IIIIZI : IIIIIEIII t": L1. fl; is pmpartional to h; and that fora tinite interval the global truncation errast at most a constant times fit. The earlier description of this method as a fourth o‘rtliit Thus i-aunstage method retlects the facts that the global truncation error is of fourth exile III 2 I III [EIII _ IIIIIIII II III IIIII II IIIIIIIIIII in the step size a and that there are four intermediate stages in the calculation.('_lli_e'_ InIIIIIII II II IIIIII calculation of fit,” . . _ . Jens }. : l + .- — _.- (I'learly the Runge—Kutta formula. Eqs. {2} and (3'). is more complicated thanaiiy Furlhermsumujingmc Rungfifimd “mud “Hm. : ,1: It : I. :IIIIHII. ..I. - III III II in Table 3.3.1. Note that the Range-44mm iiieihod iietts .i -..t...t .ll. - _ i th -' tt.ti..i~ * 1". ~-' -" _... a. "iii-lit.in 1!I- _..Tr _ of the formulas discussed previously. This is of relatively little significance. however. exact solution by only Ullltta if the step sire is ti : ii i_ .tiia' r~y t | l. ———— the latter case. the error is less than one part in an thoasaial. .tntt tat- t' aetiEsa e . tit- ' ' ' " - + - s is correct to four digits I II I I I t arl Daiitl Range tlHSo—tillil. tieririan mathematician and physicist. worked for man? Fearsulfitfl’. .. Fm IIIIIImIIIIIIIIIIIIIInI “me mm {mm “II: RunIIIEIIIthm mgrwa “uh I: : “11.. ImIII IIII . I “I I II trtiseutty the analysis of data led him to consider problems in numerical compatation. arid the RW. I EIIIIIII IIIIIIIIIIIIIIIIII IIIIIiIh III I IIIIEII IIIIIIIIIIII III“ IIIIIIIIIIIIIIIIIIII III. II. III IIIIIIIIIh II __I It ,IIIII IIIIIIIIIII II III II IIII ntial equations in lti95t' IR 1944]. Kuttasttitl tributictit'lll lSimpson‘s rule is named for Thom author. who published it in lT-l3. Hutta nictlttid originated in his paper on the numerical solution of differe iiietliotl was estcnded to systems of equations in ltltll hy l‘vl. Wilhelm Hutta {liiti'i'e l{Jami-an mathematician and aerodynaniicist who is also well known for his important con classiEttl .lll'l-t‘lll lht'tifl'. 'l — . ‘hcc. Itii' c‘ttiltiple. Il__‘l'l'€tftlt.’r 3 iii the htiiils. by Henrici ligtfll in the reftrencfli “slits-"4:5; "- '..' 3.."- i as Simpson tl7ltt-17ott. an English 2. 460 __ __ I __ d __ __n___d.ff-- .I‘ = 2 that is in error by 1.23%. Although this error may bi, aww” method i'itlldefl WW" '3“ . t . - ' ' - '. mess, it is more than t35 tunes the error yielded by the Range-guttflmglh - v nf an final-mg“ elm “I, _ _ I I ._ I _ t I Tsiiiitliiiighle computing effort. Note also that the RottgeeKutta method tytthh =0 mung 3f {ham am “fidfiyhfl'flablellui: Emmi; specific 0‘ m: {I 4“ evaluations til it PFL‘dUCES a Value at I = 2 mm an firm {If Liflfihi Which Is “nlr'siii'lillt‘ - gum-m3] mmm t-retttcr than the error in the improved Euler method tvtth h = or let} evaluatiumfifi b “I a mu“: autumn: algflmhm IE mare eminent: ll pmducfls halter W [has we see again th I'fort. or similar results with less effort. In Each . .. of P b _ . m lama 1 mrflflgh 5 fifld apprommate volts-ea of the solution of the green" Fifi value problem at l' = 0.1 02 03 i - i T 1- 0.4- l5 " . _ other methods and with the exact solution {if sham] were“ with those obtained tr,- m {a} Use the Range—Kane method with h = ill- with similar i: TABLE $3.] A Comparison of Results for the Numerical Solution of the Initial Value. [Inlhmm _t- = l — r + 4y. {Till} = I __# _ 03 U h R . -. — ——--——-#-*—'—— ' sete tt - i [ mprmw } age Kutta method with I: =. {103 Wits.-. ---_-.-.— -——--——R—‘”ig‘i‘l‘””a R L y: = 3 H ‘5“ 5"“) =1 (2 t. v =5t—~3..,J"F. vtflt = t m ' s- _ ’t = f]. h = {105 _ ' ' * ' r ”_ ill ::ll'.l_l.i- _ _ _ ir- Hits—“f. ll 1 . #2, 3. y; -- — 3L = 1 a F" 2 1' + £47.! FIB} =1 ll lthitltttith] lilithItltHJ lfllfltlfiflfl LiJthHJ-Ufl-D Loam I FF + 2:}, “I I thing-453 fl 5- J" — 3 + r": 1 = [L5 ‘2 6, y' = “t: _}.1}5ifl}._ Fle = _l "i ‘ 7" iii ll] l‘llll 5 .-- .3 J _ t2] _ s i l 3-H294145 18300354 Inieach of Problems 1 through 12 find approximate values of the solution of the given initial at s nest-iss- 5.???ft35fi srssrsss 5.7941197 53943154]. :flffinflfib‘em 3‘ i = “i 111 1-3. and 2a- Cnrnpatt the results stth those obtained by other H I:_ H_fi}.{_|_tfl 13L} _ i t 4| asst-treat tit—Hat lii‘ii haafialtl? 64.394315 5439-7303. (3) USE the Range—Kuhn method tvtth It = {11. I .-. intense: eras-t tea 479.2%?4 4?9.259'19 (70) Use the RflflgE~Kutta methtte with it = arts. '- r: than t'-."tt:'* EJLHIfiET-i 3535.8fih? 3539.38m sseozmt _ . . __ _ _ _____ fiZ. 7. y'=[}.5-r+2v. t-tai=t (2. a t- =st— t e on. — i . . . I - p - _ _ - ‘Q/ 9- .er = v” +30% i’ifl'l = 3 ll]. 3." = 2: + e‘” . ytttt : t':-. 1_ 1.; ask. tanned-Kaila method suffers fronttiltle slgtme shprtcorning as othermeths fl 1L y: = t4 _;yt;{1+y3t' Jim] : -3 ~.= an ! Iisetl -.tep sire for problems tn Whit: t e loca truncation error varies . . a. _ t _ e -- - - - . 12. :- r Zt- re. t = ,_ 'I._'|IIItI._l"-.I1l1':'i_ll liiL llllL'l". Lil til llilCFCSl. Thai-ll. l5, ill Step 512315 Eflflugl] tfl fl Jr. + l fl 1 ' I I .ti i 1'- tot-t accuracy in some parts of the interval may be much smaller than Hosea; fl 13* Cflnhrm the “35mm “1 Table 3-31 b3" aim-"me" [he ’“difaifid “‘mpumnm‘i . I[~ tiliteI mm of the interest. This has stimulated the development of adaptive fl [4. Consider the initial value problem liljt'ltitg- ieulltt methods that provide for modifying the step size automatically _ 1 1 T _ 11' = .l' -:'-- _'t". fillii = _. Mimi*Liltiiltlli proceeds. so as to maintain the local truncation error near or belting -]"I.'t.'illt:~'.i l1 tier-ance level. As explained in Section 8.2. this requires the estimation'ni the local truncation errttr at each step. One vvay it) do this is in repeat the compa— tstton with a fifth order method—which has a local truncation error proportionally {a} Dravv a direction field for this equation- {h} Use the Range—Kuhn method or another tat-rhea to and spprrrttntate sate. -_- tut 1:. solution at r : [13.1.1.9 and i195. Chemise a small enough step sire so that soc he 1* _' t -. .-'.if results are accurate to at least four digits it"—and the n to use the difference between the two results as an estimate of the errt'ir-.-. ll this is dt‘il‘tt.‘ in a straightforward {LiflE-UPl‘liSliCEtlEd) manner. then ii‘lE‘: 115E Elf lllfl “3 Try to extent! the calculations in part tini to obtain a n seem—ate app-it'd in t.. ' h- t'*-. order method requires at least five more evaluations of f at each step. in addition solution at r = 1. if you encounter dit'ticaitie a de l-lEL rats L‘HE‘lLLIE't 1v. ~ .--.i '* :ef- this to [iterate required originally by the fourth order method. However. if we make-ah happens The direction field in part lili may he tie-taint. appropriate choice of the intermediate points and the weighting coefficients iatlltt fl 15' Cflnsidgr the initial statue pmhthm espressions for huh. .. in a certain fourth order Runge—Kutta method. then Eli-ESE I. _ My 1?“; JL M. _ H expressions can be used again. together with one additional stage. in a corresponding: tiftlt order method. This results in a substantial gain in efficiency. It turns out that till! {a} Dravv a direction field for this cstflfl'it'tn- can 13': dim": in "mm than Ur“: WHY-4 The Twill-111mg adaptive Runge‘Kmm mflhfldi ' {bi Estimate how far the soiution can he :ttendcti to the right Let 't- in *1. ch endpoint ofthe interval ofesistence ot' thts solution. ‘t'i hat happens .i! :1. to a:._-t.- : -. t solution from continuing larther'? (c) Use the Runae—hlutta method nith tarious step sires to determine in .trtt‘it-hr'nttt J'Tl'lt: lirst widely used lourth and fill It order Rungcwhiutta pair was developed by Erwin Fehli‘tet'g inth'elittt Hm“ “5 Papillarin was considerath enhanced by the appearance in 19?? of its Fortran impleme'ttlflifln ; ml“: [if I” _ g I I I that: parts id} and let on tit». Fest pact- RKFJS l‘ty Lavvrence F. Shampinc and H. A. Watts. u- I. .|; ,. .- . il‘- '-a. .- r 5-1;; I."- 'I __________.._...—-—-—-__. (d) If out} continue the computation beyond to. you can continue to gen-gratewlh t- What significance. it" any. do these values have“? tut 1snppeete that the initial condition is changed to jvttlj = 1. Repeat parts it") and ' this pruhicnt. - ;.’-.- 8.4 Multistep Methods ___fl__———-———_______————fl ln Pluteltlwti sections we have discussed numerical procedures for solving the initial. value problem _t-' : ftf._t*_lt jello] = ,‘r’tts in ash-lg“ din-g at mt;- ptn‘nt r = r” are used to calculate an approximate valueoigthg: solution on” _| i at the nest tuesh point r : run-.1. in other words. the calculated era-1%. at a at anv mesh point depends onlv on the data at the preceding mesh pointt_§figfi._ methods tll‘r: called one-step methods. However. once approximate valuesnffhg siiitllit‘il‘l t” : out have been obtained at a few points beyond I“, it is natural {gflgk helitt'l' n can n‘lttix'r.‘ use of some of this information. rather than just the value-3i. the last point to calculate the value of (pttl at the next" point. Specifically, if}.1 an“ it t- . . . . i_._ .t‘. I“ are known. ht'iw can we use this information to determine yflfl 3f ‘. it. tit: that use information at more than the last mesh point are retort-edge utait ir-tep methods. In this section we will describe two types of multistep methods; mt-tlitul-t and hack ward differentiation formulas. Within each type,nne;¢ag '1-.!.t=l'I‘- . ietelsol'tteeuracv.clepcndingonthe number of preceding datapoints "en: .:.-..- tut-t: For atlt'tpiiL'iljt’. we will assume throughout our discussion that thesfep- -|' ti; |*.Id-1|i"l'-.i'lili. I Atlanta Iiiethotis. iiccall that at] 1 Millie-Fl] ‘ (mini : [ dliil’] til. t r” where drift is the solutit'iu of the initial value probietn (1). The basic idea nihil- etdams method is to approsimate out by a polynomial Pitt} of degree it and Initial: the polynomial to evaluate the integral on the right side oqu. (2). The coefficientsitts F; II] are determined hv using it + l previouslyI calculated data points. For exatiipiéi suppose that we wish to use a first degree polynomial P1 (t) = Ar + 3. Then we need. only the two data points tr,,.iv,.t and {r,,-1._v,,_1 1. Since P1 is to he an approximation- "Jolin I("ouch Adams tldl‘i—l‘dhlt. an English astronomer. is tnost famous as codiscoverer. with Leierrier. of the planet Neptune in lath. r‘tdams was also estrernelv skilled at compt.ttation;hiti 97mm for numerical integration of differential equations appeared in ltiiiii in a hook that he wrote with Basltt'tirth tin capillary action. “- .Multiltep Method; ,. to,'--.".'-:'.-_... . weflfl‘ggfimflm Pitta -,ft.tn,y.l andthat Parka = filer—1.. v._t i. Recall 1h! fi‘inl‘rlbj'fifnr an-integerj.'[hertA andBntust mmmmtm A!" + B =1;1 I Arm-l + B =fl—l' Solving for A and B, we obtain A = fit "fir—l B = fnv—lrn “flirt-1 _ {-4} h ' h Replacing do} by Pitt} and evaluating the integral in Eq. or. we find that . A .. ¢ifn+ti — one = Ernie; — er + aunt — at. Fmflflya FE'IEP1?CE ¢ii1e+r t find-oil") by y“; and v... respectively, and carry out some algebraic Simplification. For a constant step size It we obtain 1H,.“ =ivn + git}; — ghfml- [bl Equation (5) is the second order Adams—Bushflo formula. It is an esplieit formula ffll’ yn+i in firms {If ya and yang and has a local truncation error proportional to hi- We note in passing. that the first order Adams—Bashtorth formula. based on the polynomial Petr) 2 fit of degree zero. is just the original Euler formula- More accurate Adams formulas can he obtained or following the procedure out- lined ahove. but using a higher degree polynomial and correspondingh more data points. For example, suppose that a poivuomial Port of degree three is uscd. The coefficients are determined from the four points the. 3.3. i. in- . . _t'.T_ t l. tn. _3. tun; 1. and trn_3,y,l_3}. Substituting this polynomial for e'tn in Eq- {1 1. cs aluating the integral. and simplifying the result. we eventually obtain the fourth order AdamsFBashittflh formula. in.” = y" + titfl-illfififn — 591th + 3'71}:_:— vi; -~.i. rat The local truncation error of this fourth order formula is proportional to h"- A variation on the derivation of the Adams—Bashtorlh formulas gites another set of formulas called the Adams—Moulton“ formulas. To see the mil-erotica". let us again consider the second order case. .Jteaiu v... use a are. dtgree ptiiat':-.u‘:ti.'1!. Qttt‘) = ctr + {L hut we determine the coefficients to (t,.+tt}’n+t}. Thus er and B must satirljt' none the torrents it t . “it. and it follows that tilt—t2) was an .~'ltmcrtc.tn astr Itii'i"'I-".t_ ‘ ' t'Forest Rav h-toulton tldTZ i awn [he .tt-i.aat-.tt«~.t.tt-.-... , calculating ballistics trajectories during "ii-or] formula. Chapter 8.. Nahum. 3"” up II - {tine}: Method: - I I. I . e. —I-—_. -.:-- ' (2) and Simplifying, we obtain Consider-againthe initialede Substituting Gilt) fflf 1W” l“ Ell- 1 ._ EIAHPLE , FH+1 : :L'“ "l' + Ehflle-‘ltlyfl-l‘lils la; 1 I = 1 ' '- Withastepstzeefh—Dlde ' - i Ismaeanappmaimamvalueofthe I - -e4 . . . ‘ . . I " train the fun it dams- m "m l _' “n I“ " which is the second order Adams-Menus!“ fflTmUl-fl- we have Wilden fun “1151* J.) dfh all?! order)! Bashinrth formrdame {ounhorderhdams—Houhon formula. in mg [-151 [arm to emphasize that the Adams—Moulton formula is implicit! an}: e pr I reign—commi- methoi ‘ ' I ' . fll' starttn - than esplicit. since the unknown _i'a+i apps-'3“ 0“ bmh 515135 “f lhfl equation Thaw are 13:11:: I]: “11m 0‘ Firm. and vs found he the Range—finite method local truncation error for the seeund Uri-lEr Adams—“Dung” fUTmUlF-‘n 15 flblain a E 31* NEIL calculating the corresponding values of fit. _vji_ we ltt all}. . . _ y“ =11 ft} = 3 Thu “m mflgr Adgmgmflulton formula is Just the backward Euler formula-gas 1 . Ell—1U [night anticipate b}. uni-[Iggy li'elllh the Drd'fiir 3’1 "' 5039333. I] = ' Mun: ace-urine higher order forttiulas can be obtained by usmg an approximafiitjg y: = 2.5fl50062. f; _—.. mazmzi pflhnummi or higher degree. The tourth order Adams—Moulton formula, saith-319E51- y3 = 3339414; I: = ifi‘flljfiifi L p _ Then. 1fromrtlliie Adams—Bashforth formula. Ed. (at. we find that yr = 5.733605- The es- _ _ . _ 3'31 “3 Us 0 l E million at t = 0.4. correct throu h eintit di ' 's *1 tween so th ' - it. .. = a. + anaemia. + 19a e firs—1 Hit—:1. (in). —fl.flltt595. g "' gm *' " ‘3 mm ‘3' I _ _ fl _ The Adams—Mnulton formula. Eq. {Ill}. leads to the equation {thew-tr; that this is also an implicit tormula because yr,“ appears in fHH. - t, - Altltouah both the atdams—Bashforth and Adams—Moulton formulas of the-53mg 3’" = 4323135 T “'13-”- order have local truncation errors proportional to the same power of It. the Adam' from which it follows that at = SJQIIHJE. can; an mar at only (irritants, :‘sloiiiiiin formulas of moderate order are in tact considerany more accurate. Fer Finaflsuensths mill“ ffflm “1E Edema-Bflfihlflflh {mania 35 apfsdlflsd talus “he'll-4'- .tiH-tig 1.“.- 1h: fourth order formulas (6) and [10). the proportionality mama-fit we can then use En. {ID} asa corrector. Corresponding to the predicted salue oil-r. we find r. mt .ttlanis —7s-lt.iu|ton formula is less than 1/10 of the proportionality conga-m: Thelfs 7- 23-73452} flfiflcflfmm Eit- llfl‘l- ll“? sflfis‘slsd “111% 0ft: i5 5-793l‘731- The “3511” Ill; gttlanis—Bttslt forth formula. Thus the question arises: Shflttld one ls lgzrmr hyhflfl‘mbsjg' B . . _ _ hut . .- ltII'n__| tit-tiert Adams-Basht’orth formula or the more accurate but irnpli'slt. . if:in at “EA Ems— as’lifmh "1‘2.th i5 lhfiflmplm and lflflfl m IhLEL “1:1 L 3" - - -‘ 4 -i t -l-'- v ' ‘vl tulton inntuln'i The answer de ends en wheth 1;} cl since it involves only the evaluation of a single explicit formula. it is also the least accurate- I.-.~..._. at .t . ..i .s t. .:|tit~—_ t. _ IL h it . I a d h 31's I Using the Adams—Moulton fortnula as a corrector increases the amount of calculation that is Fl ‘ '1 l' “a” um: mmmlfl' Wk La” Increase HE Slap 5126‘ an t grab)! mdl‘lflgfflm required. but the method is still explicit. ln this problem the error in the corrected value ot rt ..|.,.L..-t_ ; .-i cit-pg. enough to compensate for the additional computations required-fiat" i5 [educfid by flppmgimateiwj- 3 fact-.m- of “I when compared to the error in the predicted salue. l--"' "3-"? The Adams—Mountain method alone vields its lat the best result. uith .‘in error that is about : it ' Ittll'ttcttettl tit‘tt‘tlysl‘s have attempted it} achieve bflll’l simplicity and 1/40 35 large 35 {he error from the predictoreeorrector method Remember. homes-er. that -_ I. an t“ thing the to o formulas in svlt at is called a prediflor—enrreetnr method. have the Adamseh-loulton method is implicit. which means that an equaoan must ts. sets ed at each 1.. t .. -. i . and i,_ are known. we can compute fno. fi,_s.f;,_1,andfihand [hfiflufifl step. In the problem considered here this equation is linear. so the e.ill..'tl.it"l‘t is anicsts tease- _ . . I 1 I - -. ....: .. . _ . . I g __ _ ‘IH ‘1'... '71” ._ F1:_ 11151-1] fl lltc Jadains.—Isaslil't'irtlt {predictor} tormula (at to obtain a first value for sin“. Ihen bill 1“ Gilli—‘7 PrUt‘lsmfi "JV-t Perl fll lhs star-stints mas Enrica? t...t ll . t. '1 urn In“ ‘ u _ . . ' . ."I'I -._ ' - "t i=1. a. "f .- = a. “see-- srtit an r.‘ r ‘- ~--'*' at; t-tnripute t,._t and use the Adams—Momma (corrector) tormula (10].whtchts-nu The Rm‘g‘” Rum th‘qd “l h l l lam“ i V .t . . 1 t. . . . _ - - - - - . see Table 8.3.1. ThusJor this prohlem.the tsunge—tstet' '-. cat-inert -~ to. impart-tie. v. . lonact implicit. to obtain an Itttprot'ed value oft-3m. We can. Elf course. continue-tn . . m . . - - -- v r _ - - the predictorwcnrrector method. use tlte correetoi tormuia [Hit it the change in _v,,+. is too large. Hots-eventfttn necessary to use the correclor formula more than once or perhaps twice. it means that the step size I: is too large and should he reduced. Bflfihwflrd Differentiflmfl Farming Amnhcr .15 I“: .__H-mulmmp mcihfl :t a ._-._1rt:il1l_ ' ' 'Jr J ‘ :1 ' I. i r I " ' . -. -- l---|'.'iti- t'l"'l1-l - n orderto use an} at the ntultistep methods. tt ts necessary first tocalcutatca-iet Pk“) m appnflimflm mg swung” gt“. a; [ht- lfitlgli‘.‘ s_l_l_tl.._ r -t«. . . ti t. .1 '1': hi! mum Utimr mflhfld‘ Fm Example' [he murth flrder Adams—manna“ method its derivative a'i‘t'ti. as in the Adams methods ‘i‘t'e : ‘.-.:"‘l ditttrraostc .. . ace. se: requires values for v, HHd}’j.Wl‘lllE the fourth order Adams-Bashforth method ale P. U flequal to f” H Wm “whim” fl“ imphggl “mm?” {m ,~ in L J L r-i'ui *sav' '" v. s r ' _ k "+ . ' .”'."' ' . re“. alue tor}; Ont vs a} to proceed is to use aone step method of compatflltlt _ backward differentiating. famulm _ g 1 I a I accuracy to calculate the necessary starting values. Thus. for a fourth order mulllfillflll The Simplest case uses a first degree Pt‘tl}'ttttttttttl is. t it = -t -. a la: .e at: methodonemight use the tourth order Runge—Kutta method to calculate thestartttls are Ehflsfin m match the ammuth 1.3311165. at the tolution r- and i --. I ne‘- 1 ‘ values. This is the method used in the next example. must satisfy Another approach is to use a low order method with a very small it to calculfllfiitli ' t ' B — r and then to increase gradually? both the order and the step size until enough 5W3 - r I" T B r m i "'3‘ Alan “l' : lee-l" values have been determined. 5. .. lifL—Lfil’al'ib‘. Eli”; 3. .s 5: e P”, ti] = A. the requirement that Sine Pl “"“l'” = f'i'rt'tvhlsyn-H} . ‘ iufit I "fa-I 51"53. I Z. '3] A = fitn+1.,va+i}- .. comes from subtracting the firm (if Eel-15- ’ "ti"- " .. '_ L15? .. Another espression for A second. which gives A 2 m“ ' W‘ " _ Substituting this value of A into Eq. {13) and rearranging terms, we elitain; E- ordcr backward differentiation formula :1:- 4- " 1 pm“ : “T” + thir-l-ii ytt+l 'I . Note that Eq. t H} is just the backward Euler formula'that we first saw in Seemng _r Bv using higher order polynomials and correspondingljlt more data gamma-11%;. tiht-tlt'l met; ward differentiation formulas of anv order. The second order in+l = 7H4)!” " Fit—l + “n+1 vin—l—l 1 t a and the fourth order formula is g. — hl-li‘i't'i, — Elihu—I ‘l‘ [hire—J — 3.15“; + lghfun'l—hyn'lnln ' _1l'rr.I-'l — __'_ - . - 3 s - 1-5-1- l'hest it'll-it'lllltli have local truncation errors proportional It) h and h , respecting; "i F "t- t ;h. :. -trr'tii order backward differentiation torrnula with it = 0.1 and the data-givgflm ;-.~.I--IL- ! :3: .-_i.,-t.-roiine an :ipprosiniate value of the solution_v = ditt} alt = {1.4 for-the'iititjgl i-l :I't'I'V-t'ci‘lt l i "l. “- ‘fio. r In] it tilt n : fl. is : H.l. find Wilh ,l'th - - - all? given in Example 1‘ we t. : imitate + titan-4. ilttis vi = immhlb. t.'onip.ir|ngiiie calculated value with the esoci value tfilUA] = 5.?942260. we find that [headph- i». titittjfiioh. This is somewhat better than the result using the Adams—Bashforth methodttilili not as good as the result using the predictor—corrector method. and not nearly as gootlasiflip result using the Adtinis—hltiiulton method. ' A comparison between one-step and multistep methods must take several theisti- into consideration. The fourth order RungeeKutta method requires four evalltfiv tions of f at each step. while the fourth order Adams—Bashforlh method (oueeip'fifil- the starting values] requires only one. and the predictor—correetor method only Thus. for a given step size it. the latter two methods may well be considerably fat-.151”- than Runge—Kutta. However. if Runge—Kutta is more accurate and therefflrfl'fialli use fewer steps. then the difference in speed will be reduced and perhaps eliminatfili The Adams—Moulton and backward differentiation formulas also require that-the- difficult}! in solving the implicit equation at each step be taken into account. multistep methods have the possible disadvantage that errors in earlier steps-fill (a) Use the fourth order predictor-correetor method with h = 0.l . Use the correetor formula once at each step. (b) Use the fourth order Adams—Mention method with h = 0-1. {e} Use the fourth order backward diiferentiation method with h = 0.1. i3 1+.t"=3+r-Js rt01=1 0?, l}"=5t—3fi. viola: f2, 3. if = 2}- "3!. viii} =1 #2, 4. v = 21+e"’-‘. villi = i ,2 t i + 2tv . . _ i2! 5- 3" = Ems. a Fifi} = 5.5 5. J." = if." — _v-"isin‘v. _i'tlfit = —l In each of Problems 7 through 12 find approitimate values of the solution oi the git en initial value problem at t: 0.5, 1.0. 1.5. and 2.0. using the specified method. For starting salues use the values given by the Runge—Kutta method: see Problems 7“ through 12 in Section is}- Compare the results of the various methods with each other and with the actual solution tii available]. (a) Use the fourth order predictor—corrector method with h = 0-05- Use the corrector form ula once at each step. (b) Use the fourth order Adains—hloulton method with It = uni. (c) Use the fourth order backward differentiation method with n = Hits- ‘Q, 7. _v‘=0.5—t+'2'v. villi—=1 ti- 1: :F-ieiih i. trill :2 i2; 9. _v’=vm. _l’l_fll=3 ill. _i'=2:+t"". idlizi fi 11. g = t4 —— ion] +i-Ji. as: = as 32. 12.. y“ = if + Etvlftd + ii i. villi = its 13. Show that the first order tadamsufiashforth method is the Eul.‘ larder Adams—Momma method is the backtsard Euler iiicltied -' ii‘lc‘li‘lttd .r'.‘!~t.l ll‘r.i[ ll.i.‘ lit—Hi 14. Show that the third order Adanis—Eashforth formula is as = _~.-.. + thinnest- — 1st. st. -71. 1.5. Show that the third order Adams—Moulioii formula is rm. 2g.1 + thg'l'ittif.._. + tit}- —_f.. .,i. I .. . - I ' 1. _ - i! .-. ‘- l 11-- 1h Derive the second order backward ditfereniiation tormula 1.1“ i. n b} [:q. t I "i t til l.‘lt‘~ so. it I t r_ -___...—n. - 468 erne ideas related tn the errers that . . f the initial value prebletn e 3.5 More on Errors; Stability fl 5 *t' 8.1 we discussed 5 - In fl'mn esirnatien ef the selttttert e 1-" = filly}. Fife} = .‘t'e- I} ssien and alse peint eut sente ether TABLE 3.5.1 tlttpprettirnatierts he the Selutien-ef the Initial Value In this sectien we centinue that drew that can arise. Sente et' thti Pfllnlfi that we wlfih m make am fairly diffiflultimi PDlilill‘bialfeiilltlls; 15*? I + 4’“ Fm) = 1 Ugh-IE the Ell-1H Methnd with damn, 5.3. we will illustrate them his “163115 DE Examplea 3.1-".- *TWM - i-I'i— -' N Parts Errer e’... Ermi- ' untde Errors. REC-fill that fur I116 EUIE‘Y mEthD‘d “re Shflwfid ._ hlif‘Tt‘lflllEIIitt‘igllitf: errer is prepertienal te ill: and that If" a mute Interval fig 433 “'“nt‘flfifln em“ E t” "195‘ a “instant “Hm I" In general' [m a mat-hm flfflldfillll' H.002 set}. sass tiers IeraI truncation error it arts-WI"!it""31“3"!”I and “Z”: glflli’al “Realm” “Verena. 0.001 teen sass arise asst est: interval is heunded by a censtant time‘s tit“. Te achieve high accuracy WE “km 1250 3.636 Ram 6313 1:12 a nutne rical preced ure t'er which p is tatrly large. perhaps 4 er higher. As p 0.000625 mm 3.616! 0.0% 64.35 [155 the [annals used in cenrputingttfit Tit-WITH“? bECUmES "101:3 Fflmpllflfllfldtflfldzhegge 0.0005 20m 3.??? 0.060 64.00 0.90 tnttre cttlcttlatietns are required at each step: hflweverr “TIE ‘5 “5113"? “m a mm 2500 3-507 0.205 63.40 1.50 prehletn unless _t”tt.yt is very cernplicated er the calculatten must be repeatadigggg; “30025 4000 3.231 0.431 56.7”? 3.13 [mm mm?- ”. mt. Star. size It is decreased. the glehal truncatien errer is decrease-gig; J. ilic at me [ache raised he the pewer p. Heweve r. as we ntenttened tn Sectten I L_. J... m” H“ .1 gram many steps will he required to cever a fitted interval. and the-algal. r. til-lttjd'll-l' errer may he larger than the glehal truncatien errer. The Situatten Keeping enly feur digits in enter it} shenen the calculatiens. we ehtain the data sheath in Table 3.5.1. The first twe celutnns are the step size it and the number ef steps .‘v' required te .L gtlh-E'JTZJllLfllll‘t' in Figure sit. we assume that the reurtd-eff errer R... is prepamatyfl. _ traverse thfl interval U E: E I. The“ M: and h an: approximatiflns m M151 I St?” and : a tee titlai‘tlf‘t-fi' t'tt cc-ntputatiens pert‘ernted and therefore is inversely prepertiehaié ' It’ll} = e4.90.respeetivel}t These ctttatittities appear in the third and fifth celutnns. The t‘eurth " “I”: in "m: ll‘ 0“ the other lmnd‘ [he truncmifln Errflr E” is prflpmtiflflm and sixth celurnns distiller}I the differences between the calculated values and the actual value .- meet at t't. Fretn Et}. [1?] ef Sectien 8.1. we knew that the tetal : uf the suturiflm _:".'-.1 hv -t:'... e- :H... II hence we wish he cheese it an as te minimize this quantity. Fer relatively large step sizes the-reund-etf errer is much less than the glehal truncatith 't r.- . -. entrant-n t alue el‘ h eccurs when the rate ef increase at the truncatien firm“ Cflflfiflqumlll’u the 10131 BITE“? '5 EPPTUEIWHWLF “13 5310*: filt- ihs illflhfll mint-“stile” finer- whieh fer the Euler rnethed is beurtded by a censtant times It. Thus. as the step sire is reduced- - the errer is reduced prepertiertally. The first three lines in Tattle sit shew this tjt‘t‘e et in Fill-Jun- H11 ' hehayier. Ferh : 0.001 the errer has been further reducedhul tnucit less than prepttrttrtnatiy I this indicates that reundset'ferrer is hecernittg irttpertant. Its tr ts reduced still t't‘tr.‘-t~.. the et r. ‘l begins ttt fluctuate. and lurther itnprescntents in accuracy hesitate :trehlematietf Fer values fifth less than lLflll’llfi the errer is clearly increasing. which indi. ates llt.ll rennet-er“- -.'-r-" ts ease. the derrtinant part at the it'llt'll errer. These results can else he espressed in lCtTl'It-s et‘ the netttb---- --t stees “ :'- -r it less that .mth by the rate at decrease at the reund-eff errer. as indiestetl t7 ittCt'eJtsc-st t"s [till a I abeut 1000 accuracy is imprtwed by taking ntete wept tints-'- I. ‘t grtr-ttt tl-._rt .tl'stut _“'.‘ using n‘tere steps has an adverse elt'ect. Thus far this ptr 1“Fer. -I .s “4.! -_ --.-~e an t. stunner-ere between 1000 and :liil'll Fer the caiculati-rns sltetvn tn Tattle shit tnt hes: t't_.-'-'.lli tl -' -_ t'- r eceurs fer N : 1000. while at t = i-“ the hest result is ter a 2 list: One sheuld be careful net te read tee ntuch inte the results shevv n :r. Esau-tel.- 5. ' The eptinturn ranges fer It and N depend en the differential equatten. the nunterttal h ' h methed that is used and the number at digits that are retained tn the calculatttrn. at: $ in FIGURE 3-5-1 “"3 dspsndflflsfl 0‘5 lfllnsfltifltt and mund-flff EITDTE fin “is 51W Elia-ht Nevertheless. It is generally true that it tee many sttps .trr. rtttutted an .t t. I Llll tttett 4370 then eventually round-off error is likely to I -L. .. degrades the accuracy of the - For them. any of the fourth I - 8.4 will produce good results with-a number-of ' ' owe‘fiter . . round-off error becomes Unitarian" Fm 59mg Prflblihlfclhflifih- "r dflgs became vitally important. For such problems. Id I .fim . .. _______ I I I crucial This is also one reason Wh)’ madam “ides Pm“ 6" a' Id'iflti-iirifl‘flHi-i' lili- step sire as they go along. using a larger ElEP 3E3 Wilma” Pflsmhle'is it step size only where necessary. I tit.) by (a - - ' _ - «' Thais: t —i. we anus. These. as] ea" situate- ‘ i" a I '5' m“ that iii-stasis exists at least the. S mulme the gimme and mm}; I' i 1” 5 ‘- ‘i gases-arse prablem (a). sass 1m '- -' l lution in some I. -' IIIIIIIIIIIIIIIIIIIIIIIIIIHII IIIIIIIIIIEIIE IIIIIIIIII IIIIIIIEII IIIIIII “IIIIIII IIIIIIIIIIIIIIE MIMI Olly WE g9 hflwfld II = 3’41 and Pmbafilyifigyflm i: Elm" fifiumiflfilfiat ills solution-of the initial value PTObleta axis E at I: = #31115?- 1‘4335. site-cart obtain a more accurate appraisal of difimnmamngthfl mmpmd values” ' -- fl What halilmiriffilbut???the initial valusProbletns{5) and {6) use: more accurate numerical procedure—the Rungc—Kutta method, for agggyI-III.IIIIIII III I Fm) = 1 IIIIEPIIIIIIIIIIIIIEIIIIIIIIIIIIII 3(0II9)I;I1IIIII305I Than we “main the RungeeKutta method with it = 0.1, we find the approximate value .I . .r : l.which is quite different from those obtained usingthe Euler II I . the calculations using step sizes of h = 0.05 and h = 0.01. we flbtflm the tiyfg:t_'t-Iij-It-I.I-._I _1I. . information shots-n in Table 8.3.2. I I .III_II-_y.II Vertical Asymptates. solution Ir = slit} of 1 a I1." = t" + _l"". ytfll *1 -\.'. Since the differential equation i I {Theorem 2.4.2} guarantees only that there is a so Suppose that we try to compute a solution of the int U a: t‘ 5 1 using different numerical procedures. . .. gy- L. if we use the Euler method with ft = {11. 0.05. and 0.01. we tItIIyy.I.-t-I-“,II. III- I 12.32t.l‘93.ancl 90.755131.respetctittttriliylgT ytfgiiItIg-g._.. I, I. '. 'r i. ... fl IDA-Iinlatfi- valugg 3H,? 2 12 I I I . 3;. W e convincing evidence that was; .I-.:..- ,I. fl '- 'I tial value problem-sat ll -:._I -_I 1;, -sitr}i e- ”tsatr inseam). em} = manager — ii, (a) where only five decimal places have been kept. Thus dint; —r us as t —i- a}? — 0.60100 3 {1.969% and 95:20) —r on as t -e 0.96991. We conclude that the asymp- tote of the solution-of the initial value problem (3) lies heriveen these two values. This "example illustrates the sort of information that can be obtained by a judicious TABLE 3.5.2 Calculation of the Solution {If cflmhiflatifln Inf finalfligaland numflfifiai wnIkI the Initial 'v'alue Problem y' = t3 + .122. ill” = 1 mini?- ihe Rwy—Ema Mflhfld Stability. The concept-of stability is associated with the possibility that small errors h i = 0-90 i = 1—0 that are introduced in the course of a mathematical procedure may die out as the ll 1 14.02182 735.0991 procedure continues. Conversely. instability occurs if small errors lend to increase- has 14.2711? 1.75363 it 105 perhaps without bound. For-example. in Section 2.5 we identified equilibrium so- uht 14.304178 2.0913 s: 1023'” lotions of a differential. equation as (asymptoticallle stable or unstable. depending “not 14.3fl436 on whether solutions that were initially near the equilibrium solution ieiiueu to ap- proach it or to depart from it as I increased. Somewhat more generally. the solution of an initial value problem is asymptotically stable it initially nearby solutions tend to approach the given solution. and unstable it they tend to depart trom it. h isualIly. in an asymptotically stable problem the graphs of solutions will come together. while in an- uiistable problem they will saparate. I Ifwe are solving an initial value problem numerically. the best that we can hope tor is that the numerical-approximation will mimic the behavior of the actual solution. Vl- e cannot make an unstable problem into a stable one merely solving it name riIc-aiI 1 y. "However. it may well happen that a numerical procedure will introduce instabilities that were not part ofthe. original problem. and this can cause trouble in appros must i ag the solution. Avoidance of such inStabili'ties-may require us to place restrictions on the step size ft. The values at t : 0.90 are reasonable and we might well believe that the-501g? _II. a value of about 14.305 att = 0.90- However. it is not clear what is-happenifl'gi - t 2 1.1.9 and r = 1.0. To help clarify this. let us turn to some analytical appro " I' I;-I. to the solution of the initial value problem (3)_ Nate thfltIflflU E I E II I_ I- y: Ell-l—yz 51+y2. This suggests that the solution y = sin) of yi=1+ylfi . I _ I 1 . . — . i ' ' - Inn-t '7‘ h * hfuanl-Ylmllr l'fnfll [hr I Hill-‘1' fldfllul'd I I 1 I it "hilfl'l “- n + hit. a: mi + m, V'an ‘Hnnlsuh frnin H'h: hmlha'ld Lulu ftflfllrfllfl fl Ill“ Int-1'1 "' ’rl 1' hryl-rlr v "‘ mi + m +{rhr1+r-*L f? l I Hi I; 1. u' -" -- Jill Hum. lh: uanH-IH If “H .II I... f WC "lull a. irhr’ anurutlrlh~ 9.. l-+ rh+ ~—1—— +--- _ I. r: . ...., I q: Ii n H4: mulmhwr at: that IMfifl-flflniflmm A .m-l an. n =.. im- ml I “In fnrmula an: n-l nin khan III: lhfllym 5.__:__: '_ "inm 'Hllll'flfil' lh-Jl HM" Lhdnflfl "If hllfll' r... In h, i“ it I_ I mu Hut [141 .1; rumulnlrd h'u HEW Hm: m: n1th I! t... 111mm; -. L "‘1'" ”“ “"“V‘ ”’ "*“V'm‘ "1 Hm"; um nun: “up In hm. - 'L- '11! “If [mu-I :nhllinn 1H}, [hr- Ehiflfll "I h.” d“ I” I“ I :- J_ll:.1.:- "_ n- . I- '_-.. '-|- J .' J-a 1r until: Hm Lpnlnhly' It It!“ than .5 II rllflrhl 4* Llhfll It“! i 0. m -- _- I I _ h; :' I'M :1 _ F 'i ‘ n'nlh'llulnn in I 'hnplfl I Ihfll Pal {‘3} h Mimpllflk’llb It!“ I! f I: n“ I. g H. . .11. _ I 41'“ j _ 5.9 I I H r ' n ,. j “3 “ml W: Hill‘- pm.- an: In: Ihr mum: l:qu mum-mm than: my“: infiq lldjhlhl ' *h- hr! 1' ~ "1hr quantity Hi! "In In airw- mlm and W '- lhm. ll llw dlflnwnlul Iqualhm in Ilnhle. than u: h III. M -*" __ am “him” “up um I: '3- - {In the all!" hand. Im the Eula: mun-thud. the chin;- in run ill-H 5:... mm “In. It“ rurnlllhmr r; liwmlhl ll duh! 5 l.l'lfl_ fl mus-I antitank 19 m”. “min the Huh! mollde MMHM -. +‘~ _ in Immunity umall mgr--1 'rfi—r.¢_;r;1:mfi_3.h'n-._- .- i ' "III-Id- .. _ F. _ I':r '- 1.-.... r *,,'FTfr~p-...‘C:'? £1.14.- Illi:-I " [ii-Ia; 474 _ __I_ _.-_|.- _-I--d—- e stability analysis in Eqs (9)--tlirettgh {15)5.;a1ne_.a"nt e rs.- sea-i i h i I on!“ am” “With h’l't see-need 5 - * news that form n i ._ Since r = -- [fl] fer Eq. (16).“ [e “an [m the backward Eula: I if ‘~ eerres riding restrifl'. . . . _ :____::.__'..h : hmfifliiiErtifr-ii; flhlfliEEfl {rent the Euler methflil are Shawn 1“ mlmnfilganfifi” " "'3 ""i “.1- ~ e . Th Iues fer h as t] 025 are werthless because elitist-ability. While thmfilfi: [’3 V3- 1*- "" * ' - fer thiaraeg a sis-n -. - HHWEVErImearflbIE accuracy . .. . L: iii-.1:- rite,11 _._ ffilerhickwnt’ti Euler method. as shflwn by the resulta litres-H fer it = it! by usingt -- - .. __.l J—ti' _ 1313ith ituatien is net impreved by using. instead ef the Euler methedisammfig will I E ii- i blern the Runge—Kutta methed is unstable .i_ . h's re . . . -. SUEh 33 Hung” Kuud' Fun I p by the [Egults in eeluntns 5 and 6 ef'Tabie titlEfi. as shewn hut stable fer h = _ . - _ - . E __ The results given in the table terr = {105 and fert .— [1:1 shew thrat tn the fig“? htain an accurate apprestmatten. eu-are 1:1me smaller step size is needed In e this ma'tlcr further in Preblent 3. ._ . at. . L _ _ 1;; '- _ I. " " value ' ' __ withtheaedauuhieflm it E ‘t I'tiurnerical Appreaintatiens te the Selutien ef the Initial Valenti-fit".- - i' ' I " med-3'33“- EMLM hm Emits; it”: + 1 sun = t j: iii" .' {human mm"'3"fif§‘ fifleifitfifint Die-“Wham since FJE'“ tenth In .t' = n - ‘ s ' . flea-ll - . .- ""l . - . - " " _ETT“EE— Euler Range—Kuttfl Runes-Knits " -_ Him ind? . _EFW‘FMW-‘mflmmtmmmw“New ’ “L ates nines.-. 0.0333... 0.025 i5 ' ' - ' “lit-W5 Wfllflmwm‘fifl‘ m" =hi‘l- __ _ ._ _____ P I {HIM} 1 D 1 m _ . . . Te he specialising??? thatwe use the Range—flutter method In calmlate the se- en Hill-[HRH] “saints: . Dim” ' Milan-J" = tint!) =1: ' 1%" ef the 1ra’lue rob] ens rams 2.3mm —tt.2462% m P m l. I misses swans: manna tn. . - fibula: =0, t) :1, e n -.= _.JTug_ It} It Etltltltitl 35.52% llfltlflfli 111.559 0.231257 . y fl } i i i “'4 n llsltllttl {1731241 nesnss 1.24 s '10“ 0.40097? Chemises—Knits melted. fer madam Sits-tenthth in scrim“ all-l “5‘ :_ g n till it its: Les 1e"1 tl.fit’ll}tlilll 1.38 :s: 10“: [1.600031 mg single-Mine {eight-digit) arithmetic with a step size it = 0.0!. we etaain the l. q. a s Hm“ an a 1n“ estimate 1.54 s: in“ 0.800001 __ - '. results in Table Bids-It-is-elear free: these results that the numerical appresimatien 1 e I !llltl1|tltl 1-1! ~- ltli l.t}tl{lt'lt}U Hi it 10'” 1-m I ' _ . _-I _e- 'rl ThBLE ass Exact Solutien eff — 101:} = ti. ytfl} = L y't'fl} = "filler and Numerical Appreaintatien Using the Runge—Kntta Method _ I’.‘ . 1 . -- e‘, it '.n linal esantple. censider the prebletn ef determining twe l'i'liii-l'ilij .' sulutit ans til the secend nrcler linear equatien mm h - 0'01 1: —~——- y” — Hing}? = l] _ 1 Numerical Etaet l'er r s- it. The generalizatien Eli numerical {Ethfliqu‘es fur the fifit larder ' Gigs ‘tu - l 1': Elli-“S 1“ 'i te hiehcr erder equatiens er tn systems at equatiens is discussed in 1i; it: I i “T: but lien is. net needed fer the present discussien. Twe linearly Him E 4 I“ _.i liens Hi Ett- it“) are a U I = CUE" m” l and $2“? = 5th ml”- The Willi-l" .3 tin! its-sin s in 5* terse ~.. in * mu} = cesh mitt, is generated by the initial cenditiens dam} = L d); L5 5.4%“ i m _. 33.?“ i I“ t secend selutien. digit] = sinh Mat. is generated by the initial eenditieflfiffii; 1t} 1335: a mi“ 2 s. til 'I {pain} :2 MIT. Altheugh analytically we can tell the difference between": I 2.5 rifle s. tits t _t 1 and sinh Mahlerlarge t we have cesh Jilin: as {swirl/2am sinh mt” i'lses m s numerically these twe iunctiens leek exactly the same if enl},r a fixed numhfltsfl; 4:, 33.45. x we 5. a in 1" are retained. Fer example. cerreet in eight significant figures, we find that '_:; 4:5 Ila“ t me. 1543: h m- in L " ' an ass-19s in” :am s. in 3-“ sinh Mir = cesh Me = 10.3151394- ' L. Chapter 3- - J. 1-;: -I ders ef magnitude. The reasen is the Pffififinfiet -- ital appmxtmatiea. at a small cernpene-nt ef the expeneattally than = ef'm'”. With eightsdigit arithmetic we can expect a reund-eff.Efl.l..!,‘.flfirE ' g _ . _,e _ ,_ h 51E. i Since grim“ grews by a faeter ef 3.? x 105‘u frqu ism ' “rd” m m m tan p as — t cl ce an errer ef erd It‘ll-3 r = 5. an errer eferder It] ‘ near r _— lean PIT} U i _ fir . lattg’ even if he further errers are intreduced- in the Interventng calculattens, aicen in Table 8.5.4 detnenstrate that this Is exactly what happens. i I __ ii 3 Yea shnuld bear in mind that the numerical yalues efth'e entrtes in the cnlutnn nf Table 8.5.4 are extremely sensitwe te slight yartattens 1n hew the less at such details. heweyer. the espeneatial . t e frern the exact selutien fer r s- {15! and frem it by n‘Ianjtr 01‘ liens are executed. Regard ‘ the appresimatien will be clearly eyident. I i l H Ecluatien {ltll is highly unstable. and the behayter shewn In thts example nf unstable prnhlents. One can track a selutien accurately fer a while.antl the-intan’mitig can he extended by using smaller step sizes er mere accurate metheds, hut eye-n the instability in the preblem itself taltes layer and leads tfl large errers. Same Camments an Numerical Methads. In this chapter we have intreduced several-fig ntcrical metheds fer apprnsimt—iting the selutien ef an initial yalue prebletm has'c tried tit emphasize same impertant ideas while maintaining a reasenablelayalt el' ctrntplesity. Far tine thing. we hate always used a uniferrrt step size tattered; pt..tdt_ict_it.nn entries that are currently in use preyide fer yarying the step Shaggy-{HE -.'..il-..L1lt.lllttl'l Pft ILc‘tft'JE‘i. "lie-rt.- are seteral cnnsideratiens that must be taken inte acceunt in cheesingfistgij. -1; t tt ct terse. rate is accuracy: tee large a step size leads te an inaccurate restflt. t when}. an L'l'l'trl' tnlcrance is prescribed in adyance. and the step size at each-step. I-ii.. l. e. tttsi tit—nt e. itlt this requirement. As we have seen. the step size must alseljfg" . n - that the the that] is stable. Otherwise. small errers will grew and seen reader I i. Lllts e. cu thlc‘ss. Finally. fer implicit metheds an equatien must be selyedateaeh ,t _.i er the niethttd used te suite the equatinn may impese additienal restrictitjg; ml the stcp size. in t; huts-ti n g it in c theta. LlI'lti’ must alse balance the censideratie as ef accuracy and sta- t-tilit}. :teainst the .tnteunl at time required tn execute each step. An implicitmelhetly such as the .fitdants—h-‘lnultnn methed. requires mere calculatiens fer each step..-_hilt' it its accuracy and stability permit a larger step size (and censequently fewer steps);- then this may men: than centpensate fer the additienal calculatiens. The baelswtiftl differentiatien fnrmulas ef mederate erde r. say. feur. are highly stable and are there fete indicated fer stiff prnhlems. fer which stability is the centrelling facter. Same current preductinn cedes alse permit the erder ef the methed te be yariitd, as well as the step size. as the calcalatien preceeds. The errer is estimated at each-slay; and the erder and step size are chesen te satisfy the prescribed errer telerattca-Ifl practice. Adams metheds up te erder twelye and backward differentiatien females. up In erder liye are in use. Higher erder backward differentiatien Eerrnulaa unsuitable because at a lack ef stability. Finally. we nete that the smeethness ef the functien f—that is, the nuthI'lll centinueus deriyatiyes that it pessesses—is a facter in cheesing the erder ef'fll-E' methed te be used. High erder metheds lese seme ef their accuracy if f ifi'flfll sntenth in a eerrespending erder. #2 2. Censider the initial value problem 3!! "'2 ‘2 + if"! = [l .- Usmis the Range—Knits instants «with step size a. we ehtai-a the results in Table sss. These resu ts suggest that the selutien has a vertical asymptete between i =. {31-9 and t = Ltt. (3) L31 .1“ = all) be the selutien ef-prehleai til. Further. let y = let, tn be the mlutien ef r=1+a hm=a an and let y = legit} be the selutien ef y' = e. ytfll = t}. tittl Shaw that ezttlgetrtieittt an en seme Interral. centained in [l E l' 5 l. where all three selutiens exists. ('3') Dfllfll’miflfi shit} and legit}. Then shew that am -+ a: fer seme t betneen t = in: ‘=‘ 0.69315 and: = l. (C) Selye the differential equatiens y' = e“ and y” = l + e“. respectisely. tsith the IT‘tlIlJl eendttien‘ttfflfll = 3.4298. Use the results te shew that e'tttr - n. tut-hen .' '-_= “$33. TABLE 3.5.5 Calculatien at the Seiutiea ef the Initial Value Preblen. . 1| - . _t' : t* 't'L". _t"t.ll : t! L sing the Range-Kaila ftlcthttit _._-_._ ____. _._.—...—.— __ _ h f : if it 1 ll it: ‘-* lit-w: '1 tltt! 1.42M: 1 3. Censider again the initial value preblent tlet treat Esat‘ttplc [fuel-Light. c - . - - step size It must be chesen te ensure that the errer at t : this? and .-.t r z t- ': t t- ~. 2 t. :- 0.0005. {a} Use the Euler ntethcid. [b] Use the backward Euler ntethed. {c} Use the Range-Rune ntethed. #713 IE :4. ‘Cohsisier-tlhet‘fififiataefiugpmfilfi .- - ‘e. .._~.--5_ ,. H. y =-..ioy' sent? sri'ilfitt ill is" ‘ .. .- _ _ I “£15., .r.“ (a) Find me: sat-trim: r = m thd'detflvrat’h ml.“ ii ‘ 5?? . _ [h] The s'tahilitv analysis in the'teitt suggests thflh . stable only for}: e 0.2. Confirm that this is trufl'hl' flPPIWE'-‘Ihfii”..-. " - pfflhlgm for t} 5 r 5 5 with step sizes- near 0.2. D 5 - ' tree-ti.“ ' - for er e . where” - .. A l the Run EhKllllfl-l‘flfllhfld to this. problem .— - - (a) pp y g hilitv of this rnethodiI 3.; . {d} Applv the backward Euler method to this problem for _i sizes. What step size is needed to ensure that the error at r = 5 ts leanings-g1thig- 1-- .. in each of Problems 5' and 6: :I ia) Find a formula for the solution of the initial value problem. and note tha;_;_.“;fl._ L- “I” . ' " ' I i. J oil It. {h} Use the Runge-Kutta method with it = tion for ti 5: r 5 l for various values of It such as I _ I I t ct Explain the differences. if any. between the exact solution and the I: it 0.01 to compute approximate s: 1.10.2a.sndso. - -- '— _- lififll'i- I. "J- : I _ . 2 .L _I I l ' 5. _v' - iv = l — it- _vtill = l] #2: 6. it # ly = 21‘ H in , 3(0) :91}. {I I} " _ _ H _ _..._ _ __-__ .. __—.-..— ____—__- r --I" Brawl-fl 4 1"” i hit '. E z, '. late I. 'r- - I I- . I- . _ 'IL rl._ " . ,r I II I I I -. I - r '_-_ I I-_;‘ . ., 2.. _1 _I__ __ ' “drain, it . _a__ _I” ._ - I.“ .-I -'-":t.' if - . _ .. r n. d . . _I '1‘ I I l _t_.tl.s_ I Irl “I I! 1- - "I ‘- -'-‘r I “I 'l' __1 I I- L. I' I I " -J Ferri".*ii.'t:;"lé .' “In '1 .. .- I. -'_-~ r.1- .-. . :1: lime '. " ;_I"_‘. .. I. . . . lit-for _ 2. - themed be extended to a system. the stewfrumrstatnrvstisse- " ' (lifelike + + Zita + to): ‘6) "eff-"L -'.:_._Q-..:_:?S.:. .-..j-'r_' - - - . _ strengthen , _ - where '.= ill-[till In its * kin-'5 Ht... + {ft/2}. In + {M33411}. n -—= tit. Harem... + (fifllkaal. _ the + h.-x.. + Mini}- (7) a. if bystems of First Order Equations I. ._.I The .fonnttles-far the Adams—Menitonpredictoneorrector method as it applies to HH- . I l.‘-..;.."" ' ' In the preceding. sections we discussed numerical methods for solvingtimé j. p: ilhiLFilh associated with a first order differential equation. These niet.hodst_"__ _ in: trpt'iiietl to a its-stern of first order equations Since a higher order equegii alas-tits in: reduced to a system of first order equations. it is sufficient the-rd sjsstt-rns ol' lirst order equations alone. For simplicity. we consider a systeiiitff " first order equations 4 - .r’ =ftt..r.v}. _vr = g{t.x._v). 1* with the initial conditions In“) = It]: FUD} = _l’n. The functions f and g are assumed to satisfy the conditions of Theorem iiiiiiilfu _' the initial value prohlem (1 ). (2) has a unique solution in some interval oftlf. - containing the point tn. We wish to determine approximate values shag. _ and rate. . .. _v... . . . of the solutions = satay = tit/(rt at the points t... :1" Elijit'it‘ji; n 21.1.... Iii s. In vector notation the initial value problem i l). (2) can be written as " "'" ' it’ = fin)“. um.) i: x“! -__' .-_ .- ill _ where it is the vector with components I and y. f is the vector function "iiiiit-itt‘uifi. Pflnflms f flfld 3+- flfld fin is the vector with components in] and yr}. The the _I the previous sections can he re adin generalized to handle systems of twe_:.('_f git-i1 tilt r:' I-'I_' EXAMPLE- the initial value problem (1.). (3.) are given in Ptoblem 9. The. vector equations (3). (4). (a). and ('3) are. in fact. valid in any number of dimensions. All. that-is needed is to interpret the vectors as having a: components rather than two. Determine approximate values of the solution a: = sin. v = tent at the initial value problem _v' = ‘r-t- v. tl‘il vii-i} = fl, {1-H I’ = .t: r- 4?. 110} = l. at the point t = 0.2. Use the Euler method with h = tit sad the Range-liens method with h = 0.2. Compare the results with the values of the esact solution: ‘l :4. _r __i" E + r.’ 't’ '— t. 1r — _....._._._.._' I i. gt — .__.._—.—— t..H: dttl -r 2 st 4 Let os-lirst-use the Euler method. For this problem!” =1... — 4v. and g. = -- I... fa=l—t4-ttt}t=l. et_.=r-l-rll=—l Then. from the Euler formulas {4} and [3). we obtain it. = t + {tutti} = 1.1. _r. = it + title—ti : will. At the next step fl = Ll _ {4}l-fl*n = L51 3. = “1.1+t—llllr: —l.l. .-_...__ .- - i . I r I i a- I -I_ .f_. .. I. .r---: 1. II ._ I ' _- ._ __ _ I I I .Iv ._._ .. _ ._._ __ ._ — .__. I. -" — ' ' . " . " I I .. -- Hansen-=11} n: ' ' - " i=- ' ".r q-‘-."---.-:: ‘.- _ a. — .2 2:..- .II _ _-‘E_:_f_-II::;I_{ ‘tlir -__ 1; “r”: i._:__1 -_ ll. .. I . - .. . _ ._ .- 1.15. 41.121 1-63 - ‘3”. f I “i ' ' '2'" - .. . km = (21 13 43.12)) = (-1.27): '3 : .- . -_ -- - _, . . . flfi‘gsgffiflyffl—wflh ' r — Smirk-335m - 93H) f{1.326.-—U.254} _ 2.342 ' an; . k‘“ = gt1.326.-—fl.254} ” —1.sse ‘ Then. substituting these values in Eq. (h). we ehtain _ 1 ill 9.602 _ 1.320066? 1' = (a) + ? —7.s2 “ weasessass ' _ _ i'hcse s-alucs cf .r: and y] are in errer by aheut [1.000353 and 0.000136. res r ' I percent-.1 gc c-rrers much less than ens-tenth at 1%. _ _ I. U: is e _-. ample again iiiustrates the great gains in accuracy that are ebtainabtehyli ' -e uraie .irnrusimatien meihed.such asthe Runge-Kutta methed. In the-calculatin {'51 '37:. _‘ we tun-items. the Range—Knits methed requires enly twice as many functieni '. ' ._;-1._ Euler inc-thud. but the errer in the Runge-Kutta ntethed is abeut 20E] tiara.an _' I_ . .j' ' l I. + :fififlffli + 19f: - 515...: '+ f»: I. "Ff: +- + 193.. —' 551.1 + h—fl- IE-vill’ne the flutter! at I = 0.4 her the; example value ejrehlein;atJ =.I~'_*'+ 4y.-..'_ = -_-e:.+ Jr with-rte] = 1. ya?!) = 0.- Take in =-n.1. Correct the pm dieteti- valuegnce... Enn'ghevnlnesefxr. . . . .. )5 in the values of the exact solutien mended teeis;psi-gitsass1L a; -.= 1.32042, :3 =1.srnst. y. = 4.111155. y; = hasten. W t i r F = .1 . LL huh" r mflhl d REFERENCES There are many beekeef varying degrees a-fsopflsislicafien that deal with Emfifll analysis in were! "and the numerieal-selutien'efetdinsry diflcrcnfial equations in paniaflsr. Arming these are .Aflfiet. Uti- EFL, arid Renew. _R.. Compare: Medic-d: fer animus flifl’cflm Erin-“em: and DifihrentieI-Algebreic (Philadelphia: Society fer Indteuiai and Applied Ruthenium. 1093?. Gear. C.Wt11iarn. Nem‘cricei Int-tie! we: Mince in firemen Diflrrmfifl Elam-5mm magi-anew (11th NJ: Prentieiisflel]. 1971). Henrici. Peter..Di.rcrctc-Vnniehfc stalled: in Hide-tel? Difimntiai EtfflfllicI-‘H {New hurt. W3. 1M1. Lawrence F..Nnn1criee! Solution efflrdnmq Dim-rennet 54er [New Tart Grantees and .._—. _ .. ll. In each at Prehlcrns 1 threugh 6 determine appreximate values of the L .‘r' = will at the given initial value prehlem at I = 0.2. [1.4. [1.6. 0.3, and 1.1]. -_-. . results ebtained by different rnethecis and different step sizes. " . ' ' {a} Use the Euler methed with h = 0.1. Shampiflfls {11) Use the RungewKutta methed with h = 0.2. Hall.199'4)-. {E} USE the Runge—Kutta methed with h = {1.1. A detailed expmiIien ef Edens prwiflerwnmm nicthruasindudmg practical grind-chm tie-r impa- nienta'tien'. mfi'? be feund in Shampiite. L. E. and Gerden. M- '11.. {Twitter Sumner: ef (Winery Dig?th Eqafllflns.‘ m fiend Veins Pivbicrn (San-Flancisee: Wu. 1915}. have chapters en differential equations Fer example. at In clemen- l. .r'=.r+y+r. y'=4.r—2y: xiii}: l, y{fl}=fl €33 2- I' = is + e. y’ =xyr. x10} = 1. he) =1 ‘2. 3. .r' = —t.r —y — 1. y’ =.t:'. IUD) = 1. NO} = 1 fl 4.x’=.r—_v+.cy. y’=3x-2y—J:y; xifl}=fl. y(fl)=1 (2, 5. .r' = .111 — as: — ash. y = yi—tm + ash; size) = 4. yfll) =1 ‘2. 6. x’ = carpi—x +y} — cesx. y' = sin{.r — 3y}; .110} = 1,r ya?!) = 2 Many beeks'en numerical anabGi-s nil-3|I leech-see Burden. E. L. and hires. .l. EL. Hume-i ...
View Full Document

Page1 / 21

CH08 - 44‘? Irlll'rF'Ilnl 't_I' ll 'lr'l'Ii-J' l' '5'...

This preview shows document pages 1 - 21. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online