A New Iterative Method for Solving Initial Value Problems
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1 A New Ierave Meod or Solvg Ial Value Problems Mgse Wu ad Weu Hog Dearme o Ma., Sascs, ad CS, Uvers o Wscos-Sou, Meomoe, WI 5475, USA Dearme o Maemacs, Clao Sae Uvers, Morrow, GA 36, USA Absrac - I s aer, we roduce a ew arameer erao meod (P-Ierao or sor,wc ca be aled o Adams-Moulo meods o solve al value roblems. Comared w Jacob erao meod, as ree advaages: ( ma sgcal mleme e sable rego o e erave rocess suc a we ca ecel use e large sable rego o mlc ormula; ( allows large se sze o reduce e umber o eraos; ad (3 ca elarge e sabl rego. Kewords: P-Ierao, Adams-Moulo Meod, Sable Rego. Iroduco Cosder a al value roblem: ' ( (, We s dcul o d a aalc soluo, we aurall al a umercal meod o d a aroxmae soluo o ( over a erval [, T]: ( ( or,, L. Toug ere are ma umercal meods or comug (, e ca be classed o wo es o meods : exlc ormulas ad mlc ormulas. Te Adams- Basor (A-B ormulas ad Adams-Moulo (A-M ormulas are oular mul-se exlc ad mlc meods, resecvel. For examles, e exlc wo-se Adams-Basor meod (A-B s ( ( ( (3 (, (, ( ad e mlc wo-se Adams-Moulo meod (A-M s ( ( (5 (, 8 (, (, Te geeral -se A-M ormulas ca be wre as (3 ( ( β (,., Usg a mlc meod, sce aears o bo sdes o e ormula, we ave o al erave rocess o ge (oce a drecl solvg ad comug could be dcul or ( cumbersome. Le be a al aroxmao o, e ( (5 (, 8 (, (, (4 were, m rereses e umber o eraos. I geeral seag, more eraos ca resul a beer aroxmao o e covergece codo olds (see laer dscusso. Ts erave rocess mg coue ul a ges a aroxmao w e requred error boud. O course, e se sze or e calculao mus be roerl seleced because o (a Error corol ere s rucao error a s deedg o, ad (b Sabl eac umercal ormula as a sable rego (also called absolue sable rego suc a λ mus be w e sable rego or covergece, were λ s a egevalue o e ODE. Te sable rego dscusso ca be oud rom oer resources [], we om e deals. For examle, e sable rego o A-B ad e sable rego o A- M are as sow Fgure ad Fgure, resecvel. Fgure Fgure B Comarso, e exlc meod s easer o be erormed a e mlc meod, bu s accurac, geeral, s lower
2 a a o e mlc meod. Te sable rego o e exlc meod s relavel smaller a a o e mlc meod (reer o Fgure ad aga eve oug usg e same ses. Aoer cal examle: e sable rego o e exlc Euler s ormula (, s sde o a ds ceered a (-, w radus wle e sable rego o e mlc Euler s ormula (, s e ere comlex lae exce e u dsc ceered a e o (,. Te sable rego lms e se sze wle usg a umercal meod. Te mlc meod ma allow larger se sze comaravel, bu requre a erave rocess. Almos all umercal aalss exs roduce e oular mulse A-B ad A-M meods. However, mos o em om e sabl dscusso due o comlex. A-B ad A-M ormulas are wdel used scec calculaos suc as a euroal ssem []. Somemes, eole combe wo es o ormulas ogeer o develo so called redcor-correcor meod a uses a exlc ormula as a redcor ad a mlc ormula as e correcor [3]. Te erave rocess or a correcor ca coue ul e requred accurac s aceved. Te erave ormula (4 s called e Jacob erao (also called smle erao. I addo o a a umercal meod lms e se sze, e erao rocess sel also lms e se sze as well. For sace, e sable rego o (4 s e sde o a crcle ceered a e org w radus.4. Eve A-M (3 as a relave large sable rego (see Fgure, bu ol e erseco o e sable regos o (3 ad (4 s useul or acual calculao. I s aer, we roduce a ew erave meod (amed P-Ierao or sor, wc uses a small arameer regular erao meod suc a ma sgcal mleme e sable rego o e erave rocess, ecel use e large sable rego o e mlc ormula ad large se sze a ca resul reducg e umber o eraos, ad sgcal elarge e sabl rego. Ts aer s orgazed as ollows. Te P-Ierao s roduced seco. Te sable rego s dscussed seco 3. I seco 4, we comared P-Ierao w Jacob erao b reseg umercal resuls solvg bo lear ad olear al value roblems. I seco 5, we gve some commes o urer alcaos o e P-Ierao o Mulse ormulas suc as A-M ormulas or solvg ODE ssems. Te P-Ierao meod I s seco, we wll derve our P-Ierao meod. For smlc, we wll use e oaos: (, (,, ad (,. Te small arameer erao ecque dscussed s aer ma be aled o all mul-se A-M ormulas. Le s roduce s rocess b worg w e A-M. Rewre (3 as 5 8 (, ( (, (, c Addg o bo sdes o e equao, were c s a osve arameer, we ave or 5 8 c c 8 c 5 c Tus, e A-M becomes ( c 5 c (5 8 8 Alg smle erao rocess o ge we ave 8 ( c 5 c (5 8 ( m (5 (6 I (6, e eraos ma be erormed a ew mes or reeaed ul acevg e requred accurac, sa < olerace.
3 We ca combe e wo exressos o oe: b leg 5, we ave: 5 c ( c 8 ( (6b W a roer se leg ad a seleced value o e arameer c, we ca erorm e erave rocess o A-M b (6 or (6b easl. Prooso : Ierao ormulas (6b ad (4 bo aroac as m. o e same value o Proo: I ac, s eas o cec a, ere s a value * a maes (6b exacl rue,.e., 8 ( c ( * * * Te, b relacg b 5 5 c, we ca reduce o (4: * ( * 5 8. Tus, (6b ad (4 rovde e same value o I s well-ow a egevalues o a sel-sable dereal equao ssem are all locaed o e closed le-al o e comlex lae. Te es equao ' λ s a sel-sable λ re θ w r ad π θ 3π. Te λ ma be cosdered as e egevalue o e es equao. Prooso : Ierave ormula (6b coverges ( cλ <. Proo: Le us dee 8 Φ ( (6b. Clearl, we eed o comue Φ ol oce durg e erave rocess o comug, so we sml wre e erave rocess w arameer c as ( c Φ. Alg e es equao I * ' λ, we ave ( cλ ( cλ Φ Φ (7 s a xed o o e erave rocess, e ( c Φ λ Subracg (8 rom (7 gves Ts mles a (8 m ( λ ( c. * ( * as m, *, ( cλ <. As ou ave see a e sable rego o A-M ormula s gve b e gure. Wle alg A-M o a ssem o equaos, we mus al erave rocess. Oe mg use e Newo s meod, wc, owever, requres comug verse o e marx or eac se. Ts rocess s o ol exesve, bu also volves roud-o errors. Oe mg also use e Jacob erao as sow b ormula (4. However, a smlar argume (as e roo o rooso sows a e Jacob erao o A-M requres λ < / 5. Eve A-M ormula as a relavel large sable rego, e Jacob erao would reduce e se sze < /(5 λ o a ver small umber λ s large. I ex seco, we wll sow a e ormula (6 or (6b ma overcome s roblem. 3 Sable Rego Le us sae a ver mora resul o e sable rego o e erave ormula (6b. Prooso 3: Te sable rego o (6b ca be arbrarl elarged o e closed le-al o e comlex lae as c. Proo: Accordg o rooso, ormula (6 coverges ( cλ <, were 5 5 c <. So, (6b coverges θ cλ <. Alg λ re r cosθ sθ, we ave cλ ( cr θ ( cr θ cos s <, ad (6b coverges c cosθ r <. Tus, c s roerl seleced
4 o sas cosθ c <, e e ormula (6 or (6b coverges r regardless o wa se sze s. I arcular, e ormula (6 or (6b would wor erecl λ s a real umber or close o real umber (θ s close oπ. For coveece, le c. Te (6b coverges ( λ <, or re θ <. We ave or < ( r ( r cosθ s < r < s Te lower boud s less a zero, so we ol eed o ae care o e uer boud. For a xed ad s s π 3π θ,,, ; 5,. Ts meas a e sable rego o (6b ma be elarged arbrarl o e le-al o e comlex lae as, or c. We would le o comare e sable rego o e P- Ierave rocess (6b w a o e A-M because e se sze s lmed b bo A-M ad e erave rocess. Te sable rego w dere values s gve b gure 3. I s obvous a e sable rego o A-M s a roer subse o e sable rego o e P-Ierave rocess (6b. B comarg e sable regos, we sugges o selec a value o rom [.5, ]. Fgure 3: Te sable rego o P-erave (6b w.5,.5, ad, resecvel. Te ermos crcle s e sable rego o e Jacob erao or A-M. 4 Numercal Resuls I s eresg o comare Jacob erao w our P-Ierao. We solved que a ew ODE ssems b Jacob erao ad P-Ierao. We would le o sow e resul o solvg several ssems below. Examle : Solve e x ssem d d d 3 4 d ( Te aalc soluo s ( 5e 4 e, ( 5e 6e. Ts ssem as egevalues - ad -. Sarg w exac al value o ad, we comued (, or o,b A-B ad b A-M. Te erave rocess o A-M was doe b Jacob erao ad P-Ierao searael. We le eac meod comue ses, e.g., comue o. Te 5 erave rocess sos <. We coued e oal umber o eraos (..o.. erormed or e ses. W large se sze, Jacob erao als, bu P- Ierao coverges (see e able. However, due o e rucao error, e soluo as bg error. Sce e rucao 3 error so(, large value o wll resul a bg error. Te eresg o s a P-Ierao as a larger sable rego a a o Jacob erao as we roved earler. Usg small value o, P-Ierao soluo s closer o e exac value.
5 Table. Ierao meod Se sze. Soluos..o.. Jacob Dverge 436 P-Ierao,.5 Coverge 5 P-Ierao,. W large 3 P-Ierao,. Error 7 Ierao meod Se sze. Soluos..o.. Jacob Correc 436 P-Ierao,.5 Beer a 57 P-Ierao,. Jacob s 55 P-Ierao,. 75 Ierao meod Se sze.5 Soluo..o.. Jacob Correc 96 P-Ierao,.5 74 Beer a P-Ierao,. 57 Jacob s P-Ierao,. 48 Ierao meod Se sze. Soluo..o.. Jacob Correc 53 P-Ierao,.5 7 Beer a P-Ierao,. 48 Jacob s P-Ierao,. 37 Examle : Solve e ssem o d 9 3 d d. 9 3 ( d d d Ts s a sable ssem w egevalues -, 4 ± 4. We solved s ssem a smlar maer. Te resul s sow as e able. Te rae o covergece o e P-Ierao s muc aser or large value o a a or small value o. However, we s small, e advaage o usg P-Ierao exss, bu o ver sgca. Te suggesed value wors e. Examle 3: Le s solve a o-lear al value roblem Table. Ierao meod Se sze.4 slouo..o.. Jacob 459 P-Ierao.5 Correc ad P-Ierao,.5 comable 97 P-Ierao,. eac oer. 34 P-Ierao,. 3 Ierao meod Se sze. slouo..o.. Jacob 98 P-Ierao,.5 Correc ad 97 P-Ierao,.5 comable 65 P-Ierao,. eac oer. 54 P-Ierao,. 63 Ierao meod Se sze. slouo..o.. Jacob 58 P-Ierao,.5 Correc ad P-Ierao,.5 comable 7 P-Ierao,. eac oer. 53 P-Ierao,. 43 Table 3. Ierao meod Se sze. slouo..o.. Jacob 49 P-Ierao,.5 Correc ad 7 P-Ierao,.5 comable 43 P-Ierao,. eac oer. 45 P-Ierao,. 64 Ierao meod Se sze.5 slouo..o.. Jacob 3 P-Ierao,.5 Correc ad 79 P-Ierao,.5 comable 48 P-Ierao,. eac oer. 35 P-Ierao,. 54 Ierao meod Se sze. slouo..o.. Jacob 8 P-Ierao,.5 Correc ad 8 P-Ierao,.5 comable 55 P-Ierao,. eac oer. 4 P-Ierao,. 3 d d 3, (. 3 Aga, we solved s ssem a smlar wa. Te resul s sow as e able 3. We.5, Jacob erao s dverge, owever, P-Ierao sll coverges. Te advaage o P-Ierao s sgca or larger.
6 5 Coclusos Te P-Ierao o Mul-se ormulas suc as A-M ormulas s useul sceme or solvg ODE ssems. Lmed b e sace, we coclude s aer w e ollowg commes: (a A-M ormulas w Jacob erao are ece or o-s ODEs. Te A-M ormulas w P-Ierao ca be aled o s ssems [4]. (b I e egevalues o a ssem are real umbers or close o real axs, e advaage o usg P-Ierao s ver sgca ad oe ca use muc larger se sze o do e calculao. (c I e egevalues o a ssem are close o e magar axs, e P-Ierao does o mae oo muc derece comarg w Jacob erao, bu wors e ad rovdes a more accurae soluo. (d Te erave rocess ma be aled o oer A-M ormulas. For examle, or e ree ses A-M ormula ( , ( A M 3 e erave rocess ma be doe b e ollowg sceme ( c ( Or, erave orma, ( c ( (9 Fgure 4: Te sable rego o P-Ierao or A-M3 w.5,.5, ad, resecvel. Te ermos crcle s e sable rego o Jacob erao or A-M3. 6 Reereces [] Oe o e resources: Bra Brade, A Fredl Iroduco o Numercal aalss, 6, [] Meods Neuroal Modelg: From Ios o Newors, Crso Koc & Ida Segev, 998, ISBN:63. [3] Numercal aalss (8 edo, R. Burde ad J. D. Fares, 5, ISPN: , 9 3. [4] Solvg Imlc Equaos Arsg Fromm Adams-Moulo Meods, Tam Ha & Yuua Ha, BIT, Volume 4, No., were. Wle leg c as we dscussed 9 4c earler or A-M, e value ca be ae a smlar maer. Te sable rego o (9 s gve o cure 4.
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