A Class of Lobatto Methods of Order 2s
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1 IAENG Ierol Jourl of Appled hem, :, IJA A Cl of oo ehod of Order Wg Fgog d o Xog Ar he fmle of oo RugeKu mehod h o of oo IIIA mehod, oo IIIB mehod, d oo IIIC mehod re ll of order d Ale. Ug V rformo d he mehod of udeermed oeffe, l of oo RugeKu mehod of order d Ale re orued hrough overg ly fuo o dgol Pdé ppromo o ep(. he wo umerl emple lo how h he derved ew oo mehod hve hgher ury h rdol oo mehod. Ide erm oo mehod, order, Vrformo, Pdé ppromo ge d e odered ppromo o he oluo h. oo RugeKu mehod re hrered y d. he fmle of rdol oo mehod volve oo IIIA mehod, oo IIIB mehod, d oo IIIC mehod. A ompreheve revew ou oo mehod e foud Referee [][] d e ummred le. ABE I A SUARY OF RADIIONA OBAO EHODS mehod mplfyg umpo order ly fuo I. INRODUCION oo RugeKu mehod [][] for he umerl egro of ordry dfferel equo re med fer Rehuel oo. hey re hrered y he ue of ppromo o he oluo he wo ed po d of eh uervl of egro, ]. Oe of [ wellow oo mehod he mpl rpeodl rule, whh h ee wdely ued pre. Coder yem of ordry dfferel equo (ODE d f,, < d ( ( Srg from oe ep, f, of he ge oo RugeKu mehod ppled o ( e epreed follow f h,, (, ( h h f h, ( where he ge vlue fe d he oeffe,, hrere he oo RugeKu mehod. he ermede vlue, (, re lled he erl urp reeved Jue, ; reved Augu,. h wor w uppored y he Nol Nurl See Foudo of Ch uder Gr No d 977. Wg Fgog wh College of Elerl Egeerg & New Eergy, Ch hree Gorge Uvery, Yhg, Hue Prove, Ch. (Correpodg uhor: Eml: fwg@gu.edu... o Xog wh College of Elerl Egeerg & New Eergy, Ch hree Gorge Uvery, Yhg, Hue Prove, Ch. (Eml: lgu@.om. oo IIIA B(, C(, D( (, Pdé oo IIIB B(, C(, D( (, Pdé oo IIIC B(, C(, D( (, Pdé However, hee rdol oo mehod re ll of order d Ale; hey hve he ddvge of lower lulo preo whe omprg wh Gu mehod of order. I h pper, we devoed o oru l of ew RugeKu mehod ed o oo qudrure formul [] whoe order e hghe of. II. OBAO QUADRAURE FORUAS For gve umer of ge, he vrou oo mehod hve he me oeffe d ed o he orrepodg oo qudrure formul []. he oluo of ( e ppromed y ug drd qudrure formul: h Ł f, ( d f h, ( h wh ode oeffe,,, d wegh oeffe,,. oo qudrure formul, lo ow Guoo qudrure formul he lerure, re gve for y e of ode d wegh fyg odo dered herefer. he ode re he roo of he polyoml of degree d d ( hoe ode fy < < <. he wegh ( ( (Adve ole pulo: Ferury
2 IAENG Ierol Jourl of Appled hem, :, IJA d ode fy he odo B ( where B( p :, for,,, p ( oo qudrure formul re ymmer,.e., her ode d wegh fy,, for,,, (7 he fmle of oo RugeKu mehod dered ove dffer oly he vlue of her oeffe mr A. he oeffe mr A of hee fmle e lerly mplly defed wh he help of olled mplfyg umpo [] C( q :, for,,, d,,, q (8 D( r : (, for,,, d,,, r (9 he mpore of hee mplfyg umpo ome from fudmel reul due o Buher. he oeffe,, hrerg he oo RugeKu mehod (( e rrged o he form of le lled Buher leu [] A O ( III. CONSRUCION OF A CASS OF OBAO EHODS OF ORDER S A wellow h, f he ly fuo of ge RugeKu mehod (,Pdé ppromo o ep( [7], he h mehod of order d Ale. Ipred y h, he ly fuo of ew oo mehod re overed o dgol Pdé ppromo o ep( y ug he mehod of udeermed oeffe, o urlly, h l of ew oo mehod re of order d Ale. Whou hgg he oeffe d, ge ew oo mehod defed A V AV ( where V ( O Ł A O ( O Ł Equo A V A V he fmou Vrformo [8], [9]. Suppoe h ew oo mehod fe ymmery odo PAP e A [], he, he ly fuo of ew oo mehod e luled y de( I ( e A de( I ( PAP R( de( I A de( I A ( de( I A de( I A de( I A de( I A Due o A eg mr wh pel ruure, ( e furher epreed (! ( (! R ( ( (! (! e R ( equl o (,Pdé ppromo o ep( deoed y e,.e. (! ( (! (! (! e ( Comprg he orrepodg polyoml oeffe o oh de of (, e ferred h ( ( O Ł Ł Ł ( Ovouly, (7 e furher mplfed o (7, for,, (8 ( herefore, whe he oeffe (,,, of ew oo mehod fy (8, he ly fuo (,Pdé ppromo o ep(, o urlly, h oo mehod he mehod of order. Gu mehod good emple for he propoed oruo mehod. Sge Gu mehod fe (8 d (Adve ole pulo: Ferury
3 IAENG Ierol Jourl of Appled hem, :, IJA ly fuo (,Pdé ppromo o ep(. Equo (8 h uque oluo for gve, for emple, re gve y, (,,, (,,,, ( 8,, 9 8,, Now, ge order oo mehod wh,, re led elow, Α 7, Α, A o he uhor owledge ou RugeKu mehod, he oo mehod of order d Ale dered ove hve o fr o ee repored. So, for oveee h l of oo mehod re med herefer oo IIIF mehod. Wh more, he oruo mehod ed o Vrformo d dered ove mlrly e ppled o oruo of order Rdu mehod. Ieregly, whe he oruo mehod dop oeffe d of rdol Rdu IA mehod, ew RugeKu mehod re Rdu IB mehod [], [], whe he oruo mehod dop oeffe d of rdol Rdu IIA mehod, ew RugeKu mehod re Rdu IIB mehod[], []. Rdu IB mehod d Rdu IIB mehod were fr propoed y Su Geg [], [] ug Wrformo []. However, her ly fuo re hged o (,Pdé ppromo o ep(, ed of (,Pdé ppromo. herefore, Rdu IB mehod d Rdu IIB mehod re o he mehod of order u order. IV. NUERICA EXPERIENS I h eo, we gve wo mple emple o llure our m reul oed prevou eo. Emple. Coder wodegreeoffreedom yem govered y d he e oluo of he prolem o( 8 o( o( o( o( o( ( Wh he e oluo ( referee, error reore wll e oerved d red ug four d of oo mehod (error deoed y Δ d Δ. Fg. how he dpleme error reore ompro of oo IIIA mehod d oo IIIF mehod. Fg. how he dpleme error reore ompro of oo IIIB mehod d oo IIIF mehod. Fg. how he dpleme error reore ompro of oo IIIC mehod d oo IIIF mehod. From Fg., h ee how h oo IIIF mehod h hgher lulo ury h ll he rdol oo mehod. D oo IIIA oo IIIF && Ł && Ł o( ( wh l odo ( (, & ( & ( ( Error reore of ( (Adve ole pulo: Ferury
4 D IAENG Ierol Jourl of Appled hem, :, IJA oo IIIA oo IIIF D. oo IIIC oo IIIF ( Error reore of ( Fg. Error reore ompro of oo IIIA mehod d oo IIIF mehod (, h. ( Error reore of ( Fg. Error reore ompro of oo IIIC mehod d oo IIIF mehod (, h. D D oo IIIB oo IIIF oo IIIB oo IIIF.8... ( Error reore of ( ( Error reore of ( Fg. Error reore ompro of oo IIIB mehod d oo IIIF mehod (, h. Emple. Coder hrdeg el prg. he oler dym equo & & ( ( where >, >. he e ol eergy E (& ( o e he me egro umerl mehod, he pere error erm of he eergy w rodued [], E r E E E % ( ( where E he ol eergy. For umerl reul pree here,,. he l odo re &. d. le l he mmum perege error of he ol eergy over he me duro of. he reul of four d of oo mehod re oed wh d he Newo mehod h ued olvg he oler lger equo. ABE II AXIU PERCENAGE ERRORS OF HE OA ENERGY DURING S ep e h.... oo IIIF.9... oo IIIA oo IIIC oo IIIF oo IIIB oo IIIC D I e lo ee from le h he oo IIIF mehod h hgher lulo ury h ll he rdol oo mehod. Aure reul e oed wh muh lrger ep e ug oo mehod ( Error reore of ( V. CONCUSION A l of oo mehod of order d Ale hve ee uefully orued d umerl emple hve (Adve ole pulo: Ferury
5 IAENG Ierol Jourl of Appled hem, :, IJA how h ew oo mehod re more pree h rdol oo mehod of order. he oruo mehod ed o Vrformo e lo ppled o oruo of order Rdu mehod. I h ee foud h order Rdu mehod re Rdu IB mehod d Rdu IIB mehod, whoe ly fuo re (, Pdé ppromo o ep(. ACKNOWEDGEN he uhor would le o h he oymou referee for her helpful omme d vlule uggeo, whh led o he mproveme of he murp. REFERENCES [] E. Hrer, S. P. Nøre, d G. Wer, Solvg Ordry Dfferel Equo I: Noff Prolem. d. ed. Berl: SprgerVerlg, 99. [] E. Hrer d G. Wer, Solvg Ordry Dfferel Equo II: Sff d DfferelAlger Prolem. d. ed. Berl: Sprger Verlg, 99. [] J.C. Buher, Numerl ehod for Ordry Dfferel Equo. d. ed. New Yor, Joh Wley & So, 8. [] N. Seu,. Sulem, d F. Iml, e l, A gly dgolly mpl RugeKuNyröm mehod for olvg ollory prolem, IAENG Ierol Jourl of Appled hem, vol. 9, o.,, pp.,. [] W. Guh, Hghorder Guoo formule, Numerl Algorhm, vol., pp.,. [] H. Y. u d G. Su, Impl RugeKu mehod ed o oo qudrure formul, Ierol Jourl of Compuer hem, vol. 8, o., pp ,. [7] B.. Ehle, O Pdé ppromo o he epoel fuo d Ale mehod for he umerl oluo of l vlue prolem, Reerh Repor CSRR, Dep. AACS, Uvery of Werloo. [8] F. H Chpm, Ale RugeKu proe, BI Numerl hem, vol., o., pp. 8 88, 97. [9] K. Burrge, Sly d effey propere of mpl RugeKu mehod, PhD he, Uvery of Auld, 978. [] R.P.K. Ch, O ymmer RugeKu mehod of hgh order, Compug, vol., pp. 9, 99. [] G. Su, Coruo of hgh order ymple RugeKu mehod, Jourl of Compuol hem, vol., o., pp., 99. [] G. Su, A mple wy of orug ymple RugeKu mehod, Jourl of Compuol hem, vol. 8, o., pp. 8,. [] Y.. Xe, A eme of me egro heme for oler dym equo, Jourl of Soud d Vro, vol., o., pp., 99. (Adve ole pulo: Ferury
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