Integrasi Numeris dengan Menggunakan Polinomial Lagrange. Syawaluddin H 1) Hang Tuah 2) Widiadnyana Merati 2) Leo Wiryanto 2) f (x) =

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1 Huhe, Vol. No. dkk. Aprl 5 url TEKNIK SIPIL Iegrs Numers deg Megguk Poloml Lgrge Sywludd H Hg Tuh Wddy Mer Leo Wryo Asrk Pd pper dsjk pegemg egrs umers erdsrk poloml Lgrge. Meod yg dhslk mrp deg meod Guss-qudrure, deg pered erlek pd pegml k-k egrs. Meod memerk hsl egrs yg sg k, deg keslh seesr.% -.5% pd egrs fugs susodl. Meod yg dhslk jug dp dguk uuk meyelesk persm dfferesl wku orde deg hsl yg sg k. K-k kuc : Poloml lgrge. Asrc I hs pper, developme of umercl egro mehod sed o Lgrg polyoml s preseed. The resuled mehod lke Guss-qudrure s mehod, u dffers pos of egro smplg. The mehod gves very good resul egro of susodl fuco, wh error ou.% -.5% o exc soluo. The mehod c lso e used for solvg frs order me dfferel equo, wh very good resul. Keywords : Lgrge polyoml.. Lr Belkg Meod egrs umers sudh cukup yk yg dkemgk, mul meod rpezodl smp deg Guss-qudrure. Dr semu meod yg d yg plg mudh pemky d memerk hsl yg cukup k dlh meod Guss-qudrure. Pd pper dkemgk lerf meod egrs p ermksud meel keleh upu kekurg meod Guss-qudrure. Meod egrs yg dkemgk dlh erdsrk fk hw suu fugs dp dyk dlm euk l yg leh sederh yu deg poloml Lgrge. Megg euk poloml Lgrge dlh ep pd suu jumlh k poloml, mk l poloml erseu dkerjk pd ssm koord kurvler deg dom yg ep < ξ <, mk egrs poloml erseu k meghslk hrg yg ep, yg seljuy k dseu deg koefse egrs.. Poloml Lgrge Pd psg d (f, x,,, f dp ddek deg poloml Lgrge. Adpu euk pedek erseu dlh [Burde, Dougls, 85] f (x j L (x j f ( - x ( x- x ( x - x ( x - x (. dm : L (x dlh poloml Lgrge deg euk ( x - x ( x - x ( x - x...( x - x L (x ( x - x ( x - x ( x - x...( x - x ( x - x j L (x (. ( x - x j Deg j demk egrs suu ( x - x j x...( x - xj...( x- x ( x - x... x - x... x - x ( j (. Sf Pegjr Depreme Tekk Spl FTSP-ITB.. Sf Pegjr Depreme Tekk Spl FTSP-ITB.. Sf Pegjr Depreme Tekk Spl FTSP-ITB.. Sf Pegjr Depreme Memk FMIPA-ITB. C : Usul mklh dkrmk pd Me 5 d dl oleh peer revewer pd ggl 6 Me 5 - Ju 5. Revs peuls dlkuk r ggl 5 Ju 5 hgg 8 Jul 5. Vol. No. Aprl 5 5

2 Iegrs Numers deg Megguk Poloml Lgrge fugs F(x, dm egrs secr lk dk dmugkk mk dp dlkuk lgkh erku. Berdsrk fugs F(x yg dkehu dp du psg d {F, x },,, pd < x <, dm d dlh s-s egrs. Lgkh erkuy dlh medek F(x deg poloml Lgrge megguk d se {F, x }. F(x f(x L (x Sedgk egrs mejd egrs pd Persm (. dp deg mudh dselesk megg L (x ereuk poloml. Pd ssm koord ξ, poloml Lgrge ereuk sedgk egrsy mejd. Pemeuk Koefse Iegrs F Meskpu egrs poloml dp deg mudh dlkuk, peggu Persm (. uuk meghug egrs suu fugs msh kurg prks, kre sep kl hrus memeuk poloml Lgrge d megegrs-ky. Bl segme grs < x <, drsformsk ke ssem koord kurvler (ξ, deg ervl - < ξ <, mk fugs rsforms dlh (. F(x dx f(x dx L (x F dx (. L F(ξ - F( ξ dξ (ξ F(ξ (.5 L (ξ F(ξ, u F( ξ dξ L( ξ dξ F(ξ L( ξ - - dξ F(ξ + L( ξ dξ F(ξ L( ξ dξ F(ξ x(ξ ξ + - d x dξ (. Seljuy fugs F(x, ddek deg poloml Lgrge dlm sysem koord ξ, yu F(x Iegrs F(x mejd I egrs dp deg mudh dselesk d mempuy hrg yg ep, msl C, mk Persm (. mejd dm : Uuk memh keel, dp sj ervl (, dg-g lg dlm sejumlh su ervl, hl megg hw dm < < < < m <, u pejels deg ske Iegrs umers dkerjk pd msg-msg segme, yu - - F(x dx L( ξf + L( ξf m L ( ξ F (.5 x L ( ξ ξ F - - Gmr.. Trsforms ssem koord x ke ssem koord ξ (. F(x - dx L ( ξ F dξ (. - L ( - F(x dx C - F(x dx ξ dξ C F (. L ( ξ dξ F(x dx + F(x dx... + m m F(x dx 6 Jurl Tekk Spl

3 Huhe, dkk. Jumlh k poloml pd suu segme upu su-segme dp dguk,,. k poloml, dm semk yk k poloml, k semk kecl keslhy. Tep pd umumy cukup dguk poloml Lgrge deg - 5 k poloml. Perhug koefse egrs deg, d 5 k poloml k djelsk pd g erku.. Iegrs deg k poloml Pd skem ervl grs - dg dlm ervl, yu seper erlh pd gmr erku x ξ - ξ ξ ξ Deg k poloml u k egrs erseu, poloml Lgrge k ereuk Deg memsukk hrg ξ -, ξ, d ξ, Persm (.6 mejd ( ξ ( ξ L (ξ ( (.7 ( ξ ( ξ ( ξ ( ξ L (ξ ξ (ξ + (.8 ( ξ ( ξ Deg demk dperoleh koefse egrs yu C L( ξ dξ ξ (ξ + dξ - - Cooh ( plks egrs deg k poloml pd fugs susodl Gmr.. Iegrs deg k poloml L (ξ C C - - ( ξ ( ξ -ξ ( ξ ( ξ (ξ - (.6 L( ξ dξ ξ (ξ - dξ -- L( ξ dξ - ( dξ s x dx Peyeles : x F ; x ; x F s x dx Hsl eksk dr s x dx, erlh hsl umers sg dek deg hsl eksk, deg keslh.8 %.8%. Bl (- erllu esr, mk dp dg lg dlm sejumlh su segme, d seljuy egrs dlkuk dlm msgmsg su-segme yu, msl uuk pemg dlm su-segme, egrs mejd - C F +. Iegrs deg k poloml F (C F + C F + C F F(x dx F(x dx + F(x dx - - Pd skem ervl grs - dg dlm ervl, yu ξ Deg k poloml, mk poloml Lgrge ereuk Deg memsukk hrg-hrg ξ s/d ξ seper pd Gmr (. d deg megegrsk dr d ke, mk dperoleh - - -/ / Gmr.. Iegrs deg k poloml L (ξ L (ξ L (ξ L (ξ C F ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ ( ξ -ξ ( ξ -ξ ( ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ ( ξ -ξ (. (. (.c (.d Vol. No. Aprl 5 7

4 Iegrs Numers deg Megguk Poloml Lgrge C C L( ξ dξ.5; C L( ξ dξ L.75 ; C ( ξ dξ L( ξ dξ y Cooh ( egrs fugs susodl s x dx, deg k egrs Π/ Π/ Π/ x F s ; x F s 6.5 x F s -.. Iegrs Bdg ; x F s F(x dx (C F + C F + C F + C F ( Keslh erhdp hsl eksk dlh.5 %.5%. Berdsrk [Burde, Dougls, 85], suu fugs dg F (x, y dp ddek deg perkl poloml Lgrge, yu Gmr.. Poloml Lgrge pd dg seg-emp Deg demk uuk dg seg emp seper fugs pd Gmr (., mk L (x, y L (x L (y ; L (x, y L (x L (y L (x, y L (x L (y ; L (x, y L (x L (y Segm hly deg dom su dmes, mk pd dom du dmes jug k leh mudh l dguk ssem koord kurvler (ε, η. Bl dguk poloml Lgrge deg k poloml, k pd rh ξ mupu η, k ddp poloml Lgrge deg k-k poloml (x, y y (x, y x (-, (, η Pd k (x, y pd Gmr (., F (x, y L (x L j (y F (x, y j (. m k j L, (x, y L (x L (y pooml Lgrge-y dlh x (x, y (x, y (-,- (,- Gmr. Trsforms dr dg (x,y ke dg ξ,η 7 8 ξ y (x, y 5 6 x Gmr.. Poloml Lgrge pd dg Gmr. Tk-k poloml uuk poloml Lgrge k 8 Jurl Tekk Spl

5 Huhe, dkk. seg erku (Gmr. Pd rh horzol, rh ξ, erdp poloml L (ξ d pr rh η jug erdp uh L ( η, L (ξ ξ (ξ - L yu (η η (η - L (ξ ( L (η ( - η L (ξ ξ (ξ + L (η η (η + erdsrk Persm (.7, (.8, d (. mk x x y y x - x y - y Fx dx dy - - L (ξ, η F dξ dη x - x y - y C F (. dm dx x - x y - y dξ d dy dη. Deg demk poloml Lgrge pd k-k poloml seper erlh pd Gmr. dlh L (ξ, η L (ξ L (η ; L (ξ, η L (ξ L (η L (ξ, η L (ξ L (η ; L (ξ, η L (ξ L (η L 5 (ξ, η L (ξ L (η ; L 6 (ξ, η L (ξ L (η L 7 (ξ, η L (ξ C L (ξ, η dξ dη L (η ; L 8 (ξ, η - - L (ξ L (η L (ξ, η L (ξ L (η Koefse C L (ξ, η dξ dη egrs dp - - deg mudh dperoleh deg ;C L (ξ, η dξ dη megegrsk L (ξ, η pd - - Persm (. smp 6 C 5 L 5 (ξ, η dξ dη deg (., yu - - ; C 6 L 6 (ξ, η dξ dη C 7 L 7 (ξ, η dξ dη ; C 8 C ; C - - L (ξ, η dξ dη L 8 (ξ, η dξ dη L (ξ, η dξ dη Deg megguk koefse-koefse egrs erseu mk egrs pd dom dg dp deg mudh dselesk, yu seg erku Deg cr yg sm, dp duruk koefse egrs uuk poloml Lgrge k poloml, pd dg egrs k erdp sehgg k Gmr.5. Bdg egrs deg 6 k poloml (, / (, / (/, / (/, / 7 8 (/, / 5 6 (, (/, (/, (/, / Gmr.6. Koord k poloml Vol. No. Aprl 5

6 Iegrs Numers deg Megguk Poloml Lgrge poloml seper erlh pd Gmr (.5. s x s y dy dx - - L (ξ, η dη dξ C F 6 Seljuy deg m e g e g r s k msg-msg suku pd poloml Lgrge k dperoleh hrg koefse egrs seg erku C.65 ; C.875 ; C.875 ; C.65 C ; C ; C ; C C.875 ; C.565 ; C.565 ; C s x s y dy dx C F C 6.65 ; C.875 ; C ; C 6.65 Cooh ( : egrs fugs susodl s x s y dx Tel.. Perhug cooh ( deg k poloml Tk x y s x s y F C C F. / / / / / / / /.. 8 / / / / C : F s x s y - - dy, deg d 6 k egrs. Perhug deg k poloml F(u dm F s x s y. Deg (5. hsl eksk., mk keslh meod umers dlh %.565%. Seljuy hsl perhug dp dlh pd Tel.. F (u +δ F (u +δ. Perhug deg 6 k egrs Adpu hsl perhug dsjk pd Tel. d wh Deg hsl eksk., mk keslh meod umers dlh. - %.%. 5. Peggu pd Peyeles Persm Dfferesl Wku Orde Noler Persm dfferesl wku oler orde, dp duls deg euk umum seg erku. Tel.. Perhug cooh ( deg 6 k poloml Tk x y s x s y F C C F /6 / / / /6 / / / / / / /6 / / / / /..65 / /6 / / / / /. C : F s x s y Jurl Tekk Spl

7 Huhe, dkk. (5. mejd + δ F( u F(u (5. Peyeles secr explc dlh deg megerjk Persm (5. pd s. Jd u +δ hy deuk oleh kods F (u sj. u - u δ +δ δ F(u Peyeles secr mpls dlh deg megerjk Persm (5. pd s +δ. F (u +δ Dlm hl hrg u +δ (5. deuk oleh F (u +δ. Jd k meod expls mupu meod mpls erdp suu kekurg, dm sehrusy hrg u - u δ +δ u +δ deuk oleh F(u d F (u +δ. Uuk megs kekurg F(u +δ u kelemh dr meod expls d mpls mk dkerjk meod predkor korekor, yu. Predkor u/ τ dp dyk deg skem forwrd mupu cerl dfferece. Deg +δ +δ u old + uew < ε Persm (5. erseu k dperoleh hrg. Bl dselesk deg cerl dfferece, mk Persm (5. mejd. Korekor u + δ pred u + δ pred u + δ kor Deg hrg u +δ (hsl lgkh predkor, dkerjk skem mpls yu + δ du + δ F( u d Bl dselesk deg + δ u +δ + ckwrd dfferece, F( u d mk Persm (5. Deg megguk skem ckwrd dfferece pd u/ τ, mk k dperoleh hrg Seljuy hrg u +δ dlh u u +δ + F ( u u +δ d.5 (5.5 (5.c Deg hrg u +δ yg ru erseu, dulg lgkh korekor, y u deg -δ -δ Gmr 5.. Iegrs deg k poloml megerjk Persm (5. d (5.c lg. Lgkh dulg-ulg smp dperoleh hrg u +δ yg koverge, dm + δ δ δ δ u pred u + δ F + F + F (5.6 hrg δ dperoleh dr - - ( - δ δ dm ε dlh lg yg sg kecl. u +δ u δ pred Meod predkor korekor, dp jug dkerjk deg cr egrs yu seg erku. Persm (5. dp jug duls dlm euk l yu : F -δ -δ +δ (u d Gmr 5.. Iegrs deg k poloml (5. Persm (5. dp dselesk deg meod egrs yu u +δ u -δ +δ (.5 F -δ +.75 F -δ +.75 F +.5 F +δ (5.7 hrg δ dperoleh dr δ +δ - - δ ( -δ Vol. No. Aprl 5

8 Iegrs Numers deg Megguk Poloml Lgrge Pd Persm (5. erseu d dk dp dselesk kre +δ +δ u hrg u +δ old + uew < ε. dk dkehu. Uuk megs hl erseu, mk dkerjk meod predkor-korekor, seg erku :. Predkor η + ( uh Seg lgkh predkor dlh (5.8 Iegrs (5.5 dp η dselesk + u + g d e g (5.8 egrs umers deg poloml Lgrge k, yu erdsrk Gmr 5., dm δ. Deg dkehuy u -δ, u -δ d u dr perhug seelumy, mk dp dhug F -δ F (u -δ, F -δ F (u -δ d F F (u, d d η Persm (5.5 dp - ( uh u dselesk secr umers deg koefse egrs seper d η F (u, H yg deuk pd (5. g ( uh dm F (u, H - erdhulu.. Korekor Deg dkehuy hrg G (u, η mk dp dhug F +δ (5. F(u + δ d η Persm dm G (u, η - u - g (5.5 dp dselesk secr u m e r s z x η Gmr 5.. Vrel pd persm gelomg Ary h Muk r dm deg skem k. + δ δ δ η +δ η -δ + δ F koefse F (u, H d + δ + δ δ D e g megguk egrs dr erdhulu, mk peyeles umers dr Persm (5.6 dlh ( uh seg erku Deg yg ru, δ η G -δ - u - g ; G -δ η δ - u - g ; G η - u - g dulg perhug korekor Persm 5.7, smp dcp koverges dm: u +δ Seg cooh pemk k dselesk persm gelomg Ary su dmes, yu [De, Dlrymple] :. Persm momeum F + F δ δ ( uh ( uh F -δ - ; F -δ - + δ δ G (u, η d + δ δ η +δ η -δ + F +δ - δ ( uh G F + δ F. Persm kous δ u +δ u -δ + δ.5g +.75G +.75G +.5G δ η dm G +δ - u - g η flukus muk r u kecep rus pd rh x h kedlm perr + δ ; F - ( uh δ δ + (,5 F +.75 F +.75 F +,5 F δ +δ δ + δ ( Jurl Tekk Spl

9 Huhe, dkk. g percep grvs H h + η Persm kous dp duls Sedgk persm momeum dp duls dlm euk. Predkor.. Persm kous dm (u, H -δ, (u, H -δ, d (u, H, dperoleh dr perhug seelumy. Sedgk pd peel dhug deg megguk meod selsh hgg... Persm momeum Gmr 5.. Kl uuk eksekus model dm u -δ, η -δ, u -δ, η -δ, u d η dperoleh dr hsl perhug seelumy. Sedgk u/ x d η/ x pd peel dhug deg meod selsh hgg.. Korekor.. Persm kous dm u -δ d η -δ dperoleh dr perhug hp predkor... Persm * f ξ dξ w f( ξ momeum ( - sedgk u +δ d η +δ dperoleh dr perhug pd hp predkor. Seg cooh perhug, model dkerjk pd perhug gelomg pd kl (Gmr 5.. Perod gelomg 6 dek, mpludo. m, sedgk kedlm kl h 5. m. Berdsrk kj Le Mehue [Me, Méhué, 66], persm gelomg Ary uuk gelomg pedek hy dp dguk uuk gg gelomg yg sg kecl. Pd perhug F (u, H - (uh/ x d G (u, η - u u/ x - g η/ x dhug deg megguk meod selsh hgg. Hsl perhug dsjk pd Gmr 5.. Seper erlh pd Gmr 5., model memerk solus yg sl, hl erlh pd kurv gelomg yg sl pd eksekus x perod gelomg, serg erjd kedk sl kurv erjd seelh eksekus u kl perod gelomg. Hl Muk r (m,5,,,8,5,, -, -,5 -,8 -, -, -, x (m Gmr 5.. Gelomg susodl pd kl, hsl smuls muk r (m,5,,,8,5,, -, -,5 -,8 -, -, -, Hsl model x (m Hsl eor gelomg ler Gmr 5.5. Perdg r eor gelomg ler deg model Vol. No. Aprl 5

10 Iegrs Numers deg Megguk Poloml Lgrge * ξ dp erjd k meod umers (peyeles dferesl wku yg kurg k. Sel u perhug gelomg pedek deg persm gelomg Ary dlh juh leh sul drpd perhug gelomg pjg, δ u +δ u + + δ + δ ( G 6 G + 5 G δ η +δ η + + δ + δ F 6 F + 5 F m s l perod * ξ * ξ Gmr 6.. Poss k egrs erhdp k wku (φ perhug δ ( gelomg psg suru deg jm u 86 dek. u +δ u + + δ + δ + δ ( G + G - 5 G + G Solus δ η +δ η + + δ + δ + δ ( F + F - 5 F + F lk erdsrk eor gelomg ler dp duls dlm euk susodl : η (x, A s (kx σ dm : A + δ + δ mpludo u +δ - u d f u d ( gelomg ; k f ( u d lg gelomg σ / T frekues sudu ; T perod gelomg Uuk gelomg deg perod 6 dek, d pd kedlm perr 5. m, mempuy pjg gelomg 5 m erdsrk eor geomg ler. Terlh pd Gmr 5.5., hw gelomg yg dhslk model leh pjg dr eor gelomg ler. Berdsrk [Huhe, dkk., 5], pjg gelomg dr eor gelomg ler memg msh perlu dlkuk koreks. 6. Dskus +δ Seper elh dseuk hw meod yg dkemgk dlh serup deg meod Guss-qudrure yg ereuk [Burde, Dougls, 85] dm w dseu deg koefse oo, sedgk ξ * dlh k egrs. Berdsrk [Frk, 86], k egrs pd meod Guss-qudrure dlh seper dsjk pd Tel 6.. Seper erlh pd Tel 6.. erseu, koord k egrs uk erup lg yg sederh, demk jug deg ervly. Sehgg uuk pegguy pd egrs wku sul uuk megkky Tel 6.. Tk egrs pd meod Guss-qudrure * ( ξ dξ w f( ξ f - * ξ w Jurl Tekk Spl

11 Huhe, dkk. deg poss,,.. ds. seper erlh pd Gmr 6.., mk erlh hw k egrs dk ep erd pd k wku perhug. Oleh kre u l dguk meod Gussqudrure, perlu dlkuk erpols uuk meghug hrg-hrg f (ξ *,f (ξ *,f (ξ *.. ds. Hl sel memh proses perhug jug k megurg keel. Meod predkor-korekor yg dkemgk mrp deg meod predkorkorekor dr Adms Bshforh Moulo yg dguk oleh Y.S L, dkk [L, Lu, Yu, L, ] ser Nwogu [Nwogu, ] dlm meyelesk persm Boussesq. Adpu euk persm dr Adms Bshforh Moulo erseu dlh [Burde, Dougls, 85], [L, Lu, Yu, L, ], d [Nwogu, ]. Predkor : Korekor : Pered deg meod yg dkemgk dlh pd hrg koefse egrs. Meod Adms Bshforh Moulo jug duruk erdsrk meod egrs, yu Seljuy peyeles dselesk meod ckwrd-dfferece dr Newo, d dperoleh koefse-koefse egrs. Uuk pejels leh rc, dp dlh pd ref [Burde, Dougls, 85]. Mege meod yg m yg leh k, dlur lgkup peel. Mksud dr pper hylh megemukk lerf meod peyeles persm dferesl wku oler. 7. Kesmpul Dr peel yg elh dlkuk, dp dml kesmpul hw. Meod egrs umers yg dkemgk memerk hsl yg cukup k, sg dek deg hsl eksk, deg keslh.% -.5%.. Semk yk k poloml u k egrs yg dguk dperoleh hsl egrs yg semk k. c. Persm dfferesl wku dp dselesk deg meod egrs, dm egrs wku dp dlkuk secr umers. Vol. No. Aprl 5 5

12 Iegrs Numers deg Megguk Poloml Lgrge 6 Jurl Tekk Spl