BAB I. PENGANTAR METODE NUMERIK

Metode Numerk utuk Tekk Mes BAB I PENGANTAR METODE NUMERIK Metode umerk merupk metode pemroses dr dt umerk (dskret) mejd hsl umerk, dm metode mmpu meg sstem persm besr, ketdkler d ksus deg geometr yg komplek yg bs djump d ksus rekys d sergkl sult utuk dselesk deg cr lts Als umerk dpt drtk sebg ls memperguk lgortm dr metode umerk Als umerk memuculk du ss yg merk ytu dpt mejd : IPTEK (scece) : Bg dr mtemtk dm lgortm yg dpk dkembgk dr persm-persm mtemtk tertetu Se (rt) : Berkt deg peetu cr terbk utuk meyelesk sutu persol mtemtk Peyeles persol mtemtk dpt dselesk deg cr lts (eksk), grfs d umerk Metode lts mempuy keuggul dlm hsly yg eksk, tetp bsy terbts utuk kemudh peyeles pd mslh yg deg sums ler d pd ksus geometr yg sederh Metode grfs bertuju utuk meggmbrk perlku system dlm betuk gmbr tu omogrf deg keterbts hy mmpu mksml megurk mslh deg megguk gmbr tg dmes Jes persol Mtemtk yg k dselesk deg cr umerk dpt dgologk sebg berkut: Akr-kr persm Persm ljbr ler seretk 3 Iterpols 4 Pecocok kurv (curve fttg) 5 Persm dfferesl bs 6 Persm dfferesl prsl 7 Itegrs Numerk Peyeles sutu persol mtemtk deg metode umerk umumy dpt dselesk deg lebh dr stu metode sehgg hrus dplh metode terbk yg dpt meghslk peyeles yg efse d efektf sert tdk meghslk error yg besr Cr memlh metode terbk : Megethu jes-jes metode yg d Megethu bgm metode-metode tersebut bekerj

Metode Numerk utuk Tekk Mes 3 Megethu kelemh d kelebh metode-metode 4 Mempuy fktor tus d peglm dlm meerpk metode-metode d ts Deg mempeljr metode umerk k memberk sr lgsug dlm beljr pemrogrm komputer Deg perkembg hrdwre d softwre st sgt medukug ketrmpl dlm memftk komputer sebg lt btu dlm meyelesk ksus rekys d bdg tekk mes Deg memhm d terbs deg metode umerk k memberk kemmpu lebh utuk mercg progrm sedr tp hrus membel progrm pket yg mhl Beberp bhs progrm yg umum dguk dlh FORTRAN, PASCAL, DELPHI, VISUAL BASIC, C++ d byk bhs progrm ly Cr-cr pemrogrm terstruktur hrus mempuy krter ytu : Ber Mudh Dfhm Mudh Dmodfks Slh stu yg mejd kelemh metode umerk dlh muculy glt tu error dkrek metode melbtk sutu pedekt/proksms hsl dr metode lts Glt yg umum djump melput : Glt Stkss Melggr kdh bhs pemrogrm Glt wktu rug Terjd selm eksekus progrm 3 Glt logk Keslh logk progrm 4 Glt pembult Komputer hy mmpu memperthk sejumlh gk tetp gk ber selm perhtug Glt pembult merupk glt yg plg serg djump terutm dlm perhtug metode umerk secr mul utuk lth, quz mupu uj kulh Cotoh pd blg-blg sepert d 5 tdk dpt dekspresk sebg sejumlh tetp gk ber Utuk tu yg hrus dperhtk dlm lth metode umerk dlh peggu yg kosste jumlh gk d belkg kom selm perhtug Kotrol kults progrm umerk meckup pekerj dlm : DEBUGGING Perbk glt yg dkethu PENGUJIAN Medeteks glt yg tdk dkethu

Metode Numerk utuk Tekk Mes Berkut dsjk flowchrt thp-thp dlm mercg d megembgk progrm yg berbss metode umerk RANCANG BANGUN ALGORITMA Pegembg yg medsr logk progrm KOMPOSISI PROGRAM Peuls progrm dlm bhs komputer PENCARIAN DAN PENGUJIAN Pemst bhw progrm bebs Glt d dl DOKUMENTASI Membut progrm mudh dguk d dphm PENYIMPANAN DAN PERAWATAN Meymp progrm d memperbky sesu peglm d kebutuh psr 3

Metode Numerk utuk Tekk Mes BAB II AKAR-AKAR PERSAMAAN Peyeles ksus kr-kr persm dpt dgologk mejd du metode ytu : Metode pegurug (Brcketg Methods) Mul deg terk wl yg megurug tu memut kr d kemud secr bersstem megurg lebr kurug Cotoh : Bsecto, Regul Fls Metode terbuk (Ope Methods) Iters cob-cob yg sstemts Cotoh : Newto Rphso, Sect Metode Bg Du (Bsecto Methods) Perumus mecr kr : md = Evlus : f ( md ) = f ( md ) y y = f() f( ) f( ) f( md ) md Msl : Tetuk l ol dr sutu fugs y = 3-7 + Pertm tetuk l wl utuk d sehgg ddptk f ( ) d f ( ) yg berbed td, yg berrt ttk peyeles d d sektr tu But tbel utuk mempermudh pembc prosesy No f( ) f( ) md f( md ),5,6 -,875,376,55 -,69 f( ) d f( md ) sm td = md,55,6 -,69,376,575,49 4

Metode Numerk utuk Tekk Mes f( ) d f( md ) sm td = md 3,55,575 -,69,49,56 -,7 f( ) d f( md ) sm td = md 4,56,575 -,7,49,568 -,4 f( ) d f( md ) sm td = md 5,568,575 -,4,49,57, f( ) d f( md ) sm td = md 6,568,57 -,4,,57 -,5 f( ) d f( md ) sm td = md 7,57,57 -,4,,57 -,3 Sehgg slh stu kr yg dcr dlh,57 Algortm Metode Bseks START Cr poss kr f( ) d f( + ) bed td Htug md =, f ( md ) f(), f(md) sm td? Tdk + = md f ( + ) = f ( md ) Y = md f ( ) = f ( md ) Tdk f (md) STOP Y 5

Metode Numerk utuk Tekk Mes Metode Poss Plsu (Flse Posto Methods) Berdsr terpols ler tr hrg f() yg mempuy td berbed f( ) f( ) < Koverges yg dhslk cept y y = f() f( ) f( ) 3 4 f( ) d f( ) berbed td berrt d kr tr d Perumus mecr kr : * = f( ) f( ) f( ) Evlus sutu kr : f( * ) Algortm Metode Poss Plsu = Algortm Metode Bseks hy tggl meggt rumus md = mejd * = f( ) f ( ) f ( ) No f( ) f( ) * f(*),5,6 -,875,376,57 -,5 f( ) d f( md ) sm td = md,57,6 -,5,376,57,3 Sehgg slh stu kr yg dcr dlh,57 Terlht deg metode hy dbutuhk ters sehgg koverges lebh cept dbdgk deg metode bseks 6

Metode Numerk utuk Tekk Mes 3 Metode Newto Tdk perlu mecr hrg f() yg mempuy td berbed Koverges yg dhslk cept Perlu meghtug turu fugs f () Kelemh : - tdk sellu meemuk kr (dverge) - kemugk mecr f () sukr - Peetp hrg wl sult Dr deret Tylor : f( + h) = f( ) + h f ( ) + Jk + h dlh kr f( + h) = f () = f( ) + h f ( ) h = - f' ( ) Perumus mecr kr : h f () + dbk + = + h = - f () f' ( ) Algortm Metode Newto START Plh X yg cocok Cr +, f (+) f () + = - f' ( ) f ( + ) Tdk = + STOP Y 7

Metode Numerk utuk Tekk Mes 4 Metode Sect Tdk perlu mecr fugs deg td berbed Kombs Metode Newto d Metode Poss Plsu Tp mecr turu fugs f () y y = f() 3 B C A E D d dplh = + Segtg ABC segtg DEA - f() f() - f( = ) = - f( ) - mk : = - f( ) f( ) f( ) f( ) f( ) Perumus : + = f( ) f( ) f( ) Algortm Metode Sect = Algortm Metode Newto, hy tggl meggt rumusy Peggt l dlkuk meurut urut yg kett, deg l bru + meggtk d l meggtk - Sehgg kdg du l tersebut dpt pd poss yg sm kemugk dverge SUMMARY Metode pegurug Metode terbuk JENIS KELEBIHAN KEKURANGAN Bsecto - Sellu -Lju koverge Regul Fls Koverge lmbt Newto-Rphso Sect -Lju koverge cept - Cukup stu terk wl - Turu hrus dcr secr lts - Bs dverge 8

Metode Numerk utuk Tekk Mes TUGAS : Selesk deg cr mul d But progrm komputer deg megguk metode d ts d Uj hsl progrm deg meyelesk fugs sebg berkut : y = 4 + 3 3 + + 5 Berkut lstg progrm deg megguk metode poss plsu progrm poss_plsu; uses crt; vr j,k,l,m,,mt,,,b,,,gm : rel; fucto f(,b,c,d,e, :rel):rel; beg f:=*sqr(sqr())+b**sqr()+c*sqr()+d*+e; ed; {prosedur pegs dt} procedure dt; beg clrscr; wrtel('meghtug kr persm '); wrtel('f()=a ^4 + B ^3 + C ^ + D + E wrtel('deg metode Poss Plsu '); wrtel; wrtel; wrte('msuk l A = '); redl(j); wrte('msuk l B = '); redl(k); wrte('msuk l C = '); redl(l); wrte('msuk l D = '); redl(m); wrte('msuk l E = '); redl(); wrtel; wrte('bts Error = '); redl(gm); wrte('jumlh Iters Mks = '); redl(mt); wrte('nl bwh = '); redl(b); wrte('nl ts clrscr; ed; = '); redl(); '); {prosedur perhtug poss plsu} procedure posplsu; vr ters : teger; glt,uj : rel; :rel; beg wrtel; 9

Metode Numerk utuk Tekk Mes wrtel(' =========================================='); wrte(' Iters ke-');wrte(' '); wrte('hsl');wrtel; wrte(' =========================================='); :=b; :=; ters:=; :=-((f(j,k,l,m,,)*(-))/(f(j,k,l,m,,)-f(j,k,l,m,,))); repet ters:=ters+; uj:=f(j,k,l,m,,)*f(j,k,l,m,,); f uj= the := else f uj < the beg :=b; :=; :=-((f(j,k,l,m,,)*(-))/(f(j,k,l,m,,)- f(j,k,l,m,,))); wrtel;wrte(' '); wrte(ters); wrte(' ',:3:5); ed else f uj> the beg :=; :=; :=-((f(j,k,l,m,,)*(-))/(f(j,k,l,m,,)- f(j,k,l,m,,))); wrtel;wrte(' '); wrte(ters);wrte(' ',:3:5); ed; utl (bs(f(j,k,l,m,,))<=gm) or (ters=mt); wrtel; wrtel(' =========================================='); wrtel; wrtel('persm : ',j::,'x^4 + (',k::,')x^3 + (',l::,')x^ + (',m::,')x + (',::,')'); wrtel('jumlh Iters = ',ters,' Bts Error = ',gm:3:5); wrtel('bts Bwh = ',b:3:,' Bts Ats = ',:3:);wrtel; wrte('slh stu kry dlh = ',:3:5); ed; {prosedur / progrm utm} beg dt; posplsu; redl; ed

Metode Numerk utuk Tekk Mes HASIL RUNNING PROGRAM Meghtug kr persm f()=a ^4 + B ^3 + C ^ + D + E deg metode Poss Plsu Msuk l A = Msuk l B = 3 Msuk l C = Msuk l D = 5 Msuk l E = Bts Error = Jumlh Iters Mks = Nl bwh = -35 Nl ts = -5 ========================================= Iters ke- Hsl ========================================== -34464-6648 3-7883 4-8557 5-8837 6-8957 7-978 8-98 9-936 -9395 ========================================== Persm : ()X^4 + (3)X^3 + ()X^ + (5)X + () Jumlh Iters = Bts Error = Bts Bwh = -35 Bts Ats = -5 Slh stu kry dlh = -9395

Metode Numerk utuk Tekk Mes BAB III PERSAMAAN ALJABAR LINEAR SERENTAK Betuk umum persm ljbr ler seretk : + + + = b + + + = b + + + = b dm dlh koefse-koefse kostt, b dlh kosttkostt d dlh byky persm Peyeles persm ler seretk dpt dlkuk cr : Elms Elms Guss, Guss Jord Iters Iters Jcob, Guss sedel 3 Dekomposs Dekomposs lower-upper (LU), Cholesky 3 Elms Guss Elms blg ukow deg meggbugk persmpersm Strteg : meglk persm deg kostt gr slh stu blg ukow k terelms blm du persm dgbugk Kebutuh : pemhm Opers Mtrk b b b Skem lgkh elms Guss 3 3 3 3 33 3 Forwrd Elmto b b b 3 (E ) (E ) (E 3 ) ' ' 3 3 '' 33 3 b b' b'' 3 Upper Trgulr System Bck Substtuto 3 = b 3 / 33 = (b - 3 3) / = (b - - 3 3) /

Metode Numerk utuk Tekk Mes Lgkh elms mju : Elmsk dr (E ) d (E 3 ), sums m = ; m 3 = ' ' 3 kurgk (m (E )) pd (E ) d kurgk (m 3 (E )) pd (E 3 ), sehgg : + + 3 3 = b + 3 3 = b 3 + 33 3 = b 3 NB : td petk stu berrt persm telh dmodfks stu kl Elmsk dr (E 3 ), sums m 3 = ' ' 3 kurgk (m 3 (E )) pd (E 3 ), sehgg : + + 3 3 = b + 3 3 = b NB : 33 3 = b 3 td petk du berrt persm telh dmodfks du kl Lgkh substtus mudur : 3 = b 3 / 33 Sehgg dpt drumusk : = b ( - ) ( - ) Utuk meghtug ssy : = (b - 3 3 ) / = (b - - 3 3 ) / Sehgg dpt drumusk : = b (-) j (-) (-) j NB : deg =,,, Persm (E ) dsebut Pvot Equto, dsebut koefse Pvot d opers perkl brs pertm deg / dsebut sebg Normlss 3

Metode Numerk utuk Tekk Mes Utuk kemudh dpt dpk mtrk dlm betuk kombs yg dsebut deg Augmeted Mtr (mtrk yg dperbesr) b 3 b 3 b 3 3 33 3 Mslh hrus meghdr pembg deg ol, sehgg mucul sebut utuk metode ytu Elms Guss Nf Tekk utuk memperbk peyeles Elms Guss : Pvotg Sebelum tp brs dormlk, mk dlkuk peetu koefse terbesr yg tersed Kemud brs-brs tersebut dpertukrk sehgg eleme terbesr tersebut merupk eleme pvot Sclg bergu dlm pemml glt pembult utuk ksus dm beberp persm mempuy koefse-koefse yg juh lebh besr dr ly Cotoh sol : Selesk persm smult berkut 7 + 6 3 = 85 () 6 + 5 + 3 = 7 (b) + + 54 3 = (c) Peyeles : Guk mtrk dlm betuk Augmeted Mtr (mtrk yg dperbesr) 7 6 6 5-54 85 7 E - 6/7 E E 3 - /7 E 7 6 3,667,778 -, 54,37 85 53, 6,85 E 3,778/3,667 E 7 6 3,667 -, 53,9 85 53, 3,89 deg megguk substtus mudur k dperoleh,, d 3 3 = 3,89 / 53,9 =,96 4

Metode Numerk utuk Tekk Mes 3,667 +, 3 = 53, = 3,573 7 + 6-3 = 85 =,45 3 Elms Guss-Jord Merupk Vrs dr Elms Guss deg kebutuh utuk meghtug mtrk vers Strteg : Lgkh elms meghslk mtrk stu, sehgg tdk dperluk proses substtus mudur Skem lgkh elms Guss-Jord b 3 b 3 b 3 3 33 3 (E ) (E ) (E 3 ) Elmto 3 b * b * b * NO Bck Substtuto = b* = b* 3 = b* 3 Selesk sol yg sm pd metode Elms Guss : 7 6 6 5-54 85 7 /7 E 6, 5 -,337 54 3,48 7 E 6 E E 3 E, 3,667,778 -,337, 54,37 3,48 53, 6,85 /3,667 E,,778 -,337,63 54,37 3,48 3,886 6,85 5

Metode Numerk utuk Tekk Mes E, E E 3,778 E -,73,63 53,9,85 3,886 3,88 /53,9 E 3 -,73,63,85 3,886,96 E (-,73 E 3 ),46 =,46 E,63 E 3 3,57 = 3,57,96 3 =,96 33 Iters Guss-Sedel Betuk umum persm ler seretk : + + 3 3 + + = b + + 3 3 + + = b + + 3 3 + + = b Dpt dubh betuky mejd : = ( b - + 3 3 - - ) = 3 = = ( b - + 3 3 - - ) 33 ( b3-3 + 3 - - 3 ) ( b - - - - (-) (-) ) Lgkh-lgkh Iters Guss-Sedel Asumsk = 3 = = =, sehgg dpt dperoleh : b = Hsl dr tersebut dmsukk persm utuk medptk hrg (dm 3 = = = ), mk k dperoleh : = ( b - ) 3 Lgkh d dlkuk terus smp dperoleh l d seleslh proses ters yg pertm Kemud hsl proses tersebut dmsukk kembl pd persm utuk medptk hrg ukow dr,, 3 pd proses ters kedu, ketg d seterusy 6

Metode Numerk utuk Tekk Mes 4 Proses ters berkhr bl hsl dr ters terkhr sm deg tu hmpr sm deg ters sebelumy I merupk kelemh metode ters guss-sedel ytu proses khr ters mejd merguk Cotoh sol : Selesk persm smult berkut : 7 + 6 y z = 85 () 6 + 5 y + z = 7 (b) + y + 54 z = (c) Peyeles : Persm d ts dpt dubh betuky mejd : = 7 ( 85-6 y + z ) () y = 5 ( 7-6 - z ) () z = 54 ( - - y ) () Iters pertm Asumsk y = z =, sehgg dr persm () k dperoleh 85 : = = 3,5 7 Hsl dr tersebut dmsukk persm (b) utuk medptk hrg y (sums z = ) y = 5 ( 7-6 (3,5) ) = 3,54 3 Msukk hsl d y ke dlm persm (c) z = ( 3,5 3,54) =,9 54 Iters kedu = ( 85-6 (3,54) +,9 ) =,43 7 y = 5 ( 7-6 (,43) (,9) ) = 3,57 z = (,43 3,57) =,96 54 Iters seljuty dpt dtbelk sebg berkut : Iters ke - y z 3,5 3,54,9,43 3,57,96 3,43 3,574,96 4,45 3,573,96 5,45 3,573,96 Jd hsl peyelesy dlh : =,45 ; y=3,573 ; z =,96 7

Metode Numerk utuk Tekk Mes 34 Iters Jcob Mellu proses ters deg megguk persm : (+) b j = - j (); j j Keutug metode dlh lgkh peyelesy yg sederh, sedgk kelemhy dlh : Proses tersy lmbt Terutm utuk persm ler seretk deg ordo tgg Hy dpt dguk meyelesk persm ler seretk yg memeuh syrt berkut : > j j ; j d =,,, N Cotoh sol : Selesk persm smult berkut : 7 + 6 y z = 85 () 6 + 5 y + z = 7 (b) + y + 54 z = (c) Peyeles : Persm d ts dibetuk mejd : () = ( 85-6 y () + z () ) () 7 y () = 5 ( 7-6 () - z () ) (b) z () = ( - () - y () ) (c) 54 Iters pertm Asumsk () = y () = z () =, sehgg dr persm (, b d c) k dperoleh : () 85 = = 3,48 7 y () 7 = = 4,8 5 z () = =,37 54 Iters kedu () = ( 85-6 (4,8) +,37 ) =,57 7 y () = 5 ( 7-6 (3,48) (,37) ) = 3,69 z () = ( 3,48 4,8) =,89 54 8

Metode Numerk utuk Tekk Mes Iters seljuty dpt dtbelk sebg berkut : Iters ke - X Y z 3,48 4,8,37,57 3,69,89 3,49 3,685,937 4,4 3,545,93 5,43 3,583,97 6,43 3,57,96 7,46 3,574,96 8,45 3,573,96 9,46 3,573,96,45 3,573,96,45 3,573,96 Jd hsl peyelesy dlh : =,45 ; y = 3,573 d z =,96 35 Dekomposs LU Deg cr membetuk mtrk segtg ts (Upper) d mtrk segtg bwh (Lower) dr mtrk koefse A sert membetuk vektor mtrk dr mtrk hsl deg tur tertetu Kelebhy dlh sgt efektf utuk meyelesk persm ler seretk ordo tgg, deg hsl yg sgt medekt l eksky Tetu sj kosekuesy metode memerluk cr yg cukup kompleks Dekomposs [A] {X} = {B} [U] [L] [L] {D} = {B} {D} [U] {X} = {D} {X} Mju Mudur Pesubttus 9

Metode Numerk utuk Tekk Mes Lgkh-lgkh Dekomposs LU Membetuk mtrk koefse [A], mtrk vrbel {X} d mtrk hsl {B} dr persm smult [A] {X} = {B} Mecr mtrk segtg bwh [L] d mtrk segtg ts [U] dr mtrk koefse [A] deg tur berkut : l = ; =,,, j j u j = = ; j =,3,, l - utuk j =,3,, - j l j = j - k jk l ; = j, j+,, k u kj j l j uk u jk = ; k =j+, j+,, ; l = - l l jj k 3 Mecr mtrk {B } deg tur berkut : b = b l ; b = b lj j l b' j utuk =, 3,, k u k 4 Membetuk Augmeted Mtr {UB } d peyelesy dperoleh : = b d j = b j - u jk k k j Berkut dsjk cotoh lstg progrm VISUAL BASIC utuk meghtug persm ler seretk ksus d ts deg metode Iters Jcob LISTING PROGRAM Beg VBForm Form Cpto = "MENGHITUNG INTERASI JACOBI" CletHeght = 63 CletLeft = 6 CletTop = 345 CletWdth = 95 LkTopc = "Form" ScleHeght = 63 ScleWdth = 95 StrtUpPosto = 'CeterScree Beg VBCommdButto Commd3 Cpto = "Msukk Vrbel" BegProperty Fot

Metode Numerk utuk Tekk Mes Nme = "Ndll" Sze = Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 495 Left = 384 TbIde = 38 Top = 54 Wdth = 5 Ed Beg VBCommdButto Commd Cpto = "Htug Iters" BegProperty Fot Nme = "Ndll" Sze = 5 Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 495 Left = 7 TbIde = 3 Top = 3 Wdth = 695 Ed Beg VBFrme Frme Cpto = "Msukk Agk" BegProperty Fot Nme = "Plto Lotype" Sze = Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 45 Left = 7 TbIde = Top = 4 Wdth = 6495 Beg VBTetBo Tet4

Metode Numerk utuk Tekk Mes Heght = 375 Ide = Left = 56 TbIde = Top = 8 Wdth = 975 Ed Beg VBTetBo Tet3 Heght = 375 Ide = Left = 348 TbIde = Top = 8 Wdth = 495 Ed Beg VBTetBo Tet Heght = 375 Ide = Left = 9 TbIde = Top = 8 Wdth = 495 Ed Beg VBTetBo Tet Heght = 375 Ide = Left = 36 TbIde = 9 Top = 8 Wdth = 495 Ed Beg VBTetBo Tet4 Heght = 375 Ide = Left = 56 TbIde = 8 Top = 8 Wdth = 975 Ed Beg VBTetBo Tet3 Heght = 375 Ide = Left = 348 TbIde = 7 Top = 8 Wdth = 495 Ed Beg VBTetBo Tet

Metode Numerk utuk Tekk Mes Heght = 375 Ide = Left = 9 TbIde = 6 Top = 8 Wdth = 495 Ed Beg VBTetBo Tet Heght = 375 Ide = Left = 36 TbIde = 5 Top = 8 Wdth = 495 Ed Beg VBTetBo Tet4 Heght = 375 Ide = Left = 56 TbIde = 4 Top = 36 Wdth = 975 Ed Beg VBTetBo Tet3 Heght = 375 Ide = Left = 348 TbIde = 3 Top = 36 Wdth = 495 Ed Beg VBTetBo Tet Heght = 375 Ide = Left = 9 TbIde = Top = 36 Wdth = 495 Ed Beg VBTetBo Tet Heght = 375 Ide = Left = 36 TbIde = Top = 36 Wdth = 495 Ed Beg VBLbel Lbel6 3

Metode Numerk utuk Tekk Mes Cpto = "+" Heght = 375 Ide = Left = 3 TbIde = 3 Top = 9 Wdth = 375 Ed Beg VBLbel Lbel5 Cpto = "+" Heght = 375 Ide = Left = 44 TbIde = 9 Top = 9 Wdth = 375 Ed Beg VBLbel Lbel3 Cpto = "Z" Heght = 375 Ide = Left = 48 TbIde = 7 Top = 9 Wdth = 65 Ed Beg VBLbel Lbel Cpto = "Y" Heght = 375 Ide = Left = 64 TbIde = 6 Top = 9 Wdth = 495 Ed Beg VBLbel Lbel Cpto = "X" Heght = 375 Ide = Left = 8 TbIde = 5 Top = 9 Wdth = 495 Ed Beg VBLbel Lbel6 Cpto = "+" Heght = 375 Ide = 4

Metode Numerk utuk Tekk Mes Left = 3 TbIde = 4 Top = Wdth = 375 Ed Beg VBLbel Lbel5 Cpto = "+" Heght = 375 Ide = Left = 44 TbIde = 3 Top = Wdth = 375 Ed Beg VBLbel Lbel4 Cpto = "=" Heght = 375 Ide = Left = 468 TbIde = Top = Wdth = 495 Ed Beg VBLbel Lbel3 Cpto = "Z" Heght = 375 Ide = Left = 48 TbIde = Top = Wdth = 65 Ed Beg VBLbel Lbel Cpto = "Y" Heght = 375 Ide = Left = 64 TbIde = Top = Wdth = 495 Ed Beg VBLbel Lbel Cpto = "X" Heght = 375 Ide = Left = 8 TbIde = 9 Top = 5

Metode Numerk utuk Tekk Mes Wdth = 495 Ed Beg VBLbel Lbel6 Cpto = "+" Heght = 375 Ide = Left = 3 TbIde = 8 Top = 48 Wdth = 375 Ed Beg VBLbel Lbel5 Cpto = "+" Heght = 375 Ide = Left = 44 TbIde = 7 Top = 48 Wdth = 375 Ed Beg VBLbel Lbel4 Cpto = "=" Heght = 375 Ide = Left = 468 TbIde = 6 Top = 48 Wdth = 495 Ed Beg VBLbel Lbel3 Cpto = "Z" Heght = 375 Ide = Left = 48 TbIde = 5 Top = 48 Wdth = 65 Ed Beg VBLbel Lbel Cpto = "Y" Heght = 375 Ide = Left = 64 TbIde = 4 Top = 48 Wdth = 495 Ed Beg VBLbel Lbel 6

Metode Numerk utuk Tekk Mes Cpto = "X" Heght = 375 Ide = Left = 8 TbIde = 3 Top = 48 Wdth = 495 Ed Beg VBLbel Lbel4 Cpto = "=" Heght = 375 Ide = Left = 468 TbIde = 8 Top = 9 Wdth = 495 Ed Ed Beg VBFrme Frme Cpto = "Vew Persm d Hsl Iters" BegProperty Fot Nme = "Plto Lotype" Sze = Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 5775 Left = TbIde = 3 Top = Wdth = 955 Beg VBCommdButto Commd Cpto = "CLer" BegProperty Fot Nme = "Ndll" Sze = Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 375 Left = 78 TbIde = 37 7

Metode Numerk utuk Tekk Mes Top = 58 Wdth = 85 Ed Beg VBLstBo Lst Heght = 5 Left = 4 TbIde = 33 Top = 3 Wdth = 8655 Ed Beg VBLbel Lbel9 Algmet = 'Ceter BorderStyle = 'Fed Sgle BegProperty Fot Nme = "Arl" Sze = 45 Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 735 Left = 4 TbIde = 36 Top = 6 Wdth = 8655 Ed Beg VBLbel Lbel8 Algmet = 'Ceter BorderStyle = 'Fed Sgle BegProperty Fot Nme = "Arl" Sze = 45 Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 735 Left = 4 TbIde = 35 Top = 3 Wdth = 8655 Ed Beg VBLbel Lbel7 Algmet = 'Ceter 8

Metode Numerk utuk Tekk Mes BorderStyle = 'Fed Sgle BegProperty Fot Nme = "Arl" Sze = 45 Chrset = Weght = 7 Uderle = 'Flse Itlc = 'Flse Strkethrough = 'Flse EdProperty Heght = 735 Left = 4 TbIde = 34 Top = 48 Wdth = 8655 Ed Ed Ed Attrbute VB_Nme = "Form" Attrbute VB_GloblNmeSpce = Flse Attrbute VB_Cretble = Flse Attrbute VB_PredeclredId = True Attrbute VB_Eposed = Flse Dm () As Sgle Dm y() As Sgle Dm z() As Sgle Dm () As Sgle Dm b() As Sgle Dm c() As Sgle Dm d() As Sgle Prvte Sub Commd_Clck() O Error Resume Net FrmeVsble = True FrmeVsble = Flse CommdVsble = Flse Lbel7Cpto = Tet()Tet & "X" & "+" & Tet()Tet & "Y" & "+" & Tet3()Tet & "Z" & "=" & Tet4()Tet Lbel8Cpto = Tet()Tet & "X" & "+" & Tet()Tet & "Y" & "+" & Tet3()Tet & "Z" & "=" & Tet4()Tet Lbel9Cpto = Tet()Tet & "X" & "+" & Tet()Tet & "Y" & "+" & Tet3()Tet & "Z" & "=" & Tet4()Tet If Tet()Tet = "" Ad Tet()Tet = "" Ad Tet()Tet = "" Ad Tet()Tet = "" Ad Tet()Tet = "" Ad Tet()Tet = "" Ad Tet3()Tet = "" Ad Tet3()Tet = "" Ad Tet3()Tet = "" The opto5 = MsgBo("ANDA BELUM MEMASUKKAN NILAI VARIABEL", vbokoly, "WARNING") 9

Metode Numerk utuk Tekk Mes FrmeVsble = Flse Commd3Vsble = True Ed If For I = To (I) = Tet(I)Tet b(i) = Tet(I)Tet c(i) = Tet3(I)Tet d(i) = Tet4(I)Tet Net I () = y() = z() = jumlh = For I = To jumlh = jumlh + (I) = (d() - (b() * y(i - ) + c() * z(i - ))) / () y(i) = (d() - (() * (I) + c() * z(i - ))) / b() z(i) = (d() - (() * (I) + b() * y(i))) / c() If (I) = (I - ) Ad y(i) = y(i - ) Ad z(i) = z(i - ) The GoTo Ed If Net I LstCler For I = To jumlh LstAddItem I & vbtb & vbtb & (I) & vbtb & vbtb & y(i) & vbtb & vbtb & z(i) Net I Ed Sub Prvte Sub Commd_Clck() Lbel7Cpto = "" Lbel8Cpto = "" Lbel9Cpto = "" FrmeVsble = Flse FrmeVsble = True CommdVsble = True Ed Sub Prvte Sub Commd3_Clck() FrmeVsble = True CommdVsble = True Commd3Vsble = Flse Ed Sub Prvte Sub Form_Lod() FrmeVsble = Flse Commd3Vsble = Flse Ed Sub 3

Metode Numerk utuk Tekk Mes TAMPILAN HASIL PROGRAM 3

Metode Numerk utuk Tekk Mes BAB IV INTERPOLASI Umumy dt egeerg byk yg berup tbuls Pempl dt sepert tu dkrek pd keyty dt yg bs dperoleh dlh bersft dscrete tu jug kre keterbts dlm pegukur sehgg hy sebg dt yg dpt dsmp tu dctt Cotoh dt yg dscrete y 3 5 4 4 5 78 6 93 Megterpretsk mpuls dt dscrete dpt dlkuk deg beberp cr ytu : Numercl Iterpolto Curve Fttg 3 Numercl Dfferetto 4 Numercl Itegrto INTERPOLATION 4 Ler Iterpolto Ytu terpols plg sederh, deg megggp hubug berup grs tr du ttk dt y y + y Grs Lurus y = f() Persm grs lurus yg meghubugk du ttk dt tersebut : y y y y = - - y = y + y y - ( ) + Utuk cotoh dt d ts msly g dcr utuk =,5 3

Metode Numerk utuk Tekk Mes y =, +,5,,3 -, (,5,) =,3 4 Lgrge Iterpolto Membut hubug ttk dlm tbuls berup sutu poloml dm msg-msg ttk berup smpul-smpul yg hrus dpeuh poloml Tbuls berup ttk-ttk, y dm =,,, dm terdpt + dt, k dpresetsk y() = f() pd tervl Poloml terpols mempuy betuk : P () = y b () + y b () + y b () + + y b () deg b j () = sutu poloml derjt Poloml b j () dpt dcr deg megguk + persm costrt Persm costrt dpt dbut sebg berkut : P ( ) = y ; =,,,, Sehgg : P ( ) = y y b ( ) + y b ( ) + + y b ( ) = y P ( ) = y y b ( ) + y b ( ) + + y b ( ) = y P ( ) = y y b ( ) + y b ( ) + + y b ( ) = y Utuk mempermudh peyeles persm costrt, mk dplh: ; = j b j ( ) = Plh tersebut memeuh persm costrt Betuk persm poloml b j () dlh sebg berkut : b j () = C j ( - ) ( - ) ( - ) ( - j- ) ( - j+ ) ( - ) Sesu plh d ts yg cocok deg costrt ytu : b j ( j ) = Mk kostt C j dpt dcr deg rumus berkut : C j = ( )( )( )( )( ) j ; j j j j j j j Deg demk semu poloml b j () dperoleh : b () = C ( - ) ( - ) ( - ) b () = C ( - ) ( - ) ( - 3 ) ( - ) b () = C ( - ) ( - ) ( - 3 ) ( - ) b () = C ( - ) ( - ) ( - - ) 33

34 Metode Numerk utuk Tekk Mes dm : C = ) )( )( )( ( 3 C = ) )( )( )( ( 3 C = ) )( )( )( ( 3 C = ) )( )( )( ( Jd poloml b j () dpt dtuls secr legkp : b () = ) )( )( ( ) )( )( ( b () = ) )( )( ( ) )( )( ( b () = ) )( )( )( ( ) )( )( )( ( 3 3 b () = ) )( )( )( ( ) )( )( )( ( Sehgg persm poloml dr lgrge terpolto dpt drumusk sebg berkut : P () = j y j ) )( )( )( )( ( ) )( )( )( )( ( j j j j j j j j j Atu jk : L () = ) )( )( )( )( ( j j Mk : P () = j y j ) ( ) ( j j j L L = y() = f() Cotoh Sol : Htug hrg y(5) pd dt yg dsjk pd tbel berkut y 3 4 4 8

Metode Numerk utuk Tekk Mes (,5 )(,5 )(,5 3) y(,5) = y ( )( )( ) (,5 )(,5 )(,5 3) + y ( )( )( ) (,5 )(,5 )(,5 3) + y (3 )(3 )(3 ) + y 3 (,5 (4 - )(,5 )(,5 3 3 3 )(4 )(4 y(,5) = 33 + 875 5 + 5 = 438 ) ) 43 Newto-Gregory Iterpolto Berdsrk formuls Bed hgg, dm dbut sutu poloml deg ttk-ttk dt sebg ttk smpul Betuk terpols polomly dlh : P () = C + C ( - ) + C ( - ) ( - ) + + C ( - ) ( - ) ( - - ) dm : C, C,, C sutu kostt C j ; j =,,, dpt dcr deg memk persm costrt berkut : P () = y ; =,,,, Yg k meghslk persm berkut : P ( ) = f( ) = y C = y P ( ) = f( ) = y C + C ( - ) = y C + C ( - ) + C ( - )( - ) = y C + C ( - )+ + C ( - )( - ) ( - - )= y Dr persm ler smult tersebut dpt dhtug C j ; j =,,,, D seterusy P () = f() = y() dpt dcr d hrg y utuk setp hrg dpt dhtug Hrg C j dpt drumusk sebg berkut : C = y y C C = C = C 3 = y y ( 3 C C C ( )( ( 3 C ( 3 ) )( ) ) C 3 ( 3 )( 3 )( ) 3 Metode mejd lebh mudh jk kreme dr tetp + = = h tu = + h ; =,,, ) dst 35

Metode Numerk utuk Tekk Mes Persm costrt d ts mejd : y = C y = C + C h y = C + C (h) + C (h ) y 3 = C + C (3h) + C (6h ) + C 3 (6h 3 ) y = C + C (h) + C (h)((-)h) + C 3 (h)((-)h)((-)h) + C (!)h Klu persm dselesk k ddptk : C = y y C C = y y = y = h h h C = h [ y - C h C ] = h [ y ( y C) - y h ] h = h [ (y - y ) (y - y ) ] = h [ (y ) ] y = h Secr umum hrg C j dpt drumusk : j y C j = j ( j!) h Utuk meghtug C j secr lebh mudh dpt dguk tbel sebg berkut : y = y = 3 y = 4 y = 5 y = y y + - y y + - y y + - y 3 y + - 3 y 4 y + - 4 y y y y y y 3 y y y 4 y y 3 y 5 y 3 y 3 y 4 y y 3 3 y 4 y 4 y 3 y 4 5 y 5 j y Dr tbel tersebut C j dpt dhtug deg rumus C j = j ( j!) h 36

Metode Numerk utuk Tekk Mes Mk byk tgkt C j yg dpk dlm meghtug hrg y mk mk telt terpolsy y() = C + C ( - ) + C ( - )( - ) k lebh kurg telt dr y() yg dhtug deg : y() = C + C ( - ) + C ( - )( - ) + C 3 ( - )( - )( - ) d seterusy Cob selesk sol yg sm pd ksus d Iterpols Lgrge Htug y utuk = 5 deg terpols Newto-Gregory Berkut dlh tbel bed hgg utuk ksus d ts y y y 3 y 3 4 4 4 8 Jk dpk frst order dfferece sj, mk : y y(5)= y + (5 ) h = + (5 ) = + 5 = 5 Jk dpk frst-secod order dfferece mk : y y(5)= y + y (5 )+ h h (5 )(5 ) = + (5 )+ (5 )(5 ) = + 5-5 = 375 Dpk frst-secod-thrd order dfferece mk : y y(5)= y + y (5 )+ h h (5 )(5 ) 3 y + 6h 3 (5 )(5 )(5 ) = + (5 )+ (5 )(5 ) + (5 )(5 )(5 3) 6 = + 5-5 + 6 = 437 37

Metode Numerk utuk Tekk Mes BAB V PENCOCOKAN KURVA (CURVE FITTING) 5 Curve Ler y y = f() b Persm pedekt utuk kurv ler dpt drumusk : f() = + b y y Metode kudrt terkecl Jumlh kudrt terkecl (D ) D = E = y f( ) = y b A d b dcr deg memmumk hrg D = D y b = - b y = y I - - b = y I - - b = = y I - b = y - b = y b () = b D y b = - b y = y - - b = + b = y 38

Metode Numerk utuk Tekk Mes y - b + b = y y - b + b = y b { - } = y I - y b = y y () Utuk melht derjt kesesu dr curve fttg deg cr melht hrg (Koefse Korerls) = t D D D t deg D t = y y 5 Curve No-Ler (berhrg s/d ) y = e b Proses Lerss l y = l + b l e = l + b Y = A + b y y = e b l y Y = A + b b y Y = l y Y y l y y y l y y y l y y y y y 39

Metode Numerk utuk Tekk Mes b = Y Y Y A = - b = Y b A = l = e A b y = b Proses Lerss log y = log + b log Y = A + b X y y = b log y Y = A + bx b y X = log Y = log y X Y X y log log y X Y X y log log y X Y X y log log y X Y X y X Y I X Y b = A = X Y X Y X X Y X - b = Y b X A = log = log - A c Poloml y = + + + + r r Jumlh kudrt dr keslh dlh : D r = y r Deg cr yg sm kostt dpt dcr deg memmumk hrg D 4

4 Metode Numerk utuk Tekk Mes D = - r r y = D = - r r y = D = - r r y = r D = - r r r y = Atu dpt dtuls dlm betuk mtrk berkut : r r r r r r r 4 3 r 3 r y y y y Peyeles persm k ddpt,,, r 53 Curve Mult Ler / No-Ler Hubug tr vrbel terkt deg lebh dr stu verbel bebs secr ler dpt drumusk sebg berkut : y = b + b + b + b 3 3 + + b k k dm: y = vrbel terkt s/d k = vrbel bebs D = Respose Predcted Respose Observ ed D = k k 3 3 b b b b b y Deg Metode Kudrt Terkecl, l D dturuk terhdp kostt b o s/d b k d dmmumk (dsmk deg ol), sehgg dpt dcr l dr kostt b o s/d b k

Metode Numerk utuk Tekk Mes Drumusk dlm betuk mtrk : 3 k 3 3 3 3 3 3 3 k k k 3k k k k k b b b b = 4 bk y y y 3y ky Cotoh ksus d bdg mes : Rumusk persm emprk Prmeter Pemotog Proses Pembubut Terhdp Gy Pemotog Mterl ST 4 Besry gy pemotog merupk forms yg dperluk dlm perec mes perkks, kre tu merupk ttk tolk setp htug d ls mes perkks Gy pemotog yg bereks pd pht d bed kerj, yg seljuty dterusk pd bg-bg tertetu mes perkks, k megkbtk letur Meskpu letur tu kecl tp mugk sudh cukup utuk mejd peyebb keslh geometr produk mupu sebg sumber getr yg dpt memperpedek umur pht Gy pemotog teorts telh dpt drumusk, tetp kre dy peyederh d ggp yg medsr peuru rumus tersebut, mk tdk dpt dpk dlm perec proses pemes sesugguhy Sehgg msh dbutuhk sutu betuk rumus emprk yg meggmbrk hubug tr gy pemotog deg vrbel-vrbel dlm proses pemes Deg meetpk d megubh beberp vrbel proses pemes (d eksperme dlkuk pd vrbel d f) mk dpt dcr sutu korels berup rumus emprk vrbel proses d f terhdp gy pemotog Vrbel yg dukur terdr dr : Vrbel Bebs (Idepedet Vrble) ) Kedlm potog ( =,5 ; =, ; 3 =, ) b) Kecept pemk (f =,5 ; f =,6 ; f 3 =,) Vrbel Terkt (Depedet Vrble) ) Vrbel Utm, sebg vrbel yg mejd pembhs utm ytu: gy pemotog (Fv) (N) b) Dt pedukug Dmeter sebelum d sesudh pemotog (mm) Putr spdel tp beb d deg beb (rpm) Wktu pemotog sebery (detk) 3 Prmeter yg dkostk pd eksperme dlh : ) Jes mterl kerj (ST 4) d pht (Krbd) b) Pjg pemotog (L) = 3 mm 4

Metode Numerk utuk Tekk Mes c) Putr spdel () = 494 rpm d) Poss pemotog Orthogol (kr =9 o ) Hsl eksperme berup besr Gy Pemotog (Fv) yg terjd terhdp perubh prmeter d f dpt dlht pd tbel berkut: f (mm/put) (mm) 5 6 5 83768 95 55 854 9469 3984 64 674 8544 8968 964 834 5598 38966 7679 4435 3795 444666 Model curve fttg yg dplh dlh : Fv = c b f b Model o-ler dlerss mejd model ler deg cr d-log-k sebg berkut: Log Fv = log c + b log + b log f Y = b + b X + b X No Fv f Y = Log Fv 83768 5 5 93 854 5 5 97 3 64 5 5 4 8968 5 78 5 5598 5 8 6 4435 5 95 7 95 5 6 36 8 9469 5 6 84 9 674 6 783 964 6 45 38966 6 58 3795 6 568 3 55 5 333 4 3984 5 379 5 8544 93 6 834 9 7 7679 43 8 444666 648 X = Log -3-3 - - -3-3 - - -3-3 - - X = Log f -3-3 -3-3 -3-3 -796-796 -796-796 -796-796 -699-699 -699-699 -699-699 X 9 9 9 9 9 9 X 693 693 693 693 693 693 633 633 633 633 633 633 489 489 489 489 489 489 X X X Y X Y 39 39 3 3 4 4 769 769 699 699-579 -577-5 -78-694 -688-783 -45-7 -76-93 -9-5 -494-58 -663-837 -76-835 -88-49 -68-54 -44-63 -663-35 -34-7 -85 378-786 -6775 6544 6888 775-49 -3436 Dr tbel d ts k ddptk persm berkut : 8 b + -786 b + -6775 b = 378-786 b + 6544 b + 775 b = -49-6775 b + 775 b + 6888 b = -3436 43

Metode Numerk utuk Tekk Mes Deg prosedur umerk Guss-Sedel ddptk : b = 3375 b = 793 b = 8847 Y = b + b X + b X = 3375 + 793 X + 8847 X Kre model tersebut dlerssk mk hrus dkemblk ke model o lery ytu meg-t log-k b -y sehgg c = Log - 3375 = 7785 Mk : Fv = 77,85,793 f,8847 44

Metode Numerk utuk Tekk Mes BAB VI PERSAMAAN DIFFERENSIAL BIASA Persm dfferesl bs deg ordo, merupk persm deg stu perubh (vrbel) yg dpt dtulsk dlm betuk : dy d y d y F(, y,,,, ) = d d d deg y = f () Peyeles persm dfferesl ordo stu dpt lebh dr stu, sehgg utuk mecr peyeles yg uk tu khusus memerluk forms tmbh berup syrt bts Metode utuk peyeles Persm dfferesl bs : EULER S METHOD Deret tylor orde Sgt sestf terhdp besry h y = y - + h f ( -,y - ) ; =,, 3, h = deg : = l yg dty l fugsy = l wl = blg bult MODIFIED EULER S METHOD Megurg keslh kbt pemlh h y (k+) h ( k) = y - + f (-, y- ) f(, y ) (k) Deg : y = y - + h f ( -, y - ) k =,,, d =,, 3, 3 RUNGE-KUTTA METHOD Deret tylor orde 4 Lebh telt h y y k k k 3 k 6 dm : k = f (, y ) k = f ( +,5h, y +,5 h k ) k 3 = f ( +,5h, y +,5 h k ) k 4 = f ( + h, y + h k 3 ) Cotoh : 4 d dy = 3 + 5 + y ; y() = Cr l y (,) deg Metode Euler, Modfed Euler d Ruge Kutt (pk h =,) 45

Metode Numerk utuk Tekk Mes 6 Euler s Method Dplh h = h = =, = = h Dr dt kods bts ddptk = d y = Iters Pertm ( = ) y = y(,) = y + h f (, y ) = y + h (3 + 5 + y ) = + (,) (3 + 5 + ) =,9 Iters Kedu ( = ) y = y(,) = y + h f (, y ) = y + h (3 + 5 + y ) =,9 + (,) (3, + 5, +,9) = 3,3 Jd peyeles ksus tersebut : y(,) = 3,3 6 Modfed Euler s Method Deg rumus Euler s Method () y = y + h f (, y ) = y + h (3 + 5 + y ) = + (,) (3 + 5 + ) =,9 Proses ters dlkuk pd rumus Modfed Euler s Iters Pertm ( =, d k = ) : y () h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ), = + =,5 (3 5 ) (3, 5,,9) Iters Kedu ( =, d k = ) : y () h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ), = + =,66 (3 5 ) (3, 5,,5) 46

Metode Numerk utuk Tekk Mes Iters Ketg ( =, d k = ) : y (3) h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ), = + =,68 (3 5 ) (3, 5,66) Iters Keempt ( =, d k = 3) : y (4) h (3) = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 (3) y y ), = + =,68 (3 5 ) (3, 5,,68) Kre hsl ters keempt d ters ketg (ters sebelumy) sm mk proses ters dhetk deg hsl hrg y =,68 y () = y + h f (, y ) = y + h (3 + 5 + y ) =,68 + (,) (3, + 5, +,68) = 3,5 Iters Pertm ( =, d k = ) : y () h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ), =,68+ = 3,357 (3, 5,,68) (3, 5, 3,5) Iters Kedu ( =, d k = ) : y () h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ), =,68+ = 3,44 (3, 5,,68) (3, 5, 3,357) 47

Metode Numerk utuk Tekk Mes Iters Ketg ( =, d k = ) : y (3), =,68+ h () = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 () y y ) = 3,47 (3, 5,,68) (3, 5, 3,44) Iters Keempt ( =, d k = 3) : y (4), =,68+ h (3) = y + f (, y) f(, y ) h = y + ( 3 5 ) (3 5 (3) y y ) = 3,47 (3, 5,,68) (3, 5, 3,47) Hsl ters keempt d ters sebelumy ytu ters ketg sm mk proses ters dhetk deg hsl hrg y = 3,47 Jd peyeles ksus tersebut : y(,) = 3,47 63 Ruge-Kutt Method Dplh h =, Iters Pertm ( y = y(,) ) = ; y = k = f (, y ) = (3 + 5 + y ) = (3 + 5 + ) = 9 k = f ( +,5h, y +,5 h k ) = 3 ( +,5h) + 5 ( +,5h) + (y +,5 h k ) = 3 (+,5,) + 5 (+,5,) + (+,5, 9) =,75 k 3 = f ( +,5h, y +,5 h k ) = 3 ( +,5h) + 5 ( +,5h) + (y +,5 h k ) = 3 (+,5,) + 5 (+,5,) + (+,5,,75) =,579 k 4 = f ( + h, y + h k 3 ) = 3 ( + h) + 5 ( + h) + (y + h k 3 ) = 3 (+,) + 5 (+,) + (+,,579) =,358 h y y k k k3 k4 6 48

Metode Numerk utuk Tekk Mes, = + (9+ (,75) + (,579) +,358) 6 =,44 Iters kedu ( y = y(,) ) =, ; y =,44 k = f (, y ) =(3 + 5 + y ) = (3, + 5, +,44) =,344 k = f ( +,5h, y +,5 h k ) = 3 ( +,5h) + 5 ( +,5h) + (y +,5 h k ) = 3 (,+,5,) + 5(,+,5,) + (,44+,5,,344) =,787 k 3 = f ( +,5h, y +,5 h k ) = 3 ( +,5h) + 5 ( +,5h) + (y +,5 h k ) = 3 (,+,5,) + 5(,+,5,) + (,44+,5,,787) =,3359 k 4 = f ( + h, y + h k 3 ) = 3 ( + h) + 5 ( + h) + (y + h k 3 ) = 3 (,+,) + 5 (,+,) + (,44+,,3359) = 3,558 h y y k k k3 k4 6, =,44 + (,344+(,787)+(,3359)+3,558) 6 = 3,356 49

Metode Numerk utuk Tekk Mes BAB VII PERSAMAAN DIFFERENSIAL PARSIAL Formuls mtemtk dr kebyk permslh dlm lmu pegethu d tekolog dpt dpresetsk dlm betuk persm dfferesl prsl Persm tersebut merupk lju perubh terhdp du tu lebh vrble bebs yg bsy dlh wktu d jrk (rug) Persm dfferesl dpt dbedk mejd tg jes ytu : A Persm Dfferesl Prbolk Bsy merupk persm yg tergtug pd wktu (tdk perme) d peyelesy memerluk kods wl d bts Persm prbolk plg sederh dlh permbt ps T K T t Peyeles dr persm d ts dlh mecr tempertur T utuk l pd setp wktu t B Persm Dfferesl Elptk Bsy berhubug deg mslh kesetmbg tu kods perme (tdk tergtug wktu) d peyelesy memerluk kods bts d sekellg derh tju Sepert lr r th d bwh bedug d kre dy pemomp, defleks plt kbt pembeb, dsb y C Persm Dfferesl Hperbolk Bsy berhubug deg getr tu permslh dm terjd dskotue dlm wktu, sepert gelombg kejut yg terjd dscotue dlm kecept, tek d rpt mss U C t U 5

Metode Numerk utuk Tekk Mes 7 Peyeles Persm Prbolk deg Skem Eksplst T K T t (7) deg : T = tempertur K = koefse koduktvts t = wktu = jrk Pd skem eksplst, vrbel pd wktu + dhtug berdsrk vrbel pd wktu yg sudh dkethu Deg megguk skem sepert d bwh, fugs f(,t) d turuy dlm rug d wktu ddekt oleh betuk berkut : + f (, t) = f f(,t) = t f f t - + f t (,t) = f f t f Dr skem d ts, persm (7) dpt dtuls dlm betuk berkut : T T t = K T T T tu T t = T K T T T (7) 5

Metode Numerk utuk Tekk Mes Stblts Skem Eksplst Dlm skem eksplst, sebelumy ytu: T, T d T T tergtug pd tg ttk Ked dpt meyebbk ketdkstbl dr skem tersebut, yg berup terjdy mplfks hsl htug dr kods wl Agr stbl dbutuhk sutu syrt ytu : < < / deg = t Cotoh: L = m Dm : k = =, t =, = t =,, =, <,5 (stbl) Syrt bts : pd t = T = ; ½ L d T = (-) ; ½ L L pd semu t utuk = T = Deg megguk persm (6), htug dlkuk dr = smp deg 5 d dr = smp wktu yg dkehedk (N) Utuk = d bergerk dr = smp = 6, 3 T =, +, (, +,4) =, T =,4 +, (,,4 +,6) =,4 T 4 =,6 +, (,4,6 +,8) =,6 T 5 =,8 +, (,6,8 + ) =,8 T 6 = +, (,8 +,8) =,96 utuk = d bergerk dr = smp = 6, T =, +, (, +,4) =, T 3 =,4 +, (,,4 +,6) =,4 T 4 =,6 +, (,4,6 +,8) =,6 T 5 =,8 +, (,6,8 +,96) =,796 T 6 =,96 +, (,8,96 +,8) =,98 5

Metode Numerk utuk Tekk Mes Demk perhtug terus dljutk s/d wktu yg dkehedk (N) Tbel hsl skem eksplst = 3 4 5 6 7 = 3 4 5 6 t = 4 6 8 8 t =, 4 6 8 96 8 t =, 4 6 796 98 796 t =,3 4 5996 7896 96 7896 t = N N N N N N N N 7 Peyeles Persm Prbolk deg Skem Implst Dlm skem eksplst, rus k dr persm dtuls pd wktu yg ly sudh dkethu Sedgk pd skem mplst, rus k tersebut dtuls pd wktu + d m ly belum dkethu Gmbr d bwh meujukk jrg ttk smpul dr skem mplst Deg megguk skem tersebut, fugs f(,t) d turuy dlm rug wktu ddekt oleh betuk berkut + f(,t) f f(,t) f t f(,t) f f(,t) f tu f f Δt f Δ f Δ f - + 53

Metode Numerk utuk Tekk Mes Deg megguk skem d ts, mk dpt dbetuk persm dlm betuk bed hgg : tu T Δt T K Δ A T deg A B T Δt ( Δt K Δ T K B T K Δ ( K Δ T T Δt C T ) T Δ K Δ K )T Δ ; C ; D K Δ K Δ Apbl persm (73) dtuls utuk setp ttk htug dr = smp M mk k terbetuk sutu sste persm ler yg dpt dselesk deg megguk metode mtrks T D T K Δ T Δt T T (63) T Δt T Δt Utuk : = AT + BT + CT = D = AT + BT + CT3 = D = 3 A3T + B3T3 + C3T4 = D3 = 4 A4T 3 + B4T4 + C4T5 = D4 = M AMTM- + BMTM + CMTM+ = DM Utuk peyederh peuls, vrbel T + dtuls T (tp meuls +) Persm d ts dlm betuk mtrk mejd : B C T D A B C T D A3 B3 C3 T3 D3 A4 B4 C4 T4 D4 = AM BM TM DM 54

Metode Numerk utuk Tekk Mes Persm tersebut dpt dselesk deg metode peyeles persm seretk utuk medptk l T ( =M) Peyeles deg megguk skem mplst lebh sult dbdg deg skem eksplst Kelebh dr skem mplst dlh skem tersebut stbl tp syrt, lgkh wktu Δt dpt dmbl sembrg (besr) tp membulk keslh pemotog dlm bts-bts yg dpt dterm 73 Peyeles Persm Elptk Peyeles dlkuk deg medskretss sutu persm dfferesl prsl elptk deg kods bts utuk dpt dtrsformsk ke dlm sutu sstem dr N persm deg N blg u Peyeles persm elptk dlkuk deg lgkhlgkh berkut Membut jrg ttk smpul d dlm seluruh bdg yg dtju d bts-btsy Pd setp ttk dlm bdg tersebut dbut turuturuy dlm betuk bed hgg 3 Dtuls l-l fugs pd semu ttk d bts kellg bdg deg memperhtk kods bts Dr persm betuk elptk berkut : Sehgg : y,j,j,j,j y,j,j Utuk = y, mk persm d ts mejd : 4,j,j,j,j,j 55

Metode Numerk utuk Tekk Mes Cotoh Sol : Determe the stedy stte temperture of the followg plte usg α = d Δ = ft y 4 ft T b = 4 F T = F T c = F 3 ft T d = F Jwb : T b = 4 F T = F 3 5 4 6 T c = F T d = F Node Node 3 T T b -4 3 5 4 6 T c T T b 3-4 5 4 6 T c T d + 4-4 T + T + T 3 = T d 4 + T - 4 T 3 + T 4 + T 5 = 56

Metode Numerk utuk Tekk Mes Node Node 4 T b T b T 3 5-4 4 6 T c T 3 5 4-4 6 T c T d T d + + T -4 T + T 4 = + T + T 3-4 T 4 + T 6 = Node 5 Node 6 T b T b T 3 5-4 4 6 T c T 3 5 4 6-4 T c T d T d 4 + T 3-4 T 5 + T 6 = + T 4 + T 5-4 T 6 = Sehgg hsl persm-persm tersebut dpt dbetuk dlm sutu mtrk : -4 T -5-4 T -3-4 T 3-4 -4 T 4 = - -4 T 5-4 -4 T 6 - T = 3,56 F T = 8,344 F T 3 = 5,9 F T 4 = 9,84 F T 5 =,8 F T 6 = 5, F 57

Metode Numerk utuk Tekk Mes BAB VIII INTEGRASI NUMERIK Metode Guss Qudrture merupk slh stu metode tegrs umerk yg plg dpt dterm d dpk secr tesf utuk meyelesk tegrs mtrk kekku d Metode Eleme Hgg pd eleme jes soprmetrk Dbdgk deg metode Trpezod yg hy terbts pd peggu deg froms dt yg equspced, mk metode Guss Qudrture lebh lus pegguy pd peyeles fugs yg o-equspced d memlk kekurt lebh tgg Rumus Guss Qudrture megubh bts tegrl dr sutu bts (msl s/d b) mejd - s/d + Formul dsry dlh merupk pejumlh dr perkl koefse bert d hrg dr fugs pd smplg pots b I = f ()d = k W k f( k) Deg : W k = Koefse bert (Weghtg Coeff) k = smplg (Guss) pot utuk meetuk Koefse bert (Weghtg Coeff) dlkuk deg prosedur memdh bts tegrl dr s/d b mejd - s/d + Tbel Guss Qudrture Poloml Jumlh Smplg (Guss) Koefse Bert Degree Pot Pot = W = 3 = - 577353 W = = 577353 W = = 5 3 = - 7745967 W = 5555556 = W = 8888889 3 = 7745967 W = 5555556 58

Metode Numerk utuk Tekk Mes Persm mtrk kekku utuk eleme segempt soprmetrk dpt dtuls sebg berkut: T T [ k] h [B] [C][B]dX dy h [B] [C][B] J ds dt A = W W [B(s, t )] T [C] [B(s, t )] J(s, t ) + W W [B(s, t )] T [C] [B(s, t )] J(s, t ) + W W [B(s, t )] T [C] [B(s, t )] J(s, t ) + W W [B(s, t )] T [C] [B(s, t )] J(s, t ) Deg sums polyoml degree yg dpk 3 d smplg pot mk dguk hrg : s = t = -,577 s = t =,577 W = W = Htug I = y d dy deg Guss Qudrture (pot = ) A* 59

Metode Numerk utuk Tekk Mes DAFTAR PUSTAKA Abd Muf, (995), Cr Prkts Pegus d Peggu Metode Numerk, Gu Wjy, Jkrt Chpr, Steve C, (99), Metode Numerk, Erlgg, Jkrt Soehrdjo (985), Als Numerk, ITS, Surby Trtmodjo, Bmbg, (995), Metode Numerk, Bet Offset, Yogykrt 6