FORMULAE FOR FINITE DIFFERENCE APPROXIMATIONS, QUADRATURE AND LINEAR MULTISTEP METHODS

Transcription

1 Jourl o Mtemtcl Sceces: Advces d Alctos Volume Number - Pges -9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS QUADRATURE AND LINEAR MULTISTEP METHDS RAMESH KUMAR MUTHUMALAI Dertmet o Mtemtcs D G Vsv College Arumbm Ce-66 Tml Ndu Id e-ml: rm_@oocom Abstrct We derve te derece ormule or romtg ger order dervtves wc re lcble to evel or uevel sced grds wt rbtrr order o ccurc Usg tese ormule we descrbe qudrture ormule d ler multste metods or romtg tegrls d umercl soluto o deretl equtos Itroducto Fte derece romtos re ote used or romtg dervtves to solve deretl equtos [] Te recurso reltos to clculte te wegts o te derece ormule wt rbtrr order o ccurc re gve [7] K et l ve reseted drect ormule or te elct orwrd bcwrd d cetrl derece ormule o te derece romtos wt rbtrr order or rst dervtve d Mtemtcs Subect Clsscto: A 6D 6D 6L 6L6 6M6 Kewords d rses: brcetrc dervtves te derece ormule mlct romtos qudrture ormule ler multste metods Receved Setember 6 Scetc Advces Publsers

2 RAMESH KUMAR MUTHUMALAI te cetrl derece romtos or ger dervtes [-] Geerl elct ormule or rst d ger order dervtves o te bss o te geerlzed Vdermode determt wt rbtrr order o ccurc or romtg rst d ger order dervtves re gve [] Some smle d covetol eressos o cetrl derece ormule or rst d secod dervtves re oud [6] Ts metod o udetermed coecets s lmted to evel sced grds A clss o ler multste metods suc s Adms-Bsort d Adms-Moult metods or te umercl soluto o te tl vlue roblem s bsed o te rcle o umercl tegrto [ 9] Te rge o umercl tegrto ormule bsed o deret teroltg olomls s lmted s rule to te ots o terolto I te reset stud we reset te derece ormule terms o brcetrc wegts or romtg dervtves d costructo o ler multste metods to solve deretl equtos wt smle d coveet eressos Elct Fte Derece Aromtos Recursve ormul or umercl deretto [ b] Let ( ) re dstct umbers te tervl togeter wt corresodg ( ) Let be umber dstct rom ec d [ ] deotes -t order dvded derece o t te ots We ow tt ( )! [ ] tmes () Edg rgt d sde o () b Lgrge terolto ormul te ( )! [ ] l E () tmes

3 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS were d l : E ( ) ( ξ) l ( )! were l ( ) d ξ [ b] te oter d usg recursve ormul o dvded derece [ ] we ve ( [ ] ) ( )! ( ) ( ) () tmes Substtutg () () d ter smlcto we d tt ( ) l! ( ) ( ) l ( ) E () l Settg ρr r : we obt te ollowg r ( ) recursve ormul coectg uctol vlues : d dervtves u to order t : ( ) ρ! ( ) l ( ) E () Note tt te recursve ormul () c be led ol we s deret rom ec Suose tt cocde wt oe o (e ) we c stll use te recursve ormul to estmte dervtves u to order s ollows:

4 RAMESH KUMAR MUTHUMALAI were r ( r ) l ( ) r : r! r ρr (6) ( )! ρm l ( ) m m : r (7) d l : () Elct te derece ormule wt Lgrg coecets We ote eed romto o dervtves or desgg derece scemes to solve deretl equtos Tble sows some emles o elct te derece romtos [7] Tble Emles o elct te derece ormule Cse Descrto Emles o elct te derece ormule Error level Cetered regulr grd Stggered regulr grd [ ( ) ( ) ( ) ( )] 9 [ ( ) ( ) 9 ( ) ( )] ( ) ( ) Regulr grd ( ) [ ( ) ( ) ( ) ( ) ( )] Suose oe use (6) to d t-t dervtve o t we get te ollowg orm o te elct romto:

5 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS () t t! tχ Mt s s tχ (9) s were χ () d M ( ) ( ξ ) l l () ( ) ( )! d s s uow ucto Now substtutg ll M s (9) d ter smlcto we obt () t t! l tχ ED ( ; ) () t ( ) tχ were E D ( ; ) l tχ m m ( t m ) ( ξm ) ( t m)! or To d ll s set () we t we d d t l t s ( ) s t t l Rerrgg bove equto d settg ρr r : t r ( ) we d tt

6 6 RAMESH KUMAR MUTHUMALAI t ρt () Sce ll oter uow s c be oud recursvel rom () For equll sced grds I ew secl cses (eg or elct romtos o equsced grds) te otml wegts re ow closed orm Cses to Tble sow tree suc emles Let ( m ) re equll sced grds Settg () d relcg Lgrg coecets l b ( m ) A te ( m )!! m A m!! () It ollows tt ρ r ( m ) r A r m Settg ρ r ( ) W r r b ve () d ter smlcto we were gves Wr t t b b W () ( m ) A m r Tus or evel sced grds () t () t t! ( m ) b t A bt ( ) t (6) t m

7 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 7 Elct te derece ormule wt brcetrc wegts Te ormul gve () c be moded to eve more elegt orm wt brcetrc wegts Dvde l o bot sdes o () were w l l (7) w : () We d tt ρr w r r l δ s (9) Now dvdg () b l d usg (9) ter smlcto tt t δt () Dvdg bot umertor d deomtor o () b l usg (7) d ter smlcto we obt () t tχ w E t! t tχ δ () ( ) All te uow s c be oud rom () Qudrture Formule d Ler Multste Metods Tble sows some emles o qudrture ormule d ler multste metods Cses - d Cse re emles o Newto-Cotes tegrto ormule d Adms ormule or ler multste metods

8 RAMESH KUMAR MUTHUMALAI Formul or umercl tegrto Let [ ] d re dstct umbers o te closed tervl ( C ) [ ] Te roblem o umercl tegrto s to romte te dete tegrl () t dt Sce olomls re es to tegrte b usg Tlor seres we d tt ( () ) t dt ( ) () ( )! Relcg ger order dervtves ( ) : usg () we ve χ () t dt M χ Tble Emles o qudrture ormule d ler multste metods Cse Descrto Emles o qudrture ormule Error level Smso rule Newto-Cotes oe ormul Adms-Moult corrector ormul d [ ( ) ( )] d [ ( ) ( ) ( )] [ 9 ] ( ) ( ) ( ) m Settg γ m : d ter smlcto m m we d tt

9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 9 Substtutg ll te vlues o obt () t dt γχ M γ ( ) M s rom () d ter smlcto we γ () t dt γ χ l EI ( γ; ) χ () were χ s stll deed b () d E I ( γ; ) l mχ ( m ) ( ξm ) γ ( m )! m ( ) Now dvdg l o bot umertor d deomtor o () usg dett (7) d ter smlcto we get γ () t dt γχ w E ( ; ) I γ () δ ( ) Newto-Cotes ormule χ Let ( ) re equll sced grds Now relcg b d b () d ter smlcto we obt were α d α () α m m m m Te Equto () s ( ) -t ot ormul or Newto-Cotes closed tegrto ormul

10 RAMESH KUMAR MUTHUMALAI Sce te oe qudrture ormule do ot requre te uctol vlues t te lmt ots o tegrto ssume tt d Now relcg b d b () d ter smlcto we get β d ( ) () were β m m m m Te Equto () s ( ) -t ot ormul or Newto-Cotes oe tegrto ormul Ler multste metods Let re dstct umbers o te tervl [ ] were I we tegrte te deretl equto ( ) rom to usg () d ter smlcto we d tt ξ ξ δ w (6) ( ) were ξ m m m m Te m re stll deed b () t te ots Ts d o multste ormul (6) s ow s ( ) -ot redctor ormul

11 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS I we tegrte te deretl equto ( ) rom to usg () d ter smlcto we d tt τ τ δ w ( ) were (7) τ m m m m Ts d o multste ormul (7) s ow s ( ) -ot corrector ormul Comrso wt ter Formule I ts secto we comre te ew elct te derece ormule qudrture ormule d redctor-corrector ormule wt ormer ormule o vrous ot dstrbutos Elct te derece romtos Algortm For rbtrr sced grds < < < < d ow te ucto vlues ( ) t ( ) oe les ( ) -ot ormul to estmte te -t dervtve o t te tere re ve smle stes Ste For : clculte w w Ste For m : clculte δm m ( ) r Ste Set d or r : clculte r δ r

12 RAMESH KUMAR MUTHUMALAI Ste For : clculte c tχ t ( ) were χ Ste Clculte () t t! t χ δ wc Te Algortm requres cost o los Te quttes tt ve to be comuted oertos do ot deed o te dt e c esl obt more umercl deretto ormule or rbtrr sced grds usg Algortm We rovde ollowg 6-ot umercl deretto ormule o evel sced grds s emles or rst secod d trd order dervtves wose error levels re ( ) ( ) d ( ) resectvel ( 7 ) 7 6 ( 6 6 ) ( ) ( ) 6 6 ( 6 6 ) ( 7 ) ( ) ( 6 6 ) ( ) () () ()

13 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS Comrg te ormer ormule gve [6 7 -] wt te corresodg orwrd bcwrd d cetrl derece ormule () () d () resectvel t could be oud tt te re equvlet Te bove metoed ormule re lso oud rom (7) b vrg te vlues o m d I rtculr m d m te (7) gves orwrd bcwrd d cetrl derece romtos o rst d ger order dervtves I ct te rst d secod order dervtve ormule [6] re secl cses o (7) or evel sced ots For uevel sced grds we coose Cebsev ot dstrbutos Te ollowg re some emles o te -ots d 7-ots ormule or rst secod d trd order dervtves o Cebsev ots o secod d s ollows: () () Te ormule gve [] estmte ger order dervtves ol t te smlg odes were te ew ormul gve () used to estmte dervtves eve we te uctol vlues t smlg odes re ow or ot ow Ule recurso ormule or clculto o wegts gve [7] ts metod o romto gves elct ormule tt use gve ucto vlues t smlg odes drectl d esl to clculte umercl romtos o rbtrr order t smlg dt or te rst d ger dervtves Also t eed less clcultos burde

14 RAMESH KUMAR MUTHUMALAI comutg tme d storge sce to estmte te dervtves t te oter metods stted bove It c be drectl used or desgg derece scemes o DEs d PDEs d solvg tem Qudrture ormule d ler multste metods Algortm For rbtrr sced grds < < < < d ow te ucto vlues ( ) t ( ) oe les ( ) -ot ormul to estmte te tegrl () t dt te tere re s stes Ste For : clculte w w Ste For m : clculte δm m ( ) r Ste For r : clculte r δr m Ste For : clculte γ m m m Ste For : clculte d χ γ ( ) were χ Ste 6 Clculte () t dt γ χ wd δ Te Algortm requres some quttes deedet o wt cost o ( ) los but do ot deed o te dt Usg ts lgortm oe c esl get more ormule or rbtrr sced grds wt degree o

15 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS ccurc Te ollowg re some emles o Newto-Cotes closed tegrto ormule: d d d d (6) Smlrl we c obt Newto-Cotes oe qudrture ormule some o tem re lsted below: d d d d (7) For uevel sced ots we coose Cebsev ot dstrbutos Te ollowg re te -ots -ots d -ots ormule o Cebsev ots o secod d: 7 9 d d d () Smlrl te ollowg re te -ot ots d -ots ormule o Cebsev ots o rst d:

16 RAMESH KUMAR MUTHUMALAI d d d (9) Te ollowg re te some secl ormule or ler multste metods I we te we get Adms-Bsort redctor ormule o -ots -ots -ots d -ots resectvel () Smlrl we te we obt Adms-Bsort redctor ormule o -ots -ots -ots d -ots resectvel c c c c () Also we c geerte ormule or rbtrr sced grds Te ollowg re -ots -ots d -ots redctor ormule obted b tg resectvel to redct t

17 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 7 ( 7 7 ) ( ) ( ) 76 () Smlrl te ollowg re -ots -ots d -ots corrector ormule obted b tg resectvel to correct t ( 6 ) 7 c ( 6 6 ) 9 c ( ) c 6 () It s esl oud tt tese emle ormule re te sme s to tose ow corresodg umercl tegrto ormule bsed o teroltg olomls or rbtrr sced grds [ 9] Cocluso I cocluso we ote tt te derece ormule qudrture ormule d ler multste metods terms o brcetrc wegts ve bee develoed ts rtcle Frstl we ve troduced recursve ormul or romtg ger order dervtves Usg ts recursve ormul we ve derved elct te derece ormule or romtg ger order dervtves terms o Lgrge coecets d brcetrc wegts or rbtrr sced grds Moreover usg ts ew elct ormul d Tlor seres we ve derved umercl tegrto ormul or romtg dete tegrls Newto-Cotes ormule d Adms ormule re secl cses o ts ew ormul

18 RAMESH KUMAR MUTHUMALAI Secodl we ve comred te ew elct te derece ormule wt oter ormule We ve see tt ts ew ormul or romtg dervtves requres ver less comutto tme d storge sce We ve sow tt te orwrd bcwrd d cetrl derece romtos re secl cses o ts ew ormul Smlrl we ve sow te comrso o ew tegrto ormul or qudrture ormule d costructo o ler multste metods wt ormer ormule Former ormule or tegrto d ler multste metods re lmted to evel sced grds but ew ormul reseted ere s qute useul o rbtrr sced grds Reereces [] K E Atso A Itroducto to Numercl Alss d Edto Jo Wle & Sos New Yor 99 [] J P Berrut d L N Trete Brcetrc Lgrge terolto SIAM Rev 6() () -7 [] R L Burde d J D Fres Numercl Alss 7t Edto Broos d Cole Pcc Grove CA [] S C Cr d R P Cle Numercl Metods or Egeers rd Edto McGrw-Hll New Yor 99 [] S D Cote d Crl de boor Elemetr Numercl Alss rd Edto McGrw- Hll New Yor 9 [6] M Dvorov Formule or umercl deretto JCAAM (7) 77- e-rtrv:mtna/69 [7] B Forberg Clculto o wegts te derece ormuls SIAM Rev () (99) 6-69 [] N J Hgm Te umercl stblt o brcetrc Lgrge terolto IMAJNA () 7-6 [9] F B Hldebrd Itroducto to Numercl Alss d Edto McGrw-Hll New Yor 97 [] I R K d R b Closed-orm eressos or te te derece romtos o rst d ger dervtves bsed o Tlor seres J Comut Al Mt 7 (999) 79-9 [] I R K d R b New te derece ormuls or umercl deretto J Comut Al Mt 6 () 69-76

19 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 9 [] I R K R b d N Hozum Mtemtcl roo o closed orm eressos or te derece romtos bsed o Tlor seres J Comut Al Mt () -9 [] I R K d R b Tlor seres bsed te derece romtos o ger degree dervtves J Comut Al Mt () - [] J L Geerl elct derece ormuls or umercl deretto J Comut Al Mt () 9- g