Global Journal of Engineering and Technology Review

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1 Glol Jourl of Egeerg d eholog Revew Jourl homepge: Glol Jourl of Egeerg d eholog Revew () 85 9 (06) Applto of Cojugte Grdet Method wth Cu No- Poloml Sple Sheme for Solvg wo-pot Boudr Vlue Prolems H. Juste * d J. Sulm Mthemts wth Eooms Progrmme Uverst Mls Sh Kot Klu Mls ABSRAC Ojetve Cojugte Grdet (CG) method s used to solve two-pot oudr vlue prolems together wth opoloml sple pproh t u degree. Methodolog/ehque o develop sstem of ler equtos mtrx form u o-poloml sples re used to desretze the two-pot oudr vlue prolems so tht the pproxmto e omputed usg CG method. Se m prevous reserhers ttempt to ot the pproxmte soluto for the two-pot oudr vlue-prolems t dfferet degree of o-poloml sples ol the the preset pper ms to look to method whh s est used wth the u o-poloml sples order to pproxmte the soluto of these prolem Fdgs Aordg to the performe lss results term of tertos umer exeuto tme d mxmum solute error t dfferet grd szes the pplto of CG method together wth the u o-poloml sple gve the est pproxmto to the soluto of two-pot oudr vlue prolems ompred to the pproxmto show Suessve Over Relxto (SOR) method d Guss-Sedel (GS) method. Novelt the performe of CG tertve method s foud to e superor respet of tertos umer exeuto tme d mxmum solute error o vrous grd szes. pe of Pper: Emprl Kewords: Cu No-Poloml Soluto; wo-pot Boudr Vlue Prolems; Cojugte Grdet; Suessve Over Relxto; Guss-Sedel.. Itroduto here re vst umer of solutos d pprohes hve ee mplemeted to ot the pproxmte soluto for solvg two-pot oudr vlue prolems rrespetve of ler pot d o-ler pot (Kelle 995). owg to ts rod futo the feld of sees eooms d egeerg. Pror to the ud of pst ltertures tod s reserhers re trggered to dg more to omputto of the erl solutos whh hve led them to tte vrous pprohes sed o modfto or ehemet to the prevous oe. Amog severl * Pper Ifo: Revsed: Jul 06 Aepted: Novemer 06 * Correspodg uthor: E-ml: herrjust0@gml.om Afflto: Mthemts wth Eooms Progrmme Uverst Mls Sh Mls ISSN Glol Adem of rg & Reserh (GAR) Eterprse. All rghts reserved.

2 H. Juste d J. Sulm of the remrkle methods hve ee tted tods re EADM method (Jg 008) d PI method (Che et l. 006). I 00 H me up wth soluto lled s oler shootg method followed other mplemetto of soluto the ltter ers whh s kow s me weght Mohse d El-Gmel (008). Moreover Fg et l. (00) emphszed fte dfferee fte elemet d fte volume. Furthermore other pproh whh hs sgft otruto to solve these prolems s sple pproh s hghlghted Als d Hosk (969). I ft Rmd et l. (007) lso shred mutul vew. I respose to the evolvg pprohes from pst reserhes the preset pper ms to solve the sstem of ler equtos geerted from dsretzto of the two-pot oudr vlue prolems wth u o-poloml sple sheme usg CG SOR d GS methods. Importt to tke ote tht SOR d GS re oduted for omprso purpose. I order to derve the pproxmto equto the u o-poloml sple geerl futo s used to dsretze these prolems sed o sple pproxmto equtos. As result ths sple pproxmto equto leds to the formto of ts orrespodg lrge d sprse ler sstem. here re vrous tertve methods e used to solve ler sstem d hve ee preseted d dsussed thoroughl Youg (97 04) Hkush (0) Sd (996) Hestees d Stefel (95) Kell (995) d Burgersetrum (0). Prevous studes o tertve methods hve show the exstee of severl fmles of tertve methods wth dfferet oept lled s oept of lok terto whh trodued Evs (985) d hve ee further explored Yousf d Evs (995) d Irhm d Adullh (995). I regrds wth the dvtges of CG tertve method o solvg sstem of ler equtos s proposed Hestees d Stefel (95) the preset pper ms to odut ths tertve method together wth the u o-poloml sple sheme for solvg the two-pot oudr vlue prolems. As for omprso purpose term of performe SOR d GS methods re set s otrol methods.. Methodolog. wo-pot oudr vlue prolems Geerll the equto of two-pot oudr vlue prolems e expressed s follows: '' + f ( x) ' + q( x) g( x) x [ ] () sujet to the oudr odtos ( ) A ( ) A () where A s ostt d futos f (x) d g (x) re kow futo wth oudr [ ].he futos f (x) d g (x) ot e reted rdoml to solve for () s the ltl soluto for prolem () s depedg o the oudr odtos (). Moreover for the ese of dsretzg prolem () lter the dom of the soluto s eg restred to the restrto s show Fg. through uform seprto of the set of odes. Fgure. Dstruto of ode pot for dom soluto m 8 p B osderg postve teger m p d lettg the soluto dom [ ] to e dvded uforml to m sutervl the the legth of the uform sutervls x e defed s Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 86

3 H. Juste d J. Sulm x h m () m As show Fg. the grd etwork hs m umers of teror grd pots. A uform grd etwork of the soluto dom s the developed s Fg. usg the grd sze Fg. Hee the grd pots the soluto dom [ ] e lelled s x + h 0 m. Other th tht the vlues of the futo (x) t the grd pots re deoted s ( x ). he teror grd pots s show Fg. re used to formulte d mplemet GS SOR d CG tertve methods utl the overgee test e stsfed. After tht the formulto d mplemetto proesses for the three proposed pot tertve methods o the sme tpe of deepest ode pots s well s the overgee test for the terto e doe ordg to the fte grd etwork s show Fg. oe the soluto of dom for prolem () hs ee dvded to uform tervl.. Cu o-poloml sple pproxmto equto he u o-poloml sple sheme s used to dsretze prolem () order to ostrut the u opoloml pproxmto equto. he dsretzto s doe lettg (x) to e the ext soluto for prolem () d S e u o-poloml sple pproxmto to ( x ) whh s oted from the segmets of Q (x) tht pssg through the pots ( x ) d ( x + + ). herefore the o-poloml sple pproxmto e expressed geerl form s S ( x) Q ( x) x [ x x ] 0 + (4) he the geerl form for the u o-poloml sple from equto (4) s defed Q ( x) s ( x) k( x x ) + s k( x x ) + ( x x ) d Q os + (5) for 0 where d d re ostt d k s the freque for the trgoometr futo. Equto (5) s kow s the geerl form of u o-poloml sple d t terpoltes (x) t the pots x depedg o k d redug to u sple [ ] s k 0. As metoed erler the dsretzto proess s show Fg. s ver mportt to e oduted frst order to formulte the pproxmto equto for the u o-poloml sple. I ths pper the pproxmto of prolem () hs ee dsretzed usg the u o-poloml sple sheme s defed equto (5). Fgure. Illustrto of u o-poloml sple futo for dom soluto m 8 Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 87

4 H. Juste d J. Sulm Next s ssumed s urte soluto whh oted from dfferet segmets of the sple futos tht pssg through the pots x ) d x ). he eh of the sple futos oted re defed s follows: ( ( + + Q ( x ) Q ( x + + '( x ) D ) Q '( x + ) D + order to get the expresso of ostt vrles Q " ( x ) S d Q Q "( ) x + S + d form of + After strghtforwrd lulto s doe the vlue for ll the ostts followg form h h S+ + S os( ) s( ) + S h d + h ( S+ + S ) + h S D D+ S + d S. d re oted the where kh d 0.. N. m m he odto Q ( x) Q ( x) where m 0 hs ee osdered fter ll the pots d d whh pssg through the pot x ) re oted. he ths prt s solved smulteousl to get the followg ( u o-poloml sple pproxmto equto os ( ) ( ) + ( ) 0 h S h S + h S (6) s s s where.... Equto (6) e smplfed s h + [ α S + βs + αs ] 0 (7) where os α [ ] β [ ] d.. N. s s he equto of etrl fte dfferee kwrd fte dfferee d forwrd fte dfferee shemes e expressed s f q g S f ' q + g S f q g (8) S ' ' where ' ' h h + ' h + d re used to solve (7) susttutg (8) to (7). he ths elds to the equto the followg form + F (9) where Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 88

5 H. Juste d J. Sulm p p p q µ 0 + γ + + γ h h h p p q µ 0 γ + + γ h h p µ + γ + p p + γ + 0 q+ F γf f + γf+ h h h h γ µ h φ µ h µ s µ s os. B usg equto (9) sstem of ler equtos mtrx form e ostruted s A F (0) where A F [ ] [ F F F F F ] 0 ( ) +. Dervto of CG method As metoed erler CG method s tertve method trodued Hestees d Stefel (95) whh termtes t most steps f there re o roudg-off errors ourred gve tht the mtrx A equto (0) s ssumed to e smmetr d postve defte. I ddto to tht Hestees d Stefel lso stted tht ths method s superor to elmto method suh s Guss elmto method muh smpler to ode d less storge spe requred the orgl dt e used to ts mxmum s the mtrx gve rem ultered durg the method mplemetto d mprovemet of the soluto our t eh step of estmto. B referrg to equto (0) where mtrx A s ssumed to e x smmetrl mtrx the the formulto of CG tertve method e omputed fdg the sequee of vetors p0 p... p whh re the ss R d fulfll the odto s follows: p Ap j 0 j. () As for the formulto for method SOR d GS the mtrx A equto (0) hs ee deomposed s A D + L + U () where L D d U re lower trgulr mtrx dgol mtrx d upper trgulr mtrx respetvel. he the deomposto equto () hs ee mposed to equto (0). herefore the formulto oted for SOR tertve method s s follows: ( k+ ) ( k) ( k) ( ω ) + ω( D + L) U + F () wheres the formulto for GS tertve method e wrtte s Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 89

6 H. Juste d J. Sulm ( k + ) ( D + L) U + ( L + D) F (4) I order to fltte the overgee rte of SOR tertve method the vlue for prmeter ω must e determed orretl d prte the rge of optml vlue for ω s ω < whh oted odutg severl omputer progrms. he the est pproxmte vlue for ω s hose sed o ts smllest umer of tertos. As for GS tertve method t e formed redug the SOR tertve method wth the vlue of. ω. Se SOR d GS tertve method re ssged s otrol methods ths stud thus ol the lgorthm for CG tertve method s preseted Algorthm. Algorthm : CG Sheme. Itlze x 0.. Compute the resdul r0 f Ax0 d hoose dreto of p0 r0 fax0.. Ot the ew x r f Ax d the dreto p the ompute the ew estmte x + d ts resdul r+ usg the formuls r r α p Ap x+ x + α p r r α Ap +. v. Next fd the dreto of p + usg the formuls d repet step () r + r + p + r + β p where β r r v. Chek the overgee. If es go to step (v). Otherwse go k to step (). v. Dspl pproxmte solutos..4 Numerl performe lss he performe lss of the u o-poloml sple pproxmto usg the three proposed pot tertve methods re vestgted respet of ther terto umers exeuto tme d mxmum solute error. he effetveess of the three methods s verfed odutg umerl test o the followg equto '' 4 4 osh() x [0] (5) where the ext soluto for prolem (5) s gve s ( x) osh(x ) osh(). he the results for the performe lss hve ee tulted le. le. Comprso of tertos umer exeuto tme d mxmum solute error for GS SOR d CG tertve method Numer of Itertos m GS SOR CG Exeuto me (Seod) GS SOR CG Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 90

7 H. Juste d J. Sulm GS SOR CG Mxmum Asolute Error e e e e-06.00e e-06.45e e e e e-06.4e e-07.5e e-08. Coluso he pproxmte solutos to the two-pot oudr vlue prolems hve ee formulted wth opoloml sple sheme d solved usg the three proposed methods. he summrzto wthdrw sed o le eles us to exme whh methods gve the most fvorle pproxmte soluto. Bsed o the performe lss le SOR tertve method performed etter term of umer of tertos exeuto tme d mxmum solute error ompred to the GS tertve method. But the whe omprg GS d SOR tertve methods wth the CG tertve method the performe of CG tertve method s foud to e superor respet of tertos umer exeuto tme d mxmum solute error o vrous grd szes. Referees Als E. L. & Hosks W. D. (969). Cu sple solutos to two-pot oudr vlue prolems. he omputer jourl () 5-5. Burgersetrum J. M. (0). Itertve soluto methods. Appled Numerl Mthemts 5(4) Che B. og L. & Gu Y. (006). Prese tme tegrto for ler two-pot oudr vlue prolems. Appled Mthemts d Computto 75() 8-. Evs D. J. (985). Group explt tertve methods for solvg lrge ler sstems. Itertol Jourl of Computer Mthemts 7() Fg Q. suh. & Ymmoto. (00). Fte dfferee fte elemet d fte volume methods ppled to two-pot oudr vlue prolems. Jourl of Computtol d Appled Mthemts 9() 9-9. Hkush W. (0). Itertve soluto of lrge sprse sstems of equtos (Vol. 95). Sprger See & Busess Med. Hestees M. R. & Stefel E. (95). Methods of ojugte grdets for solvg ler sstems (Vol. 49 p. ). NBS. Adullh A. R. & Irhm A. (995). Solvg the two-dmesol dffuso-oveto equto the four pot explt deoupled group (edg) tertve method. Itertol jourl of omputer mthemts 58(-) 6-7. Jg B. (008). wo-pot oudr vlue prolems the exteded Adom deomposto method. Jourl of Computtol d Appled Mthemts 9() 5-6. Kelle C.. (995). Itertve Methods for Ler d Noler Equtos (Froters Appled Mthemts vol 6) (Phldelph: SIAM). Mohse A. & El-Gmel M. (008). O the Glerk d olloto methods for two-pot oudr vlue prolems usg s ses. Computers & Mthemts wth Appltos 56(4) Rmd M. A. Lshe I. F. & Zhr W. K. (007). Poloml d opoloml sple pprohes to the umerl soluto of seod order oudr vlue prolems. Appled Mthemts d omputto 84() Sd Y. (996). Itertve methods for sprse ler sstems. PWS Bosto 60. H S. N. (00). A oler shootg method for two-pot oudr vlue prolems. Computers & Mthemts wth Appltos 4(0) Youg D. M. (04). Itertve soluto of lrge ler sstems. Elsever. Youg D. M. (97). Seod-degree tertve methods for the soluto of lrge ler sstems. Jourl of Approxmto heor 5() Yousf W. S. & Evs D. J. (995). Explt de-oupled group tertve methods d ther prllel mplemettos. Prllel Algorthms d Appltos 7(-) 5-7. Glol Jourl of Egeerg d eholog Revew () 85 9 (06) 9