Solving Third Order Boundary Value Problem Using. Fourth Order Block Method

Transcription

1 Applied Matematical Scieces, Vol. 7,, o. 5, HIKARI Ltd, Solvig Tird Order Boudar Value Problem Usig Fourt Order Block Metod Amad Sa Abdulla, *Zaaria Abdul Maid, ad Norazak Seu, Istitute or Matematical Researc Uiversiti Putra Malasia, Serdag, Malasia Departmet o Matematics, Facult o Sciece Uiversiti Putra Malasia, Serdag, Malasia *zaaria@sciece.upm.edu.m Coprigt Amad Sa Abdulla et al. Tis is a ope access article distributed uder te Creative Commos Attributio Licese, wic permits urestricted use, distributio, ad reproductio i a medium, provided te origial work is properl cited. Abstract A ourt order two-poit block metod is developed or solvig oliear tird order boudar value problems BVPs directl. Te two-poit block metod will solve te oliear tird order BVPs at two poits simultaeousl witi te block. Te sootig teciue will use te Newto s metod or ceckig o te coverget ad te guessig values or te ext block. Te proposed metod will be implemeted usig costat step size ad te PECE mode. Numerical examples is preseted to illustrate te applicabilit o te propose metod. Te results clearl sow tat te proposed block metod is able to produce good results compared to te existig metod. Matematics Subect Classiicatio: 65L6, 65L Kewords: Boudar value problem, Liear sootig metod, Block metod Itroductio Boudar value problems BVPs maiest temselves i ma braces o sciece. Some o tem are i te ield o egieerig, tecolog ad optimizatio

2 6 Amad Sa Abdulla et al teor. Sice te boudar value problem as wide applicatio i sciece researc, tereore aster ad accurate umerical solutio o boudar value problem are ver importace. I literature cotais several metods as bee proposed to solve BVPs. Logmai ad Amadiia [] use a tird degree B-splie uctio to costruct a approximate solutio or tird order liear ad oliear boudar value problems coupled wit te least suare metod. Quartic opolomial splie metod was proposed b El-Daa [7] or te umerical solutio o tird order two poit boudar value problems. El-Salam et al.[] are preseted secod ad ourt order coverget metods based o uartic opolomial splie uctio or te umerical solutio o a tird order two-poit boudar value problem. Wile Pag et al. [5] ad solved secod order boudar value problem usig two step direct metod b sootig teciue. Te ourt order two poit block metod also use sootig teciue to solve te boudar value problem directl. I tis paper, we propose a ourt order block metod or solvig boudar value problems o te orm as ollows wit boudar coditios,, a x b a, ' a, ' b were a, b,,, are te give costat. Te Newto metod will be implemeted as te iterative metod to estimate te guessig values. Te advatage o tese metods is to solve BVPs witout reduce it to te sstem o irst order ordiar dieretial euatios ODEs. Te give euatios i will be treated i teir origial secod order orm ad tereore te reuiremet o te storage is lower. Formulatio o te metod Figure : Two-Poit Block Metod

3 Solvig tird order boudar value problem 6 Te iterval [a, b] is divided ito a series o blocks wit eac block cotaiig two poits as sow i Figure. Two value poits will be oud simultaeousl usig te same back value i.e. ad.te poit at x ca be obtaied b itegratig E. over te iterval [ x, x ] oce, twice ad trice tat sow i E. 5: Itegrate oce: x x Itegrate twice: x x dx,, dx x x x x Itegrate trice: x x x x x x x x x x dxdx, ', dxdx x x x x x x dxdxdx, ', dxdxdx. 5 x x x Te same process will be applied to id te secod poit x oce, twice ad trice gives, itegrated over te iterval [ ] Itegrate oce: x x Itegrate twice: x x x, x. E. will be x dx, ', dx 6 x x Itegrate trice: x x x x x x x x x x dxdx, ', dxdx 7 x x x x x x dxdxdx, ', dxdxdx 8 x x x

4 6 Amad Sa Abdulla et al Te uctio, ', i E. 8 will be approximated usig Lagrage iterpolatig polomial, P as i E. 9. Te iterpolatio poits ivolved are x,,,, ad x, we will obtai te Lagrage iterpolatig polomial: x - x - x - x x - x P x - x x - x x - x - - x - x - x - x x - x 9 x - x x - x x - x x x x - x - - x - x - - x - x x - x x x - - x x - x - x x - x x x - x x - x - - x x x Takig s ad replacig dx ds, cagig te limit o itegratio rom - to - or E. 5 ca be writte as: P ds s P ds s P ds! ad rom - to or E. 6 8 ca be writte as: P ds -

5 Solvig tird order boudar value problem 6 sp ds s P ds! 5 Evaluate E. 5 usig MAPLE ad te corrector ormulae ca be obtaied. Te metod is te combiatio o predictor ad corrector. Te predictor is oe order less ta te corrector. Te same process as above ca be applied to id te predictor ormulae. Predictor First poit: '' '' 5 6 ' ' '' 9 6 ' '' 7 57 Secod poit: '' '' 7 9 ' ' '' 7 ' '' Corrector First poit:

6 6 Amad Sa Abdulla et al '' '' '' ' ' '' ' Secod poit: '' '' '' ' ' 9 5 '' '. Te ormulae 8 ad 9 ma be rewritte i te orm o matrix dierece euatio as ollows:

7 Solvig tird order boudar value problem Te order o tis developed metod is idetiied b reerrig to Maid et al. [8]. Te two-poit block metod or ODEs ca be writte i a matrix dierece euatio as ollows: Y m Y m Y m δf, m were,,, ad δ are te coeiciets wit te m-vector Y m, Y m, Y m ad deied as F m be [,,,, ] T, [,,,, ] T Y m, Y m [,,,, ] T, [ F,F,F,F,F ] T Y m F m B applig te ormulae or te costats C, i Fatula [6], te ormulae is deied as k C, k C,

8 66 Amad Sa Abdulla et al k C!, δ k C!!, M δ k C!!!!, were K,5,6, Tereore, te order ad error costat o te two-poit block metod ca be obtaied b usig. For,! C. For,!! C. 5

9 Solvig tird order boudar value problem 67 For,!!! C 6. For, δ!!!! C 7 6 δ δ δ δ δ

10 68 Amad Sa Abdulla et al For,

11 Solvig tird order boudar value problem 69 δ!!!! C. 8 For 5, δ 5 5!!! 5! C. 9 For 6, δ 5 6 6!! 5! 6! C. For 7,

12 6 Amad Sa Abdulla et al C7 7! 6! 5!! δ Te metod is order p i C C K ad C is te error costat. C p p Tus, we coclude tat te metod i 6 to 9 is o order ad te error costat is C p C T Te startig iitial poits i te code will cosider te direct Adams- Basort explicit metod o oe-step to compute te startig values, but te metod will solved te problem directl witout reducig to irst order euatios. Te iitial poits will be compute ol oce at te begiig o te step. Te, te iitial poits will be used or startig te predictor ad corrector block metod. Te predictor ad corrector block metod ca be applied util te ed o iterval. Tis block metod will be adapted or solvig te boudar value problems via sootig teciues. Sootig teciue will allow or ew guessig ad or eac ew guessig o te direct Adams-Basort explicit metod will be used agai to id te startig iitial poits. I order to get better approximatio or te iitial poits we usig te direct Adams-Basort explicit metod, te value o will be reduced to. Wile te predictor ad corrector block metod will remai usig te 8 coosig step size.

13 Solvig tird order boudar value problem 6 Implemetatio o te metod Sootig teciue will be applied i te propose block metod. Te code will start wit te iitial guess, tat determies solutio o te derivative gives,, ', a x b a, ' a, ' b, a, t t. Dieretiate E. wit respect to t, ad it is simpliied ollows: ''' t t, ' t, '' t t x t, ' t, '' t t, ' t, '' t t x ' '' t, ' t, ' ' t t t, ' t, ' ' t t ' '' ' t, ' t, '' t t t, ' t, ' ' t t ' '' t, ' t, '' t t. ' ' Usig z t to deote / t, we ave te iitial-value problem z, ', z, ', z', ', z a x b, 5 ' ' ' z a, z ' a, z a. 6 For te irst iitial guessig, we cosidered t. 7 b a See Faires ad Burde 998. Te solutio o ' rom E. 9 is determied we, ϕ t ' b, t. 8

14 6 Amad Sa Abdulla et al Newto metod will be used to get a ver rapidl covergig iteratio. We compute t deied as: te { } k ϕ t tk tk. ϕ' t 9 From E. ad 5, we ma obtai te solutio or ' b, tk ad z ' b, tk respectivel. Te solutios were applied i Newto s metod to id te ext guess, t. k ' b, t k t k tk. z' b, tk Bot E. ad E. 5 will be solved simultaeousl usig te block metod. Te process will stop util te error ' b, t k tolerace, were tolerace 5. Te algoritm o te proposed metod were developed i C laguage. Results ad discussio We ow cosider tree umerical example illustratig te comparative perormace o te propose metod over oter existig metods. All calculatios are implemeted b Microsot Visual C 6.. I problem ad, we obtaied te maximum errors at dieret values o step size i.e.,, ad. Te maximum errors were compared wit Al-Said ad Noor []. For problem, te results were compared wit El-Daa [7]. Problem : x x x x 5x e, x,, ', ' e Exact solutio: x x x e Source: El-Salam et al.. x

15 Solvig tird order boudar value problem 6 Problem : x si x x cos x,, ', ' si Exact solutio: x x x si x Source: El-Salam et al.. Problem : 7 x cos x x 6x cos x,, ', ' si Exact solutio: x x si x Source: El-Daa 8. Te ollowig otatios are used i te tables: FOBM Fourt order block metod proposed i tis researc Step size TS Total step Table : Compariso o maximum errors or Problem at dieret values o FOBM TS Al-Said ad TS Noor[] x x x x - 6. x x x x -5 8

16 6 Amad Sa Abdulla et al Table : Compariso o maximum errors or Problem at dieret values o FOBM TS Al-Said ad TS Noor[] 6.85 x x x x x x x x -7 8 Table : Compariso o maximum errors or Problem at dieret values o FOBM TS El-Daa [7] TS -.7 x x x x x x x x x x -9 5 I Table ad sow te maximum errors o FOBM are better compared to te results i Al-Said ad Noor []. Te results eve better as te step size decrease. I Table, te maximum errors or bot metods are comparable, but te result o FOBM is better ta te result i El-Daa [7] we - ad -7. Te FOBM ol eed al total steps compared to te oter two metods because FOBM will solved te problem at two steps simultaeousl i a block. 5 Coclusio I tis researc, we coclude tat ourt order block metod wit sootig teciue usig costat step size is suitable to solve tird order oliear boudar value problems. Tis proposed metod is simple, eiciet ad ecoomicall.

17 Solvig tird order boudar value problem 65 Ackowledgemet Te autor grateull ackowledged te iacial support o Graduate Researc Fud GRF rom Uiversiti Putra Malasia ad MMaster rom te Miistr o Higer Educatio. REFERENCES [] D. Faires ad R.L. Burde, Numerical Metods. d Ed. Paciic Grove: Iteratioal Tomso Publisig Ic, 998. [] E.A. Al-Said ad M.A Noor, Numerical solutios o tird-order sstem o boudar value problems. Applied Matematics ad Computatio, 9, 7, pp [] F.A. Abd El-Salam, A.A. El-Sabbag, ad Z.A. Zaki, Te Numerical Solutio o Liear Tird Order Boudar Value Problems usig Nopolomial Splie Teciue. Joural o America Sciece, 6,, pp. -9. [] G. B. Logmai ad M. Amadiia, Numerical Solutio o Tird-order Boudar Value Problems, Iraia Joural o Sciece & Tecolog, Tra. A, Volume, Number A, 6, pp [5] P. S. Pag, Z. A. Maid ad M. Suleima, Solvig Noliear Two Poit Boudar Value Problem usig Two Step Direct Metod. Joural o Qualit Measuremet ad Aalsis, 7,, pp. 9-. [6] S. O. Fatula, Block metods or secod order ODEs. Iteratioal Joural o Computer Matematics, vol., 99, pp [7] S. Talaat El-Daa, Quartic Nopolomial Splie Solutios or Tird Order Two-Poit Boudar Value Problem. World Academ o Sciece, Egieerig ad Tecolog, 5, 8, pp [8] Z. A. Maid N. Z. Moktar ad M. Suleima, Direct Two-Poit Block Oe- Step Metod or Solvig Geeral Secod-Order Ordiar Dieretial Euatios. Matematical Problems i Egieerig,, pp. -6. Received: Februar,