Solving third order boundary value problem with fifth order block method

Transcription

1 Matematical Metods i Egieerig ad Ecoomics Solvig tird order boudary value problem wit it order bloc metod A. S. Abdulla, Z. A. Majid, ad N. Seu Abstract We develop a it order two poit bloc metod or te umerical solutio o oliear boudary value problems (BVPs) directly. Most o te existece researc ivolvig BVPs will reduce te problem to a system o irst order Ordiary Dieretial Equatios (ODEs). However, te proposed metod will solve te tird order BVPs directly witout reducig to irst order ODEs. Tese metods will solve te oliear tird order BVPs by sootig tecique usig costat step size. Numerical example is preseted to illustrate te applicability o te propose metod. Te results clearly sow tat te propose metod is able to solve boudary value problems (BVPs). Keywords Boudary value problem, sootig tecique, two poit bloc metod. I. INTRODUCTION OUNDARY value problems (BVPs) are used i may Bbraces o sciece. Some o tem are i te ield o optimizatio teor egieerig ad tecology. Sice te boudary value problem as wide applicatio i sciece researc, tereore aster ad accurate umerical solutio o boudary value problem are very importace. Tereore, it as may boudary value problems (BVPs) solutio tecique is proposed. I literature cotais several metods as bee proposed to solve BVPs.Logmai ad Amadiia (006) use a tird degree B-splie uctio to costruct a approximate solutio or tird order liear ad oliear boudary value problems coupled wit te least square metod. Quartic opolyomial splie metod was proposed by El- Daa(008) or te umerical solutio o tird order two poit boudary value problems. El-Salam et al.( 00) are preseted secod ad ourt order coverget metods based o Quartic opolyomial splie uctio or te umerical solutio o a tird order two-poit boudary value problem. Wile Pag et al. (0) ad solved secod order boudary value problem usig two step direct metod by sootig tecique. Te it order two poit bloc metod also use sootig tecique to solve te boudary value problem Te autor grateully acowledged te iacial support o Graduate Researc Fud (GRF) rom Uiversiti Putra Malaysia ad MyMaster rom te Miistry o Higer Educatio. A. S. Abdulla is wit te Istitute or Matematical Researc, Uiversity Putra Malaysia, 00 Serdag, Malaysia. ( a_sa@lyaoo. com). Z. A. Majid is wit te Istitute or Matematical Researc, Uiversity Putra Malaysia, 00 Serdag, Malaysia. (poe: ; ax: ; am_zaa@upm.edu.my). N. Seu is wit te Departmet o Matematics, Faculty o Sciece, Uiversity Putra Malaysia, 00 Serdag, Malaysia. ( oraza@upm.edu.my). directly. I tis paper, we propose a it order bloc metod or solvig boudary value problems o te orm as ollows = ( xyy,,, ), a x b () wit boudary coditios y( a) = γ, y '( a) = α, y '( b) = β () were a,b,α,β,γ are te give costat. Te guessig values estimated by implemet te Newto metod. Te advatage o tese metods is to solve BVPs witout reduce it to te system o irst order ordiary dieretial equatios (ODEs). II. FORMULATION OF THE METHOD I tis researc, te direct metod o multistep metod is developed or te umerical solutio o oliear boudary value problems (BVPs) directly. Fig : Two Poit Direct Bloc Metod Te iterval [a,b] is divided ito a series o blocs wit eac bloc cotaiig two poits as sow i Fig.. Two value poits will be oud simultaeously usig te same bac value i.e. y ad y.te poit y at x ca be obtaied by itegratig Eq. () over te iterval [, x ] oce, twice ad trice tat sow i Eq. -5: Itegrate oce: dx = ( x, y, ) dx () Itegrate twice: ISBN:

2 Matematical Metods i Egieerig ad Ecoomics x x dxdx = ( x, ) dxdx () Itegrate trice: x x x x dxdxdx = ( x, ) dxdxdx. (5) Te same process will be applied to id te secod poit x, y Eq. () will be itegrated over te iterval [ ]. oce, twice ad trice gives, Itegrate oce: x dx = ( x, ) dx (6) Itegrate twice: x x dxdx = ( x, ) dxdx (7) Itegrate trice: x x x x dxdxdx = ( x, ) dxdxdx (8) x Taig s = ad replacig dx = ds, cagig te limit o itegratio rom - to - or Eq. -5 ca be writte as: = ) y y = ( s )! P ds (0) y y = ( s P ds () y P ds () ad rom - to 0 or Eq. 6-8 ca be writte as: 0 = P ds () 0 y y = sp ds () y 0 y y = ( s)! P ds (5) Evaluate tese itegral usig MAPLE ad te corrector ormulae ca be obtaied. Te metod is te combiatio o predictor oe order less ta te corrector. Te same process is applied to id te predictor ormulae. Te uctio ( x, y ) i Eq. -8 will be approximated usig Lagrage iterpolatig polyomial, P. Te iterpolatio poits ivolved are (, ), (, ), (, ), ( x, ) ad x, ) ( iterpolatig polyomial: P = we will obtai te Lagrage ( 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 ) (9) Fit Order Bloc Metod: Predictor: = ( ) y y = ( ) y y y = ( ) = ( 8 7 ) y y = ( ) ( ) y y y = ( ) (6) (7) ISBN:

3 Matematical Metods i Egieerig ad Ecoomics Corrector: '' '' y y = ( ) ' ' '' y y y = ( ) ' '' y y y y = ( ) '' '' y y = ( 9 ) ' ' '' y y y = ( ) ' ( ) '' y y y y = ( ) (8) (9) For te begiig, te direct Adams Basord metod will be used to calculate te startig iitial poits. Te, te iitial poits we will be used or startig te predictor ad corrector direct metod. Te predictor ad corrector direct metod ca be applied util te ed o iterval. Te sootig tecique is used or solvig te boudary value problems. I order to get better approximatio or te iitial poits, te value o will be reduced to. However, te predictor ad corrector 8 direct metod will remai usig te coosig step size. III. IMPLEMENTATION OF THE METHOD Sootig tecique was applied i te propose metod ad it is a aalogy to procedure o irig objects at a statioary target. We start wit te iitial guess, tat determies te y a as te ollowig: solutio o te derivative ( ) = ( xyy,, ', ), a x b (0) y (a) = γ, y '( a) = α, y ( b ) = β, y ( a, = t0 () Dieretiate Eq. (0) wit respect to t, ad it is simpliied as ollows: ( xt, ) = ( x, yxt (, ), y'( xt, ), ( xt, )) x = ( x, yxt (, ), y ( xt, ), ( xt, )) x ( x, yxt (, ), y ( xt, ), y ( x, yxt (, ), y ( xt, ), y' ( x, yxt (,), y (,), xt (,)) xt (,) xt y '' = ( x, yxt (, ), y ( xt, ), y ( x, yxt (, ), y ( xt, ), ( x, yxt (, ), y ( xt, ),. Usig z ( x, to deote ( y / )( x,, we ave te iitialvalue problem z = ( xyy,, ', ) z ( xyy,, ', ) z' ' ( xyy,, ', ) z, a x b '' z ( a) = 0, z ( a) = 0, ( a) =. For te irst iitial guessig, () z () we cosidered β α t0 = () b a See Faires ad Burde (998). Te solutio o y' rom Eq. (9) is determied we, ϕ( t ) = y' β = 0 (5) Newto metod will be used to get a very rapidly covergig t deied as: iteratio. We compute te { } ϕ( t = t. (6) ϕ' ( ISBN:

4 Matematical Metods i Egieerig ad Ecoomics ' From Eq. (5), we ow = z( x,, so = z' ( x,, ad we id te solutio or y' t ) rom Eq. (9). Te solutios were applied i Newto s metod to id te ext guess, t x x0-5. x x0-5 t = t y' t ) β. (7) z' t ) Bot Eq. (0) ad Eq. () will be solved simultaeously usig te direct metod. Te process will stop util te error β y'( b, t ) tolerace, were tolerace =0 5. Te algoritm o te proposed metod was developed i C laguage. IV. RESULT AND DISCUSSION We ow cosider tree umerical example illustratig te comparative perormace o te propose metod over oter existig metods. All calculatios are implemeted by Microsot Visual C 6.0. Notatio: Step size Fit order bloc metod Problem : x y = xy ( x x 5x ) e, 0 x, y ( 0) = 0, y' (0) =, y' () = e x Exact solutio: y ( x) = x( x) e Source: El-Salam et al. (00). Problem : y = y ( x )si x ( x) cos x, 0 x, y ( 0) = 0, y' (0) =, y '() = si Exact solutio: y( x) = x( x ) si x Source: El-Salam et al. (00). Problem : y = y (7 x ) cos x ( x 6x ) cos x, 0 x, y ( 0) = 0, y' (0) =, y '() = si Exact solutio: y( x) = ( x ) si x Source: El-Daa (008). Table : Te observed maximum errors or Problem. F (Al-Said ad Noor, 007) x x x0-6.8 x0 - Table :Te observed maximum errors or Problem. (Al-Said ad Noor, 007) x x x x x x x x0-7 Table :Te observed maximum errors or Problem. (El-Daa, 008) x x x x x x x x x x0-9 I problem ad, te maximum errors will be obtaied we te step size, =,, ad. Te maximum errors were compared wit Al-Said ad Noor (007). For problem, te results were compared wit El-Daa (008). Table ad sow te maximum errors or are better ta te results i Al-Said ad Noor (007). I Table, te maximum errors or bot metods are comparable. Te results are more precise we te umber o is reduced. V. CONCLUSION I tis researc, we coclude tat it order bloc metod wit sootig tecique usig costat step size is suitable to solve tird order oliear boudary value problems directly. ACKNOWLEDGMENT Te autor grateully acowledged te iacial support o Graduate Researc Fud (GRF) rom Uiversiti Putra Malaysia ad MyMaster rom te Miistry o Higer Educatio. ISBN:

5 Matematical Metods i Egieerig ad Ecoomics REFERENCES [] G. B. Logmai ad M. Amadiia, Numerical Solutio o Tird-order Boudary Value Problems, Iraia Joural o Sciece & Tecolog Tra. A, Volume 0, Number A, 006, pp [] S. Talaat El-Daa, Quartic Nopolyomial Splie Solutios or Tird Order Two-Poit Boudary Value Problem, World Academy o Sciece, Egieerig ad Tecolog 5, 008, pp [] F.A. Abd El-Salam, A.A. El-Sabbag, ad Z.A. Zai, Te Numerical Solutio o Liear Tird Order Boudary Value Problems usig Nopolyomial Splie Tecique, Joural o America Sciece, 6(),00,pp [] E.A. Al-Said ad M.A Noor, Numerical solutios o tird-order system o boudary value problems, Applied matematics ad computatio,, 007, pp.- 8. [5] P. S. Pag, Z. A. Majid ad M. Suleima, Solvig Noliear Two Poit Boudary Value Problem usig Two Step Direct Metod, Joural o Quality Measuremet ad Aalysis, 7(), 0, pp [6] Z. A. Majid N. Z. Motar ad M. Suleima, Direct Two-Poit Bloc Oe-Step Metod or Solvig Geeral Secod-Order Ordiary Dieretial Equatios, Matematical Problems i Egieerig, 0, pp. -6. [7] S. O. Fatula, Bloc metods or secod order ODEs, Iteratioal Joural o Computer Matematics, vol., 99, pp [8] D. Faires ad R.L. Burde, Numerical Metods. d Ed. Paciic Grove: Iteratioal Tomso Publisig Ic, 998. ISBN: