On orthonormal Bernstein polynomial of order eight

Transcription

1 Oen Science Junal f Mathematic and Alicatin 2014; 22): Publihed nline Ail 20, 2014 htt:// On thnmal Bentein lynmial f de eight Suha N. Shihab, Tamaa N. Naif Alied Science Deatment, Univeity f Technlgy, Baghdad, Iaq adde alawy1978@yah.cm Suha N. Shihab), m197575@yah.cm Tamaa N. Naif) T cite thi aticle Suha N. Shihab, Tamaa N. Naif. On Othnmal Bentein Plynmial f Ode Eight, Oen Science Junal f Mathematic and Alicatin. Vl. 2, N. 2, 2014, Abtact In thi ae, we eent the new thnmal bae,, 0,1,,8 thugh the Gam-Schmidt Othnmalizatin ce n Bentein lynmial f de eight. Bentein lynmial and thei etie ae emlyed t deive exlicit fmula f deivative and integatin eatinal matice f thnmal Bentein lynmial f de eight. Cnvegence citeia i al included in thi ae. The elatinhi between the deivative f, and, themelve with me the imtant etie ae deived in thi wk. All the ed eult ae f diect inteet in many alicatin. Keywd Bentein Plynmial, Gam- Schmidt Othnmalizatin Pce, Oeatinal Matix f Deivative and Integatin 1. Intductin Bentein lynmial have many imtant etie and have been ued f lving diffeent blem uing me aximate methd. Bentein lynmial have been ued f lving Fedhlm Integal equatin [1], bunday value blem [2], diffeential equatin [3], nnlinea diffeential equatin with cllcatin methd [4], the Emden-Fwle equatin which i aiing in athyic [5] and the ee [6-10]. The Bentein lynmial ae nt thnmal thei ue in the leat quae aximatin ae limited. T vecme thi difficulty, Gam- Schmidt thnmalizatin ce can be ued t cntuct the thnmal Bentein lynmial. In thi ae, we fit cntuct an thnmal family, f lynmial f degee eight. Then thnmal Bentein eatinal matice f bth deivative and integatin ae deived. In additin, the cnvegence citeia f, with me imtant eult between, and, ae eented. 2. Bentein Plynmial and Thei Petie The Bentein bai lynmial f degee ae defined by [1], 1 1) By uing binmial exanin f 1, we can get, 1 2) Then, in Hilbet ace!0,1" i a cmlete bai. That i any lynmial f degee can be exanded in tem f linea cmbinatin f,,,,,,. Sue that the functin % &'!0,1", and ) *+, -,,,. If '. be the bet aximatin % ut f ) then [11] 2 /% '. 0!," 1, whee 6 2 max! 5 :&!," ;%.

2 16 Suha N. Shihab and Tamaa N. Naif: On Othnmal Bentein Plynmial f Ode Eight 3. Othnmal Bentein Plynmial f Ode Eight Given a et f linea indeendent functin,, 0,1,,8, if we define the inne duct and the nm % =%,%? One can deive the thnmal bae - thugh the Gam-Schmidt thnmalizatin ce a the fllwing whee we equie B, B / B, B, =B,? =,?D, =,? 0 Which yield, =,? 1 Hence,, =,? B =,?, B / B By inductin, ne ha whee G B E, F F, B / B F, F =, F?. The nmalized Bentein lynmial ae 17I1 60!8II1 G 1 2 I1 " 15 13!8II1 G D I1 D28I I1 O " ! II1G 2 13 I1 42I I1 O 56I 5 I1 P " Q 143R3! II1G D I1 D I I1 O D112I 5 I1 P D70I Q I1 Q " P 99 7! II1G 7 99 I I I1 O 1400I 5 I1 P 175I Q I1 Q 56I P I1 5 " O !4II1G D 2 33 I1 D50I I1 O D200I 5 I1 P D300I Q I1 Q D168I P I1 5 D28I O I1 " G 15 3T 14 3 II1G 8I G I I1 63I I1 O 280I 5 I1 P 490I Q I1 Q IP I1 5 98I O I1 U 9T8II1 G D32I G I1D 1 9 I1 D112I I1 O D I5 I1 P D980I Q I1 Q D784I P I1 5 D IO I1 DI U The cefficient f t WX t1 X f the nmalized Bentein lynmial f de eight ae lited in table 1). Table 1). Cefficient f I I1 f the nmalized Bentein lynmial f de eight. Y Z Y [ Y\ Y ] Y\^ Y _ Y\` Y a Y\ a Y`Y\ _ Y^Y\ ] YY\ [ Y\ Z G e e e e e O e e e e e P e e e e Q e Cnvegence Citeia f bc d If the functin % i exanded in tem f thnmal Bentein lynmial e % % 3) It i nt ible t efm cmutatin an infinite numbe f tem, theefe, the eie in 3) mut be tuncated. In lace f 3) we take: % % 4) S that %% D % %% f 5)

3 Oen Science Junal f Mathematic and Alicatin 2014; 22): whee f g % The cefficient in 4) and 5) mut be elected uch that the nm f the eidual functin f i le than me cnvegence citeia &, that i ; f ;=& We have Then % E% " A % " A % "! E % "A E E % % F A F A h 1 % i 0 % ji f E % 6) 5. Exanin f bc Z in tem f c Z Thee i a elatin between and in the fm k 1 2 l k 4 D 4 15 l k D l Q 143R3 k 14 Q 2 5 D D l P 99 7 k P 5 2 QD D l O k O3 P D 30 7 Q D D 2 33 l G 15 3k G 3.5 O D6.3 P 7 Q D D l 9k 4 G D 28 3 O14 P D14 Q D4 D 1 9 l The abve equatin can be ewitten in matix fm a m 7) whee! 5 Q P O G ".! 5 Q P O G ". E % =& m n q 6. Oeatinal Matix f Deivative f bc Z In thi ectin, we eent a geneal fmula f finding the eatinal matix f deivative f 8 th degee thnmal Bentein lynmial. Let t be an 8u8 eatinal matix f the deivative, then v t wx 8) The fit deivative f yz degee Bentein bai lynmial can be witten a a linea cmbinatin f Bentein bai lynmial f degee,f 8 we have : v t x 9)

4 18 Suha N. Shihab and Tamaa N. Naif: On Othnmal Bentein Plynmial f Ode Eight whee v= [v v v vg v ". = [ G ". and t x i the eatinal matix f the deivative f 8 th degee Bentein lynmial given by t wx t x n q By uing eq.9), we have 9u9 matix t wx named eatinal matix f deivative f the 8 th degee thnmal Bentein lynmial and can be btained a n q 7. Oeatinal Matix f Integatin f bc Z Let { wx be an 9u9 eatinal matix f integatin, { wx, whee the 9u9 matix { wx i the eatinal matix f integatin f 8 th degee thnmal Bentein lynmial n the inteval!0,1", and can be witten a : { wx P Q O P Q O P Q n O P Q Q q 8. Cncluin In thi tudy, at fit, Othnmal Bentein lynmial f degee eight wee btained. Then me fmula wee demntated the elatin between and Bentein lynmial f degee eight.we deived the eatinal matice f bth deivative and integatin. The given eult will be ued t lve timal cntl blem thughut next wk. [3] Aximate Slutin f diffeential equatin by uing the Bentein lynmial, Odkhani Y. & Davaeifa S.,ISRN Alied mathematic vlume [4] Daciglu A.A. & Ile N., Bentein lynmial cllcatin methd f lving Nnlinea Diffeential equatin Mathematical and cmutatinal alicatin, vl. 18, N.3, , [5] Paand K. & Hayni S.A., The alicatin f the exact eatinal matice f lving the Gmden Flwe equatin, aiing in athyic, Januay 3, Refeence [1] Maleknejad K. & Mheny Zadeh M., Hybid thnmal Bentein and Blck ule functin f lving Fedhlm integal equatin, WCE vl.1 July 3-5, 2013, Lndn, U.K. [2] Agentini G., numeical elutin f me BUP uing Bentein lynmial. [6] Maleknejad K. & Baiat B., A New Methd Baed n eatinal matice f Bentein lynmial f Nnlinea Integal Equatin. [7] Duha E. H.,Bhawy A. H.,On the Deivative f Bentein lynmial : An Alicatin f the lutin f High Even de Diffeential Equatin, Hindawi ublihing catin,2011,bunday value blem a inge en junal.

5 Oen Science Junal f Mathematic and Alicatin 2014; 22): [8] Qian W. & Riedel M.D., Unifm aximatin and Bentein lynmial with cefficient in the unit inteval, Euean Junal f cmbinatin ), [9] Dixit S. &Singh V. K., Bentein Diect methd f lving Vaitinal Pblem, Intenatinal Mathematical Fum, N.48, , [10] N. Mikv and B. Ranu, A Bentein Plynmial cllcatin Methd f the Slutin f Ellitic Bunday value Pblem. axiv : VI {math. NA} 15,Nv,2012. [11] Aliu M. & Rtamy D., Bentein lynmial f lving Abel' integal equatin, junal f Mathematic and cmute cience TJMCS vl.3, N ),