A New Operational Matrix of Orthonormal Bernstein Polynomials and Its Applications

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1 Global Journal of ur and Applid Mathati. ISSN 9-68 Volu Nubr 6 pp Rarh India ubliation A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation Abdlri Bnhih * Lahdar Chitr and Abbai Hoin * Dpartnt of Mathati Kadi Mrbah Univrit of Ouargla - Algria. Dpartnt of Mathati Frhat-Abba Univrit Stif Algria. Dpartnt of Mathati Univrit of Ouargla Algria. Corrponding author: Abtrat In thi wor w introdu a nw gnral produr of finding oprational atri of intgration diffrntiation D and produt Ĉ for th orthonoral Brntin polnoial in th a wa a in [] S. A. oufi M. Bhroozifar Oprational atri of Brntin polnoial and thir appliation Int. J. St. Si A an appliation th atri an b ud to olv intgral and intgro-diffrntial quation diffrntial quation and in o probl in th alulu of variation and optial ontrol. h ffiin of thi approah i hown b appling thi produr on o illutrativ apl. Kword: Orthonoral Brntin polnoial Oprational Matri Intgro-diffrntial quation Diffrntial quation. MSC: 4K8 4G 4D.. Introdution In th lat thr dad approiation b orthonoral fail of funtion hav plad a vital rol in th dvlopnt of phial in nginring and thnolog in gnral and athatial anali. h hav bn plaing an iportant part in th valuation of nw thniqu to olv probl uh a idntifiation anali and optial ontrol. h ai of th thniqu i to obtain fftiv algorith that ar uitabl for th digital oputr. h otivation and philooph bhind thi approah i that it tranfor th undrling diffrntial quation of th probl to an algbrai quation thu iplifing th olution pro of th probl to a grat tnt. h Brntin polnoial ar not

2 4 Abdlri Bnhih Lahdar Chitr and Abbai Hoin orthonoral o thir u in th lat quar approiation ar liitd. Alo to th bt of our nowldg th oprational atri for orthonoral Brntin polnoial for hort wa not invtigatd. o ovro thi diffiult Gra-Shidt orthonoralization pro an b ud to ontrut th orthonoral Brntin polnoial. In thi papr a gnral produr of foring th oprational atri of intgration. diffrntiation D and produt Ĉ for th orthonoral Brntin polnoial ar givn t dt d D d Cˆ whr.. atri D and C ar of ordr. and i an arbitrar vtor and th Spial attntion ha bn givn to appliation of Walh funtion [6] blo-pul funtion [] [6] and [] Lagurr polnoial [] Lgndr polnoial [4][9] Chbhv polnoial [] [4] alor ri [] and Fourir ri [] []. hi papr i truturd a follow. In Stion and w drib th bai forulation of orthonoral Brntin polnoial and panion of in tr of alor bai. In Stion 4 to 6 w plain gnral produr of oprational atri of intgration diffrntiation and produt rptivl. In Stion w dontrat th validit th aura and th appliabilit of th oprational atri b onidring o nurial apl. In Stion 8 w onlud.. Orthonoral Brntin polnoial h pliit rprntation of th orthonoral Brntin polnoial of th dgr ar dfind on th intrval [ ] in [8] b In addition an b writtn in a iplr for in tr of original non-orthonoral Brntin bai funtion a in [8] B h polnoial atif th following orthogonalit rlation

3 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 4 i t t dt i i whr i th Kronr dlta funtion. For apl with w hav i Figur : Orthonoral Brntin polnoial with n=8 A funtion L a b pandd b Brntin orthonoral polnoial a follow [] [6] blong to th pa hr if < : > b th tandard innr produt on L thn B trunating th ri 4 up to + th tr w an obtain an approiation for a follow 4 6

4 4 Abdlri Bnhih Lahdar Chitr and Abbai Hoin Whr and..... ail n that th lnt of in th [. ] ar orthogonal. It an b. Epanion of in tr of alor bai B uing and w hav : whr i. i r i r r in i i r r i i a i r r i i i i i i i i i 8 9 Equation an b diplad in th following atri for whr M i i i i i in M i and a. For apl with = w hav M

5 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 4 4. oprational atri of intgration oprational atri of intgration thn Lt b an t dt B w hav t dt M MX atri: whr i Now w approiat th lnt of vtor X in tr of b w hav M Whr M i + th row of So w ut nd to approiat Whr thn for. and X M M for that i M M M M b uing w hav dt 4 t t

6 44 Abdlri Bnhih Lahdar Chitr and Abbai Hoin M M M M and thn X. Lt B w hav M M t dt MB and thrfor w hav th oprational atri of intgration a MB For = th atri i dnotd b and i givn a follow: oprational atri of Drivativ In thi tion w want to driv an pliit forula for orthonoral Brntin polnoial of -th-dgr oprational atri of diffrntiation. Suppo that D i an oprational atri of diffrntiation thn d D whr 6 d M and thn Fro w hav d d M MX

7 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 4 whr i atri Now w pand vtor X in tr of and B uing w an writ X B whr M M B M X hu d MX d and thrfor w hav th oprational atri of diffrntiation a D MX For apl with = w hav D oprational atri of produt In thi tion w want to driv an pliit forula for orthonoral Brntin polnoial of th dgr oprational atri of produt. Suppo that i an arbitrar atri thn produt whnvr C i an oprational atri of C 8

8 46 Abdlri Bnhih Lahdar Chitr and Abbai Hoin B and in w hav M M M 9 Now w approiat all funtion in tr of. Lt ; b w hav. hu w obtain i i ˆ E i i i. i. i whr Eˆ i B dfining atri Eˆ Eˆ Eˆ Eˆ.. Eˆ hav So C ˆ EM ˆ and inrting 9 into w t EM ˆ t. Appliation of th oprational atri of. Appliation to th Edn-Fowlr quation Conidr th Edn-Fowlr quation givn in Wazwaz [] b

9 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 4 r n ni with initial ondition W au that th unnown funtion i approiatd b Uing and th initial ondition w hav d h 4 d and h d Whr and b 4 and 8 w hav h h hh ˆ h hh ˆ hˆ h So b indution w n hˆ n h ni whr ĥ i oprational atri of produt. W an pr th funtion and r Subtituting and 6 in w obtain hˆ n h a 6 r Uing 8 w hav and Eˆ 8 Kˆ 9 Subtituting 9 and 8 in w gt ˆ E or ˆ ˆ ˆn E Kh h Kh ˆ ˆ n h Equation i a t of algbrai quation whih an b olvd for. Now w appl th abov prntd thod with and for olving in Equation with r and n whih ha th at olution.

10 48 Abdlri Bnhih Lahdar Chitr and Abbai Hoin In abl a oparion i ad btwn th approiat valu uing th prnt approah togthr with th at olution. W found th approiatd olution for = and = a follow: abl. Etiatd and at valu rnt thod with = = Eat olution : : : : : : : : : : Figur : Graph of at olution and approiat olution at = Figur : Graph of abolut rror for =

11 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 49. Appliation to th Linar Frdhol Intgro-Diffrntial Equation W onidr th following linar Frdhol intgro-diffrntial quation d f whr th funtion L f th rnl L F f thn K whr i i hn Subtituting into w hav F K K I K F K F d K F B olving th abov linar t w an find th vtor o or Now W onidr th following linar Frdhol intgro-diffrntial quation d with at olution W found th approiatd olution for = and = a follow abl. Approiat and at olution for apl. rnt thod with Eat olution = = d d

12 4 Abdlri Bnhih Lahdar Chitr and Abbai Hoin Figur 4: Graph of at olution and approiat olution at = Figur : Graph of abolut rror for = Conluion n thi papr w hav firt ontrutd orthonral polnoial of dgr n b appling Gra-Shidt orthonoralization pro on th Brntin polnoial B. hn w hav ud anothr diffrnt nurial produr to driv th oprational atri of intgration diffrntiation D and produt Ĉ. A gnral produr of foring th atri i givn. h atri an b ud to olv probl uh a th alulu of variation intgro-diffrntial quation diffrntial quation optial ontrol and intgral quation li that of othr bai. h thod i gnral a to iplnt and ild vr aurat rult. Morovr onl a all nubr of ba ar ndd to obtain a atifator rult. Nurial tratnt i inludd to dontrat th validit and appliabilit of th oprational atri. What i nw in thi wor bid it diffiult opard to Brntin polnoial B i that w an writ in tr of alor bai. Hr th advantag in th for of i th air oputation of th offiint opard to that of B. W hav tabilihd th gnral for of and oputd th oprational atri of drivation D and w hav givn th gnral for of th atri drivd fro th produt. h ffiin of thi thod i hown on apl uh a th Lan-Fowlr quation and on o intgro-diffrntial quation. h firt apl how th ffiint u of th atri b tranforing th probl to a t of algbrai quation whih ar a to olv. h rult obtaind ar vr pri opard to

13 A Nw Oprational Matri of Orthonoral Brntin olnoial and It Appliation 4 tho obtaind in []. Eapl how how it i air to olv intgral and intgrodiffrntial quation. Rfrn [] M. I. Bhatti. Bran Solution of diffrntial quation in a Brntin polnoial bai J. Coput. Appl. Math [] A. Db A. Dagupta G. Sarar A nw t of orthogonal funtion and it appliation to th anali of dnai t J. Fran. Int [] Z. H. Jiang W. Shauflbrgr Blo ul Funtion and hir Appliation in Control St Springr-Vrlag 99. [4] F. Khllat S. A. oufi h linar Lgndr wavlt oprational atri of intgration and it appliation J. Fran. Int [] F. C.Kung and H. L Solution and paratr tiation of linar ti invariant dla t uing Lagurr polnoial panion Journal on Dnai St Maurnt and Control [6] I. Lazaro J. Anzurz M. Roan M aratr tiation of linar t bad on walh ri h Eltroni Roboti and Autootiv Mhani Confrn [] M. H. Farahi M. Dadhah Solving Nonlinar i Dla Control St b Fourir ri Int. Journal of Enginring Rarh and Appliation Vol. 4 Iu [8] S. Javadi E. Babolian Z. ahri Solving gnralizd pantograph quation b hiftd orthonoral Brntin polnoial J. Coput. Appl. Math 6-4 [9] F. Marllan and W.V. Ah Orthogonal olnoial and Spial Funtion a Coputation and Appliation Springr-Vrlag Brlin Hidlbrg 6. [] H. R. Marzban and M. Shahiah Solution of piwi ontant dla t uing hbrid of blo- pul and Chbhv polnoial Opti. Contr. Appl. Mt. Vol. pp [] A. Rihard. A. Brnatz. Fourir ri and nurial thod for partial diffrntial quation. John Wil & Son In. Nw or. [] S. A. oufi M. Bhroozifar Oprational atri of Brntin polnoial and thir appliation Int. J. St. Si [] M. Szr and A. A. Daioglu alor polnoial olution of gnral linar diffrntial-diffrn quation with variabl offiint Appl. Math.Coput [4] M. Shaban and S. Kaz and J. A. Rad A odifiation of th hootop anali thod bad on Chbhv oprational atri Math. Coput. Modl in pr.. [] A. M. Wazwaz Adoian dopoition thod for a rliabl tratnt of th Edn-Fowlr quation Appl.Math.Coput

14 4 Abdlri Bnhih Lahdar Chitr and Abbai Hoin [6] X.. Wang Nurial olution of optial ontrol for ti dla t b hbrid of blo-pul funtion and Lgndr polnoial Applid Mathati and Coputation

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