The Bernstein Operational Matrix of Integration

Transcription

1 Appled Mahemacal Sceces, Vol. 3, 29, o. 49, he Berse Operaoal Marx of Iegrao Am K. Sgh, Vee K. Sgh, Om P. Sgh Deparme of Appled Mahemacs Isue of echology, Baaras Hdu Uversy Varaas -225, Ida Asrac A accurae mehod s proposed o solve prolems such as defcao, aalyss ad opmal corol usg he Berse orhoormal polyomals operaoal marx of egrao. he Berse polyomals are frs orhogoalzed, ormalzed ad he her operaoal marx of egrao s oaed. A example s gve o llusrae he proposed mehod. Mahemacs Sujec Classfcao: 4A, 49J5 Keywords: he Berse polyomals; Operaoal marx; Dffereal equao. Iroduco Approxmaos y orhoormal famly of fucos have played a val role he developme of physcal sceces, egeerg ad echology geeral ad mahemacal aalyss parcular sce log. I he las hree decades, hey have ee playg a mpora par he evaluao of ew echques o solve prolems such as defcao, aalyss ad opmal corol. he am of hese echques has ee o oa effecve algorhms ha are suale for he dgal compuers. he movao ad phlosophy ehd hs approach s ha rasforms he uderlyg dffereal equao of he prolem o a algerac equao, hus smplfyg he soluo process of he prolem o a grea exe. he asc dea of hs echque s as follows: Correspodg Auhor. E-mal : sghom@gmal.com (O. P. Sgh), vkshu@gmal.com (V. K. Sgh).

2 2428 A. K. Sgh, V. K. Sgh, O. P. Sgh () he dffereal equao s covered o a egral equao va mulple egrao. () Susequely, he varous sgals volved he egral equao are approxmaed y represeg hem as lear comaos of he orhoormal ass fucos ad rucag hem a opmal levels. () Fally, he egral equao s covered o a algerac equao y roducg he operaoal marx of egrao of he ass fucos. he key dea of he echque depeds o he followg egral propery of he ass vecor ϕ () a ϕ m+ a k k L ( σ )( dσ ) P ϕ( ), () where ϕ( ) = [ ϕ ( ), ϕ( ), K, ϕ m ( )] whch he elemes ϕ ( ), ϕ( ), K, ϕ m ( ) are he ass fucos, orhogoal o a cera erval [ a, ] ad P m+ s he operaoal marx for egrao of ϕ (). Noe ha P m+ s a cosa marx of order ( m + ) ( m + ). Usg he operaoal marx of a orhoormal sysem of fucos o perform egrao for solvg, defyg ad opmzg a lear dyamc sysem has several advaages: () he mehod s compuer oreed, hus solvg hgher order dffereal equao ecomes a maer of dmeso creasg, () he soluo s a mul-resoluo ype ad () he soluo s coverge, eve hough he sze of creme may e large. Ul ow, he operaoal marx of egrao has ee deermed for several ypes of orhogoal ass fucos, such as he Walsh fuco [-2], lock- pulse fuco [3-4], Laguerre seres [5-7], Cheyshev polyomals [8-9], Legedre polyomals [-], Fourer seres [2-3] ad Bessel seres [4]. Laer Gu ad Jag [5] derved he Haar waveles operaoal marx of egrao followed y Razzagh ad Yousef [6] who gave he Legedre waveles operaoal marx of egrao. he am of prese paper s o derve he Berse orhoormal polyomals marx of egrao P m+. he marx P m+ may e used o solve prolems of sysem aalyss ad syhess a maer smlar o hose of he oher orhogoal fucos. he Berse polyomals are frs orhoormalzed ad he operaoal marx of egrao s he derved. A umercal example s gve o llusrae he effcecy of he proposed mehod.

3 Berse operaoal marx of egrao he Berse polyomals: A Berse polyomal, amed afer Serge Naaovch Berse, s a polyomal he Berse form, ha s a lear comao of Berse ass polyomals. he Berse ass polyomals of degree are defed y B, ( ) = ( ), for =,, 2, L,. (2) h here are ( + ) degree Berse ass polyomals formg a ass for he lear space V cossg of all polyomals of degree less ha or equal o R[x]-he rg of polyomals over he feld R. For mahemacal coveece, we usually se B, = f < or >. Ay polyomal B (x) R[x] may e wre as B x) = β B ( x). (3) (, = he B (x) s called a polyomal Berse form or Berse polyomal of degree. he coeffces β are called Berse or Bezer coeffces.bu several mahemacas call Berse ass polyomals B, ( x) as he Berse polyomals. We wll follow hs coveo as well. hese polyomals have he followg properes: ( ) B, () = δ ad B, () = δ, where δ s he Kroecker dela fuco. ( ) B, ( ) has oe roo, each of mulplcy ad, a = ad = respecvely. ( ) B, ( ) for [,] ad B, ( ) = B, ( ). (v) For, B, has a uque local maxmum [,] a = / ad he maxmum value ( ). (v) he Berse polyomals form a paro of uy.e. B ( ) =. (v) I has a degree rasg propery he sese ha ay of he lower-degree polyomals (degree < ) ca e expressed as a lear comaos of polyomals of degree. We have, + B, ( ) = B, ( ) + B +, ( ). (v) Le f ( x) C [,] (he class of couous fucos o [,] ), he =,

4 243 A. K. Sgh, V. K. Sgh, O. P. Sgh B f )( x) = f B ( x) coverges o f (x) uformly o [,] as. = (, (k ) (k ) (v) Le f ( x) C [,] (he class of k mes dffereale fuco wh f couous), he ( k ) ( ) k ( k ) ( k ) ( k ) B ( f ) f ad f B ( ) k f as k, where. s he ( ) k k sup. orm ad = L k s a ege value of B ; he correspodg ege fuco s a polyomal of degree k. 3. he orhoormal polyomals: Usg Gram- Schmd orhoormalzao process o B, ad ormalzg, we oa a class of orhoormal polyomals from Berse polyomals. We call hem orhoormal Berse polyomals of order ad deoe hem y,, L,. o For = 5 he fve orhoormal polyomals are gve y 5 5 ( ) = ( ) ( ) = 6 5( ) ( ) 2 (4) ( ) = ( ) ( ) + ( ) ( ) = ( ) 5( ) + ( ) ( ) ( ) = 7 3 5( ) 2( ) + 8( ) 4( ) + ( ) ( ) = 6 ( ) + ( ) 25( ) + 5( ) ( )

5 Berse operaoal marx of egrao 243 A fuco f L 2 [,] may e wre as f ( ) = lm c ( ), (5) = where, c = f, ad, s he sadard er produc o L 2 [,]. If he seres (5) s rucaed a = m, he m f c = m m = C B( ), (6) where, ad = [ c m, cm, L, cm m (7) C ] B )] ( ) = [ m ( ), m ( ), L, m m (. (8) 4. he operaoal marx of egrao. he orhoormal Berse polyomals operaoal marx of egrao of order ( m + ) ( m + ) wll e derved ow. o acheve hs, cosder he followg egral m ( x) dx = ϕ ( ), <, =,, L, m. = m j= c j m j m ( ), = c, c, L, c ] B( ), for m. (9) [ m m m m Usg equaos (8) ad (9), we oa B x) dx = Pm + B( ) (, ()

6 2432 A. K. Sgh, V. K. Sgh, O. P. Sgh where he operaoal marx P m+ of egrao assocaed wh orhoormal Berse polyomals s gve y m Pm ( c j m ), j= + = () ad c = ϕ,. (2) j m j m For m = 5, he marx P 6 s deoed y P ad s gve as follows: P := (3) 5. Numercal example he followg example shows he compuaoal power of he Berse polyomal operaoal marx of egrao. Cosder a lear me-varyg sysem a y& ( ) + y( ) = u( ), wh y() =, (4) / a where u() s he u sep fuco. he aalyc soluo of (4) s y( ) = e. Gu ad Jag [5] cosdered hs prolem wh a =. 25 ad gave a approxmae soluo y usg Haar waveles wh four, sx ad e ass fucos. Paraskevopoulos e all. [2] cosdered he same prolem wh a = ad used Fourer seres operaoal marx of egrao of orders ( ) ad ( 2 2) o oa approxmae soluos. I 2, Razzagh ad Yousef [6] used Legedre waveles operaoal marx of order ( 6 6) o solve hs prolem. We oa approxmae soluo of (4) usg he Berse operaoal marx of egrao P m+ y akg m = 4, 5 ad compare he soluos. Iegrag (4) from o, we ge a y( ) + y( x) dx = u( x) dx. (5)

7 Berse operaoal marx of egrao 2433 Usg (6), he ukow fuco y() ad u sep fuco u () are approxmaed as y( ) = C B( ) ad u( ) = d B( ), (6) where = [ c m, cm, K, cm m s o e deermed. Susug (6) (5), we oa C ] Usg (), we ge a C B( ) + C B( x) dx = d B( x) dx. (7) ac B ) + C P B( ) d P B( ). (8) ( m+ = m+ As (8) holds for all [,), reduces o a C. (9) + C Pm + = d Pm + akg raspose of (9), oe oas ( + a I + Pm + ) C = Pm d. Wrg Q a I + P m + =, ad E = P d, we ge m+ Q C = E, (2) where I s he ( m + ) ( m + ) u marx. Eq. (2) s a se of algerac equaos whose soluo gves c, m. Solvg (2), oe ges m C = Q E. (2) Fally, he soluo y () s oaed y susug (2) o (6).For m = 5, he gve y [.55277,.5,.44959, , ,.66667] d s d =. (22) I fgures ad 2, graphs of he exac soluo as well as hose of he approxmae soluo y( ) = C B( ) for m = 4 ad 5 are gve akg a =. 25, respecvely. Fgure 3 depcs he correspodg errors ewee he approxmae soluos. From Fg.

8 2434 A. K. Sgh, V. K. Sgh, O. P. Sgh 3, ca e see ha he accuracy creases que fas as we go from level m = 4 o m = y() Y() Fg.. he exac soluo y () Y ( ) (doed le) rucaed a level m = 4. (sold le) ad he approxmae soluo deoed y.75 y() Y () Fg.2. he exac soluo y () (sold le) ad he approxmae soluo deoed y Y () (doed le) rucaed a level m = 5.

9 Berse operaoal marx of egrao y() Y () y( ) Y() Fg.3. Comparso ewee he errors a m = 5 (sold le) ad m = 4 (doed le). 6. Cocluso. he uform approxmao capales of Berse polyomals coupled wh he fac ha oly a small umer of polyomals are eeded o oa a sasfacory resul makes our mehod very aracve. I gves eer approxmao compared o ha of paraskevopoulos e all. [2], Gu ad Jag [5], ad Razzagh ad Yousef [6]. Refereces []. C. F. Che, C. H. Hsao, Walsh seres aalyss opmal corol, Ieraoal Joural of Corol, 2(975) [2]. C. F. Che, C. H. Hsao, A Walsh seres drec mehod for solvg varaoal prolems, Joural of he Frakl Isue, 3 ( 975) [3]. C.F. Che, Y.. say,.. Wu, Walsh operaoal marces for fracoal calculus ad her applcao o dsrued- parameer sysems, Joural of he Frakl Isue, 53(977),

10 2436 A. K. Sgh, V. K. Sgh, O. P. Sgh [4]. P. Sau, Aalyss ad syhess of dyamc sysems va lock pulse fucos, Proceedgs of he Isuo of Elecrcal Egeers, 24(977) [5]. R-E. Kg, P. N. Paraskevopoulos, Paramerc defcao of dscree me SISO sysems, Ieraoal Joural of Corol, 3(979) [6]. C. Hwag, Y. P. Shh, Parameer defcao va Laguerre polyomals, Ieraoal Joural of Sysems Scece, 3(982) [7]. C. Hwag, Y. P. Shh, Soluo of egral equaos va Laguerre polyomals, Joural of Compuer ad Elecrcal Egeerg, 9(982) [8]. P. N. Paraskevopoulos, Cheyshev seres approach o sysem defcao, aalyss ad corol, Joural of he Frakl Isue, 36(983) [9]. I. R. Horg, J. H. Chou, Shfed Cheyshev drec mehod for solvg varaoal prolems, Ieraoal Joural of Sysems Scece, 6(985) []. R. Y. Chag, M. L. Wag, Parameer defcao va shfed Legedre polyomals, Ieraoal Joural of Sysems Scece, 3(982) []. P. N. Paraskevopoulos, Legedre seres approach o defcao ad aalyss of lear sysems, IEEE rasacos Auomac Corol, 3 (6) ( 985) [2]. P. N. Paraskevopoulos, P. D. Spars, S. G. Mourousos, he Fourer seres operaoal marx of egrao, Ieraoal Joural of Sysems Scece, 6(985) [3]. M. Razzagh, M. Razzagh, Fourer seres drec mehod for varaoal prolems, Ieraoal Joural of Corol, 48(988) [4]. P. N. Paraskevopoulos, P. Sklavouos, G. Ch. Georgou, he operaoal marx of egrao for Bessel fucos, Joural of he Frakl Isue, 327(99) [5]. J. S. Gu, W. S. Jag, he Haar waveles operaoal marx of egrao, Ieraoal Joural of Sysems Scece, 27(996) [6]. M. Razzagh, S. Yousef, he legedre waveles operaoal marx of egrao, Ieraoal Joural of Sysems Scece, 32 (4) (2) Receved: Feruary, 29