Bernstein Direct Method for Solving. Variational Problems

Transcription

1 Ieraioal Maheaical Foru, 5,, o. 48, Bersei Direc Mehod for Solvig Variaioal Probles Sadeep Dixi*, Viee K. Sigh**, Ai K. Sigh*, O P. Sigh* *Depare of Applied Maheaics, Isiue of echology, Baaras Hidu Uiversiy ad DS-Cere for Ierdiscipliary Maheaical Scieces, Baaras Hidu Uiversiy, Varaasi 5, Idia **Depare of Maheaics, Birla Isiue of echology ad Sciece, Pilai-Goa Capus, Zuariagar, Goa, Idia Absrac A siple algorih for solvig variaioal probles via Bersei orhooral polyoials of degree six is proposed. he sixh order Bersei polyoials are orhooralized ad heir operaioal arix of iegraio is derived. Usig his operaioal arix of iegraio, he variaioal probles are reduced o he soluio of syses of algebraic equaios. he ehod of Lagrage ulipliers is used o solve illusraive probles wih free ad cosraied boudaries. Maheaics Subjec Classificaio: 65K, 4999 Keywords: Bersei polyoials; Variaioal probles; Direc ehod; Operaioal arix Correspodig auhor: O Prakash Sigh, E- ail: sigho@gail.co, sadydixi7@gail.co (Sadeep Dixi, vksibhu@gail.co (Viee K. Sigh), aikibhu@gail.co (Ai K. Sigh)

2 35 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh Iroducio May probles of aheaical physics are coeced wih he calculus of variaios, oe of he very ceral fields of aalysis. Soe of is uses i boh physics ad aheaics are as follows. Uificaio of diverse areas of physics-usig eergy as a key cocep. Sarig poi for ew, coplex areas of physics ad egieerig. I geeral relaiviy he geodesic is ake as he iiu pah of ligh pulse or he free fall pah of a paricle i curved Rieaia space. Variaioal priciples have bee applied exesively i oder corol heory ad hey also appear i quau field heory. Variaioal aalysis provides a proof of he copleeess of he Sur-Liouville eigefucios ad esablishes a lower boud for he eigevalues. Siilar resuls follow for he eigevalues ad eigefucios of he Hilber-Schid iegral equaio. Maheaical odels of several probles i he syse ad corol heory, oder echaics, ecooics ay be reduced o syses of differeial, differeialiegral or iegral equaios as boudary-iiial value probles. hese equaios are oliear i geeral ad hece aalyical soluios are difficul o obai i orivial cases. Variaioal priciples provide a aleraive o search for aalyical soluios of such probles. he idea is o forulae variaioal fucioals whose saioariy codiios lead o equaios ha describe he probles. he Euler-Lagrage equaios obaied by applyig he well kow procedure i he calculus of variaio [4], usually leads o equaios ha are difficul o solve. May auhors [,3,6,8] have ried various rasfor ehods o overcoe hese difficulies i he proble of exreizaio of fucioal syses. he ai idea of a direc ehod for solvig variaioal probles is o cover he proble of exreizaio of a fucioal io oe ivolvig a fiie uber of variables. he Riz ehod [4], usually based o he subspaces of kieaically adissible coplee fucios, is he os cooly used approach i direc ehods of solvig variaioal probles. Orhogoal fucios are special fucios i he space of which approxiae soluios of variaioal probles are sough. Usig he Riz ehod ad orhogoal fucios wih heir operaioal arices of iegraio, oe ca reduce a variaioal proble o syses of algebraic equaios. Several orhogoal fucios ad waveles such as Walsh fucios [3, 5], Laguerre [8], Legedre [], Chebyshev polyoials [6], Fourier series [], Haar waveles [7], Legedre waveles [] were used o solve variaioal probles. Babolia e. al. [], Razzaghi e. al. [9] used riagular orhogoal fucios ad raioalized Haar fucios respecively for he purpose.

3 Bersei direc ehod 353 I his paper, we solve variaioal probles usig Bersei polyoials of degree six. Firs, soe siple properies of Bersei polyoials are give, ad he hese seve polyoials of degree six are used o cosruc a faily of seve orhooral polyoials of he sae degree which are aed as Bersei orhooral polyoials. he operaioal arix of iegraio is derived ad a direc ehod for solvig variaioal probles is preseed. he ehod is based o (i) assuig represeaios of adissible fucios by orhooral Bersei polyoials wih coefficies o be deeried; (ii) usig he operaioal arix of iegraio for perforig iegraio; (iii) fidig he ecessary codiio for exreizaio; ad (iv) solvig for he algebraic equaios obaied fro he previous seps o evaluae Bersei coefficies. he above four seps are he sadard oes followed by he oher auhors [,3,5,6,7,8,], as well. o he auhors kowledge, he Bersei operaioal arix has o bee derived before ad i is for he firs ie his arix has bee derived ad is beig used o solve he variaioal probles. he calculaio procedures for Laguerre polyoials [8], Legedre polyoials [], ad Chebyshev polyoials [6] are usually oo edious, ad soe recursive forulae are sill waiig for developes [7]. I [7,], auhors had derived Haar produc arix ad riagular fucios (F) produc arix, respecively o solve he variaioal probles. Ad i [], he order of arix is oo large, of he order 5, o achieve accuracy coparable o he oes we obai by usig arix of order 5. Hsiao [7] used he produc arix of order 8 bu he error is quie large copared o our ehod, proposed i his paper. Also, he low order of he arix akes he copuaios very siple i his paper. he Bersei polyoials A Bersei polyoial, aed afer Sergei Naaovich Bersei, is a polyoial i he Bersei for ha is a liear cobiaio of Bersei basis polyoials. he Bersei basis polyoials of degree are defied by i i Bi, ( = (, for i =,,, L,. () i h here are ( + ) degree Bersei basis polyoials ad for a basis for he liear space V cosisig of all polyoials of degree less ha or equal o i R[x]-he iegral doai of polyoials over he real field R. For aheaical coveiece, we usually se B i, = if i < ori >. Ay polyoial B (x) of degree i R[x] ay be wrie as

4 354 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh B x) = β B ( x). () ( i i, i = he B (x) is called a polyoial i Bersei for or Bersei polyoial of degree. he coefficies β i are called Bersei or Bezier coefficies. Bu several aheaicias call Bersei basis polyoials B i, ( x) as he Bersei polyoials. We will follow his coveio as well. hese polyoials have he followig properies: ( i) B i, () = δ i ad Bi, () = δ i, where δ is he Kroecker dela fucio. ( ii) Bi, ( has oe roo, each of ulipliciy i ad i, a = ad = respecively. ( iii) Bi, ( for [,] ad Bi, ( = B i, (. (iv) Fori, B i, has a uique local axiu i [,] a = i / ad he axiu i i value i ( i). i (v) he Bersei polyoials for a pariio of uiy i.e. B ( =. (vi) I has a degree raisig propery i he sese ha ay of he lower-degree polyoials (degree < ) ca be expressed as liear cobiaios of polyoials of degree. We have, i i + Bi, ( = Bi, ( + B i+, (. (vii) Le f ( x) C [,] (he class of coiuous fucios o [,] ), he i B ( f )( x) = f Bi, ( x) coverges o f (x) uiforly o [,] as. i = (k ) (k ) (viii) Le f ( x) C [,] (he class of k ies differeiable fucio wih f coiuous), he ( k ) ( ) k ( k ) ( k ) ( k ) B ( f ) f ad f B ( ) k f as k, where. is he ( ) k sup. or ad k = L k is a eigevalue of B ; he correspodig eigefucio is a polyoial of degree k. i = i, 3 he orhooral polyoials Usig Gra- Schid orhooralizaio process o B, ad oralizig, we obai a i

5 Bersei direc ehod 355 class of orhooral polyoials fro Bersei polyoials. We call he orhooral Bersei polyoials of order ad deoe he by b o, b, L, b for = 6 he orhooral polyoials are give by b = + (3) 6 6 ( 3 ( 5 b ( = ( + ( + 3 ) 6 4 b 6 ( = 3( + ( b 36 ( = 7 ( + ( b 4 6 ( = 5 ( + ( b5 6 ( = 3 ( b 6 6 ( = ( A fucio f L [,] ay be wrie as ) i= 3 ) 4 ) f ( = li c b (, (4) i i where, c = f, b ad, is he sadard ier produc o L [,]. If he series (4) is i i rucaed a =, we ge a approxiaio f of f as, where, f f = i = ci bi = C C ] 6 ) 6 B(, (5) = [ c, c, L, c (6) ) ad B )] ( = [ b (, b (, L, b (. (7) 4 he operaioal arix of iegraio I his secio, he Bersei operaioal arix of iegraio is derived. So, le us sar wih he iegral propery of he basic operaioal arix

6 356 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh ϕ + a a ( ) [ ϕ (, ϕ(, K, ϕ ( ] k k L ( σ )( dσ ) P ϕ(, (8) where ϕ = i which he elees ϕ (, ϕ(, K, ϕ ( are he basis fucios, orhogoal o a cerai ierval [ a, b] ad P is he operaioal arix for iegraio of ϕ (. Noe ha P is a cosa arix of order ( + ) ( + ). he orhooral Bersei polyoials operaioal arix of iegraio of order will be derived ow. o achieve his, cosider he followig iegral b ( x) dx = ϕ i (, <, i =,, L,. i = j= c i j b j (, i i i = c, c, L, c ] B( ). (9) [ Usig equaios (7) ad (9), we obai B x) dx = P + B( (, () where he operaioal arix P + of iegraio associaed wih orhooral Bersei polyoials is give by i P + = ( c j ) i, j= ad c = ϕ, b. i j i For = 6, he operaioal arix P 7 (deoed as p ) is give as: j p :=

7 Bersei direc ehod 357 As, B( B ( d = I ; where I is 7 7 ideiy arix, i akes he subseque copuaios very siple copared o he ehods of Hsio [7], Babolia e. al. [] ad Razzaghi e.al. [9]. 5 he Bersei direc ehod Cosider he proble of fidig he exreu of he fucioal [, x(, x( ] ( x) = F & d. () he ecessary codiio for x ( o exreize (x) is ha i should saisfy he Euler- Lagrage equaio F d F = x d x& wih appropriae boudary codiios. However, he above differeial equaio ca be iegraed easily oly for siple cases. hus uerical ad direc ehods such as he well-kow Riz ad Galerki ehods [4], have bee developed o solve variaioal probles. I his paper he Bersei orhooral polyoials are used o esablish he direc ehod for variaioal probles. Suppose, he rae variable x& ( ca be expressed approxiaely as x& ( x& ( = c b = C B(. () i = Iegraig equaio () fro o ad usig (), we represe x ( as i We ca also express i ers of B ( as i x( = x( ) d + x(), C P B( x() + + = x ( ). (3) d B(, where d = [ d d,, d ]. (4), L

8 358 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh he oher ers i he fucioal of equaio () are kow fucios of he idepede variable ad ca be expaded io Bersei orhooral polyoials via equaio (4). Subsiuig Eqs. ()- (4) i Eq. () he fucioal (x) becoes a fucio of c i, i =,, L,, ad we fially have = c, c, K, c ). (5) ( he origial exreizaio of he fucioal proble () becoes he exreizaio of a fucio of a fiie se of variables i Eq.(5). Hece, o fid he exreu of (x), we ake parial derivaives of wih respec o c i ad se he equal o zero. hus, we obai c i =, i =,, L,. (6) Solvig for c i, ad subsiuig i Eq.(3), we ge he soluio. We esablish he deailed procedure via several classical probles. 6 Nuerical exaples 6.. Firs order fucioal exreal wih wo fixed boudary codiios Cosider he proble of fidig he exreal of he fucioal [7] ( x) = [ x& ( + x& ( ] d. (7) he boudary codiios are he iiial ad he fial codiios x ( ) =, (8) x ( ) =. (9) 4 o solve his proble by he proposed ehod, we assue x& ( ca be expaded i ers of he Bersei orhooral polyoials as i Eq.(). We solve his ad he subseque exaples by akig = 6 ad oe ha a very high degree of accuracy is achieved. Subsiuig Eqs. () ad (4) io Eq. (7), we have

9 Bersei direc ehod 359 As, [ C B( B ( C + C B( B ( d]d. () B( B ( d( = I, () C C + C d. () Subsiuig he iiial boudary codiios (8) io (3), we ge x( C P B(. (3) Hece he fial boudary codiio (9) subsiued i (3) yields x( ) = C P B() = 4. (4) We iiize () subjec o (4) usig Lagrage uliplier. Suppose = + λ C P B() 4 = C C + C d + λ C P B() 4, (5) where λ is Lagrage uliplier. o iiize, he parial derivaives of wih respec o C ad λ are ake ad se o zero, hus, C =, =. (6) λ C + d + λ P B() =, (7) ad C P B( ) =. (8) 4 Eqs.(7) ad (8) cosis of six siulaeous liear equaios which are used o deerie c 6, c6, L, c66 ad λ.

10 36 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh herefore, x & ( = [.5347,.4864,.8986,.4746,.9965,.555,] B(, (9) ad x ( = [.865,.7387,.8986,.894,.7897,.6859,.3574] B( ) (3) If he Euler equaio is used o solve he above proble, he exac aswer is obaied as x& ( = (, (3) x( = ( ). (3) where ( is he Le E = x( x ( = ( ) ) x x ( d, (33) x is he exac soluio (3) ad x ( ) is he approxiae soluio (3), ad L -or. 7 he, E =.9. (34) Whereas, whe riagular fucio ehod [] is applied, we have / 8 4 E = 7.3, 56 7 E = 7 ad 5 7 E =.. (35) I his ad he subseque exaples approxiaio is perfored by usig E will deoe he error i L -or whe arix i riagular fucio ehod. Figs., 3, 5 ad 7 copare he exac soluios x( wih he approxiae soluios x ( ) (deoed by x( i he Figs.) ad Figs., 4, 6 ad 8 show he errors E( = x ( x(. ables are give a ed coparig our resuls wih ha of oher auhors [7,, 9] ad exac soluios.

11 Bersei direc ehod x () x() Fig.. Copares he exac soluio wih he approxiae soluio. E () Fig.. Shows he error bewee he. 6.. Firs order fucioal exreal wih a fixed ad a ovig boudary codiios We cosider he sae fucioal exreal of Eq.(7) bu wih uspecified x (), aely x ( ) =, (36) x ( ) = uspecified. (37) Aoher codiio ay be foud fro F [, x(, x& ( ].

12 36 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh = F x & =, x& ( ) =. [8] (38) he codiio x& ( ) = alog wih x& ( = C B( C B() =, follows fro (3). (39) Subsiuig Eqs.(), (3), (4) ad (36) io Eq.(7), we ge C C + C d. (4) is exreized subjec o he cosrai (38). Le λ be he Lagrage uliplier, so ha, we defie as + λ C B() +. (4) Equaig parial derivaives of wih respec o C ad λ equal o zero, we have C + d + λ B() =, (4) ad C B( ) + = C66 =. (43) 4 Solvig (4) subjec o (43), we ge C = [.39,.88838,.5,.4737,.39754,.8563,.749], (44) Givig he approxiae soluio x ( ) as x( = [.3577,.645,.3574,.5495,.678,.5674,.3574] B( ) (45) ad x& ( = C B(, where C is give by (44). Aalyic soluio via Euler s equaio is x& ( =, (46)

13 Bersei direc ehod 363 x( =. (47) 4 he leas squares error E for his proble is 7 E =.5, (48) which is for superior ha he errors E obaied i []. We quoe hree values of E for =8, 56 ad E = 7.3, E = 7 ad E =. (49) he copariso of he soluios via Euler s aalyic ehod ad via Bersei s direc ehod is show i Fig.3 ad he correspodig error i Fig x () x() Fig.3. Copares he exac soluio wih he approxiae soluio E () Fig.4. Shows he error bewee he.

14 364 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh 6.3. Firs order fucioal exreal wih wo fixed boudary codiios Le us cosider he proble of searchig for he exreu of his fucioal [, 5, ] = [ x& ( + x& ( + x ( ] d, (5) wih he followig boudary codiios; x ( ) =, x ( ) = (5) 4 he exac soluio of his proble is e [( + e )( e e e + e x( = 4( + e ) Argues siilar o he previous probles lead o + )] (5) = C C + C d + C P P C, (53) = + λ C P B() 4. (54) Ad x & ( = [.9937,.3394,.9838,.653,.45,.668,.3886] B(, (55) x ( = [.583,.6496,.8389,.863,.75535,.6857,.3574] B( ). (56) he leas squares error E is give as 7 E =. (57)

15 Bersei direc ehod x () x() Fig.5. Copares he exac soluio wih he approxiae soluio for exaple E () Fig.6. Shows he error bewee he for exaple Secod-order fucioal exreal wih wo fixed ad wo ovig boudary codiios Cosider he followig fucioal exreal proble

16 366 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh wih = [ & x ( + 4( x& ( ] d, (58) x ( ) = (59) x& ( ) = (6) ad x (), x& () uspecified. (6) he aural boudary codiios are foud fro [7] d Fx & ( F&& ) = = 4( &&& x = = && x x& () = (6) d F& x = = & x () =. (63) Expadig & x&( ) io Bersei orhooral polyoial series ad rucaig i a = 6, we ge 6 & x&& ( = Ci 6 bi 6 = C B(. (64) i = Iegraig (64) repeaedly wice gives & x ( = C P B( + & x (), (65) x& ( = C P B( + & x (), as x& ( ) =, (66) ad ( ) 3 x = C P B( + && x() d P B(, x ( ) = (67) where = d B(, fro Eqs.(65) ad (63), we ge & x ( ) = C PB(). (68) Expressig 4 4 = f B(, ad usig Eqs. (65) o (68), we ge = C P P C C P B() B () P C + (69) C P B() B () P C + f ( P ) C C P B() d f Differeiaig wih respec o C ad equaig i o zero, we have C = P P C P B() B () P P B() B C + () P C + P f P B() d f = (7)

17 Bersei direc ehod 367 Solvig (7) forc, we ge x& ad x. C = [ , , 5.857, , , , ] (7) x( = [.76,.49534,.9648,.5988,.33,.6845,.749] B( ). (7) he aalyic soluio via Euler equaio is & x& ( = + 4, 3 x& ( = +, x( = +. (73) 6 3 he leas squares error E is give as 7 E = 4.98 (74). x () x() Fig.7. Copares he exac soluio wih he approxiae soluio for exaple 6.4.

18 368 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh E () Fig.8. Shows he error bewee he for exaple 6 able (For Exaple 6.) Bersei soluio Haar soluio [7] Aalyical soluio able (For Exaple 6.) Bersei soluio Haar soluio [7] Aalyical soluio

19 Bersei direc ehod 369 able 3 (For Exaple 6.3) Bersei soluio RH fucios [9] Aalyical soluio able 4 (For Exaple 6.4) Bersei soluio Haar soluio [7] Aalyical soluio Coclusios he uifor approxiaio capabiliies of Bersei polyoials coupled wih he fac ha oly a sall uber of polyoials (seve o be precise) are eeded o obai a saisfacory resul akes our ehod very aracive. I [], auhors have used operaioal arix of order 5 o achieve he accuracy coparable o ours as illusraed by Ex. 6. ad 6.. ables -4 esablish he superioriy of our proposed ehod over he oher exisig ehods, oably, he ehods recely proposed by Hsiao [7], Babolia e. al. [] ad Razzaghi e. al. [9].

20 37 S. Dixi, V. K. Sigh, A. K. Sigh, O. P. Sigh Refereces [] E. Babolia, R. Mokhari, M. Salai, Usig direc ehod for solvig variaioal probles via riagular orhogoal fucios, Applied Maheaics ad Copuaio 9 (7) 6-7. [] R. Y. Chag, M. L. Wag, Shifed Legedre direc ehod for variaioal probles, oural of Opiizaio heory ad Applicaios 39 (983) [3] C. F. Che, C. H. Hsiao, A Walsh series direc ehod for solvig variaioal probles, oural of he Frakli Isiue 3 (975) [4] I. M. Gelfad, S.V. Foi, Calculus of Variaios, Preice-Hall, Eglewood Cliffs, N, 963. [5] W. Glabisz, Direc Walsh-wavele packe ehod for variaioal probles, Applied Maheaics ad Copuaio 59 (4) [6] I. R. Horg,. H. Chou, Shif Chebyshev direc ehod for solvig variaioal probles, Ieraioal oural of Syses Sciece 6 (985) [7] C. H. Hsiao, Haar wavele direc ehod for solvig variaioal probles, Maheaics ad Copuers i Siulaio 64 (4) [8] C. Hwag, Y. P. Shih, Laguerre series direc ehod for variaioal probles, oural of Opiizaio heory ad Applicaios 39 (983) [9] M. Razzaghi, Y. Ordokhai, A applicaio of raioalized Haar fucios for variaioal probles, Applied Maheaics ad Copuaio () [] M. Razzaghi, M. Razzaghi, Fourier series direc ehod for variaioal probles, Ieraioal oural of Corol 48 (988) [] M. Razzaghi, S. Yousefi, Legedre waveles direc ehod for variaioal probles, Maheaics ad Copuers i Siulaio 53() [] A. K. Sigh, V. K. Sigh, O. P. Sigh, he Bersei operaioal arix of iegraio, Applied Maheaical Scieces, vol. 3, o.49, (9) [3] V. K. Sigh, R. K. Padey, O. P. Sigh, New sable uerical soluios of sigular iegral equaios of Abel ype by usig oralized Bersei polyoials, Applied Maheaical Scieces, vol. 3, o.5-8, (9) Received: Ocober, 9