Modified chain least squares method and some numerical results

Transcription

1 IJST (2015) 39A1: Iri Jourl of Sciece & Tecology ttp://ijsts.sirzu.c.ir Modified ci lest squres metod d some umericl results F. Goree 1, E. Boli 1 * d A. Adolli 2 1 Deprtmet of Mtemtics, College of Bsic Scieces, Ter Sciece d Reserc Brc, Islmic Azd Uiversity, Ter, Ir 2 Deprtmet of Mtemtics, Mrge Brc, Islmic Azd Uiversity, Mrge, Ir E-mil: fgoree@sriu.c.ir, oli@ku.c.ir &.dolli@iu-mrge.c.ir Astrct Recetly, i order to icrese te efficiecy of lest squres metod i umericl solutio of ill-posed prolems, te ci lest squres metod is preseted i recurret process y Boli et l. Despite te fct tt te give metod s my dvtges i terms of ccurcy d stility, it does ot ve y stoppig criterio d s ig computtiol cost. I tis rticle, te ttempt is to decrese te computtiol cost of ci lest squres metod y itroducig te modified lest squres metod sed o stoppig criterio. Numericl results sow tt te modified metod s ig ccurcy d stility d ecuse of its low computtiol cost, it c e cosidered s efficiet umericl metod. Keywords: Ci lest squres; Lgrge multipliers metod; Ill- posed prolem; Itegrl equtios; Sigulr secod order iitil vlue differetil equtios 1. Itroductio Lest squres metod is oe of te efficiet metods i te umericl solutio of my egieerig d pysics prolems (Aks et l. 2006; Alexder d George, 1990; Cig d Su- Yu, 2002; Jglur-Mo et l. 2013; Jike d Hugo, 2012; Jimig, 2012; Kig d Krueger, 2003; Leli d Mlek, 2012). I order to icrese te efficiecy of tis metod i umericl solutio of some ill-posed prolems, te ci lest squres metod is preseted i recurret form y Boli et l. (2014). I tis pproc, y reducig term lest squres prolem to te ( 1)-term oes d cotiutio of tis tred up to te lst stge (1-term prolem), te efficiecy of te lest squres metod i umericl solutio of ill-posed prolems s ee sigifictly icresed (Boli et l. 2014). Tus, for solvig -term prolem y ci lest squres metod, we ve to cotiue te recurret process up to te lst stge. I tis rticle, te ttempt is to prevet te cotiutio of te recurret process up to te lst stge y providig logicl d experimetl stoppig criterio. Besides decresig te computtiol cost of ci lest squres metod, te defiitio of te stoppig criterio mitis te stility d ccurcy of tis metod. Tis stoppig *Correspodig utor Received: 12 April 2014 / Accepted: 8 Octoer 2014 criterio is sed o te covergece of itermedite mtrix elemets of lest squres metod to zero. Tis is ispired y te covergece of te Glerki metod i umericl solutio of Fredolm itegrl equtio of te secod kid (Delves d Momed, 1985). It sould e metioed tt y itermedite mtrices, we me te coefficiet mtrix of system of equtio correspodig to te ci lest squres metod i turig k-term prolem (k,,2) to (k 1)- term oe. I te secod step, cosiderig te mi role of te rtificil trjectories i te defiitio of ci lest squres metod (Boli et l. 2014), i order to decrese te computtiol cost of tis metod, ew process is itroduced i defiig of rtificil trjectories. Accordig to te kids of prolems solved y te ci lest squres metod, t lest oe of te rtificil trjectories is decresed. I te ew tred, isted of reducig -term prolem to ( 1)-term oe, te ttempt is to cge -term prolem to ( l)-term oe (l 2) i suc wy tt te computtiol cost of tis metod is decresed. By presetig umericl exmples i ec sectio, te stility d ccurcy of te ew metod will e sow. Firstly, review of ci lest squres metod s ee give, te te modified ci lest squres metod is preseted. Filly, te efficiecy of te modified metods is ivestigted y solvig severl ill-posed fuctiol equtios.

2 IJST (2015) 39A1: A Review of te Ci Lest Squres Metod Let f L 2 [, ] d L i } e sis of L 2 [, ] d f (s) i L i (s), s [, ], e ordiry lest squres pproximtio of f i te sis L i }. For determiig te ukow coefficiets i } we must solve te followig miimiztio prolem mi 1,, e( 1,, ). (1) I order to determie te solutio of (1), it is sufficiet to solve te followig orml equtios i e( 1,, ) 0, i 1,,, (2) e( 1,, ) [ i L i (s) f(s)] ds. It is possile tt solvig (2) ecomes ill-posed prolem. I oter words, te coditio umer of (2) is lrge d its solutio is determied wit lrge error (Dtt, 2010). I order to get te pproximte solutio wit ig ccurcy i ci lest squres metod (Boli et l. 2014), it is supposed tt solutio of (2) is true for te followig coditios g i ( 1,, ) 0, i 1,, 1. I wic, rtificil costris g i re defied s 1 follows (for sclrs r i } elogig to R) (Boli et l., 2014) g i ( 1,, ) i i1 r i, i 1,, 1. (3) Terefore te miimiztio prolem (1) is equivlet to mi e( 1,, ) s. t g i ( 1,, ) 0, i 1,, 1. 2 (4) By te Lgrge multipliers metod (Ito d 1 Kuisc, 2008) tere exist rel sclrs λ i } suc tt te prolem (4) is equivlet to 1 e λ i g i g i ( 1, ) 0, i 1,, 1. (5) I wic, is Grdi opertor. From (5) oe gets 2c c c 1 2f 1 λ 1 2c c c 2 2f 2 λ 2 λ 1 2c 1 1 2c 2 2 2c 2f λ r 1 r 1, c ij L i (s)l j (s)ds, i, j 1,, }, f i L i (s)f(s)ds, i 1,,. By summig te first equtios, (for removig λ i } 1 ) we ve d d d r 1 r 1, f i, d j c ij, j 1,, }. (6) Filly, y (6) te coefficiets i } re determied s follows: Let DR, ( 1,, ) T, R (r 1,, r 1, 1) T, D 1 N ( N t 1 t 1 t 1 t 1 N N t 2 N t 2 t 2 t 2 d i, N t 1 N t 1 N t 1 t 1 t 1 d 1, t i t i1 d i, i 1,, 1. L(s) (L 1 (s),, L (s)),, ) By te ove ssumptios, te -term miimiztio prolem (1) trsforms to ( 1)- term oe s follows: i wic mi E(r 1,, r 1 ), r 1,,r 1 E(r 1,, r 1 ) [ 1 r i p i (s) f (s)] 2 ds,

3 93 IJST (2015) 39A1: d p i (s) L(s)D i, i 1,,, f (s) f(s) p (s), D i i t colum of D. I ci lest squres metod (Boli, 2014), te ( 1)-term prolem is tured ito te ( 2)-term oe y te sme tred. Tis process s ee cotiued up to te fil stge (1-term prolem). 3. Coditiol Ci Lest Squres Metod Suppose tt te purpose is to determie te lest squres pproximtio of f L 2 [, ] o te sis of L i }. If tis prolem is to e solved y ci lest squres metod, te we ve to solve -term, ( 1)-term,, 1-term prolems. I oter words, te ci lest squres metod will e cotiued up to te lst stge d tis process s very ig computtiol cost. I tis sectio, te purpose is to preset experimetl stoppig criterio for ci lest squres metod i suc wy tt its recurret process is fiised efore recig to 1-term prolem. By tis tred, ot oly is te computtiol cost of tis metod decresed, ut lso its stility will e icresed sigifictly. For expliig stoppig criterio, suppose tt te purpose is to determie te ci lest squres of te followig exmples o te sis of s i } 14 i0. Exmple 1. f(s) e s, s [0,1]. Exmple 2. f(s) si(s), s [0,1]. I order to determie tese pproximtios, we ve to solve (15)-term, (14)-term,, 1-term prolems for ec exmple. Suppose tt te lier system of equtios of k-term prolems (k 15,,1) is sow y A k k F k. A k d F k mtrices re clled itermedite mtrices of ci lest squres metod. Also, we defie MA k Mx 1 i,j k (A k ) i,j, k 15,,1, MF k Mx 1 j k (F k ) j, k 15,,1. 15 To sow te evior of MA k } k1 d MF k } 15 k1, tese prmeters re computed for Exmples 1 d 2 d te umericl results re preseted i Tles 1 d 2, respectively. Tle 1. Vlues of MA k d MF k for Exmple 1 k MA k MF k Tle 2. Vlues of MA k d MF k for Exmple 2 k MA k MF k Accordig to te preseted umericl results i Tles 1 d 2, it is cocluded tt lim k 1 MA k 0, lim k 1 MF k 0. I oter words, te followig coclusio is otied experimetlly. Coclusio 1. I determiig te ci lest squres pproximtio of f L 2 [, ] i te sis L i }, te elemets of itermedite mtrices A k d F k coverge to zero. It sould e metioed tt suc stte occurs i pproximtig te solutio of te Fredolm itegrl equtio of secod kid o ortogol sis y Glerki metod (Delves d Momed, 1985). I oter words, if B is te lier system of equtios correspodig to tis prolem, te we ve lim 0, lim B j 0, j 1,,, i wic, B j } j1 re te elemets of 't row of mtrix B. Now suppose tt te purpose is to determie te ci lest squres pproximtio of f L 2 [, ] i te sis L}. Also suppose tt A k k F k, k 1,,1, is te itermedite lier system of equtios of tis

4 IJST (2015) 39A1: metod. By te coclusio 1, it is logicl tt we cotiue te lgoritm of ci lest squres up to te stge tt (eps ) MA k > eps d MF k > eps. Becuse oterwise, te cotiutio of ci metod will ot oly improve te solutio of te prolem, ut lso te ccurcy of te otied pproximtios will e decresed y itroducig roud off errors d dditiol oises to te prolem's solutio. Note 1. I te ci lest squres metod, for determiig te pproximtio of fuctio f i te sis of L i }, we ecouter ( 1)-term prolem i te first stge, -term prolem i te secod stge,, ( k 1)-term prolem i te k't stge. I oter words, tis metod is doe i ( 1) stges. By te ove expltios d ccordig to te fct tt i every stge MA k MF k (Tles 1 d 2), te followig stoppig criterio is preseted for ci lest squres metod. Coclusio 2. I determiig te pproximtio of f L 2 [, ] i te sis L i } y ci lest squres metod, te lgoritm of tis metod i te k't stge will e cotiued we MA k eps. We me tis ew metod, coditiol ci lest squres metod (CCLSM). Terefore, if te ci lest squres metod is cotiued up to te k't stge (k N), te term, ( 1)-term,, ( k 1)-term prolems will e solved y tis metod d we will ot eed to solve ( k)-term,,1-term prolems. So its computtiol cost will e decresed sigifictly. I order to compre te ccurcy of coditiol ci lest squres metod wit ordiry lest squres metod (OLSM) d ci lest squres metod (CLSM), te exmples 1 d 2 re pproximted y tese metods i te sis s i } i0. Te umericl results re give i Tles 3 d 4. Tle 3. Mximum solute errors for Exmple 1 OLSM CLSM CCLSM Tle 4. Mximum solute errors for Exmple 2 OLSM CLSM CCLSM For exmple, i determiig te pproximtio of f(s) e s, s [0,1] i te sis s i 14 } i0 y CCLSM, tis metod is cotiued up to te 11't stge. So solvig 4-term,,1-term prolems re voided. Accordig to te umericl results i Tles 3 d 4, it is cler tt te coditiol ci lest squres metod esides decresig te computtiol cost, s te desirle ccurcy d stility. 4. Modified Ci Lest Squres Metod Suppose tt te im is to oti te umericl solutio of ill-posed prolem. Fredolm itegrl equtio of te first kid (Bitsdze, 1995; Delves d Momed, 1985) d determitio of te lest squres pproximtio of ritrry fuctio o te sis s i } i0 (Dtt, 2010; Kicid d Wrd, 2002) re some exmples of tis kid of ill-posed prolem. If ill-posed prolems re pproximted y lest squres metod o te sis s i } i0 te we will ecouter ( 1)-term prolem. Let A F e te correspodig lier system of equtios of tis prolem i te sis s i } i0, y icresig, te coditio umer of te mtrix A will e elrged (Dtt, 2010; Kicid d Wrd, 2002) d te solutio of te correspodig system is determied y lrge errors. I te ci lest squres metod for solvig -term prolem (Boli et l. 2014), ( 1) rtificil trjectories re defied. By cotiuig tis process up to 1-term prolem, we overcome te ill-posedess d ccurte pproximte solutios re otied. I umericl solutio of ill-posed prolem, te first equtios of te system A F do ot ve mi role i ill-posedess of tis system (Delves d Momed, 1985). Te ill-posedess ppers mily o fil equtios of tis system. By tis rgumet, i ci lest squres metod, rtificil trjectories re defied i suc wy tt te structure of te few first equtios of te system

5 95 IJST (2015) 39A1: A F is mitited i decresig te dimesio of lest squres prolem d isted, some of te rtificil trjectories re elimited. For exmple, if te first s rtificl trjectories re elimited, te i ci lest squres metod, term prolem will e cged to ( s 1)-term prolem. I oter words, i te ew metod, we will ot ve y eed to solve ( 1)-term,, ( s)-term prolems. So te computtiol cost of ci lest squres metod will e decresed sigifictly. If tis work is doe y mitiig ccurcy d stility of tis metod, it will e very vlule. To expli tis metod, we ct s follows. Let f e te lest squres pproximtio of f L 2 [, ] i te sis L i }, i.e., f (s) i L i (s), s [, ] Now, we must solve te followig miimiztio prolem mi 1,, e( 1,, ), (7) e( 1,, ) [ i L i (s) f(s)] 2 ds. (8) Assume tt te ukow coefficiets i } olds i te followig trjectories (for sclrs s1 r i } elogig to R) g i ( 1,, ) 0, i 1,, s 1, (9), s N, 0 s 1 d g i ( 1,, ) si si1 r i, i 1,, s 1. I oter words, we omit te followig s-rtificil trjectories 1 s 2 s1 Te y (9) d (7) we ve mi 1,, e( 1,, ), g i ( 1, ) 0, i 1,, s 1. ow y te Lgrge multipliers metod (Ito d Kuisc, 2008) oe gets s1 e λ i g i g i ( 1,, ) 0 i 1,, s 1 λ i } s1 re rel sclrs. Sice r 1 r s we ve c 11 1 e ( e e,, ), 1 c 12 2 c 1 c s1 1 c s2 2 c s f s 2(c s1,1 1 c s1,2 2 c s1, f s1 ) λ 1 2(c s2,1 1 c s2,2 2 c s2, f s2 λ 2 λ 1 2(c 1 1 c 2 2 c f λ s1 s1 s2 r 1 I wic 1 c ij L i (s)l j (s)ds f 1 r s1, i, j 1,, }, f i L i (s)f(s)ds, i 1,, }, y summig equtios (s 1),, (for removig λ i } s1 ) we ve c 11 1 c 12 2 c s1 1 c s2 2 d 1 1 d 2 2 s1 s2 I wic 1 c 1 c s d f 1 f s r 1 r s1 f i, d j c ij, j 1,,. is1 is1 (10) if D e te iverse of te coefficiet mtrix of system (10) te i } re determied s follows DF, (11) Let ( 1,, ) T, F (f 1,, f s,, r 1,, r s1 ) T. L(s) (L 1 (s),, L (s)) T from (8) d (11) we ve e( 1,, ) [L(s) f(s)] 2 ds Sice [L(s)DF f(s)] 2 ds (12)

6 IJST (2015) 39A1: L(s)DF L(s)D 1 f 1 L(s)D s f s L(s)D s1 L(s)D s2 r 1 L(s)D r s1, d y tkig D i i t colum of D, p i (s) L(s)D s1i, i 1,, s 1. From (11) we ve E(r 1,, r s1 ) e( 1,, ) s1 [ p i (s)r i f (s) s f (s) f(s) L(s)D i f i L(s)D s1. So te miimiztio prolem (7) reduces to mi E(r 1, r 2,, r s1 ). r 1,r 2,,r s1 ] 2 ds, Now, for solvig te miimiztio prolem wit ( s 1)-term we use ci lest squres metod wic is itroduced i sectio 3. Defiitio 1. We cll, "trsformtio of te term lest squres prolem wit preseted lgoritm i tis sectio to te ( s 1)-term lest squres prolem d solvig y coditiol ci lest squres" s Modified Ci Lest Squres metod (MCLSM). We expect tt for smll d logicl vlues of s, te ccurcy d stility of te ove metod is te sme s CCLSM, ut computtiol cost of MCLSM is less t CCLSM. To cofirm tis, we compre te lest squres pproximtios of Exmples 1 d 2 i te sis s i } i0 y OLSM d MCLSM d CPU times of tese metods (CLSM, CCLSM, MCLSM) re compred for Exmple 1. We tke s 3 for MCLSM. Tles 5, 6 d 7 sow tt, te modified ci lest squres metod is lmost equl to coditiol ci lest squres metod ut te CPU time of MCLSM is less t CLSM d CCLSM. 5. Numericl Solutio of Some Ill-posed Fuctiol Equtios wit Modified Ci Lest Squres Metod I tis sectio, two cses of ill-posed fuctiol equtios re ivestigted d some exmples re solved y MCLSM. Tle 5. Compriso of mximum solute errors of OLSM d MCLSM for Exmple 1 OLSM MCLSM Tle 6. Compriso of mximum solute errors of OLSM d MCLSM for Exmple 2 OLSM MCLSM Tle 7. Compriso of CPU times (CLSM, CCLSM, MCLSM) for Exmple1 CLSM CCLSM MCLSM It sould e metioed tt ll of tese computtios re doe i Mtl 2011 wit 16 sigifict digits d Gussi qudrture rule of order 16 is used for computig te relted itegrls Cse 1. I tis cse, Fredolm itegrl equtios of te first kid re ivestigted. Tese equtios pper i my pysicl prolems (Blis, 1989). Becuse of te ill-poseess of tese fuctiol equtios, tese equtios re ivestigted y my resercers (Boli d Delves, 1979; Boli et l. 2007; Groetsc, 1984; Mlekejd et l. 2006; Nsed, 1976). Te geerl form of te itegrl equtios of te first kid is s follows. k(s, t)x(t)dt f(s), s [, ]. (13)

7 97 IJST (2015) 39A1: i wic f d k re kow fuctios d x is ukow fuctio. For solvig tese equtios wit lest squres metod o te sis t i } i0, let x(t) i0 i t i, t [, ], (14) y puttig pproximte solutio (14) i (13) we ve i q i (s) f(s) r (s), s [, ], i0 r (s) is residul fuctio d q i (s) k(s, t)t i dt, s [, ]. For determiig ukow coefficiets i } i0, it is sufficiet to determie lest squres of f i te set q i } i0. I tis sectio, we determie ukow coefficiets i } i0 wit (OLSM), (CLSM), (MCLSM) metods. We tke s 1 for MCLSM. Exmple 3. Exmple 4. 1 e st x(t)dt 0 e s1 1, s [0,1]. s 1 2 cos(st) x(t)dt 1 cos(s)2cos 2 (s)s(si(s)4 cos(s) si(s)1), s [1,2]. s 2 wit te exct solutios e t d t respectively. Te mximum solute errors of exmples 3 d 4 re reported i Tles 8 d 9. Tle 8. Mximum solute errors of (OLSM), (CLSM), (MCLSM) for Exmple 3 OLSM CLSM MCLSM Tle 9. Mximum solute errors of (OLSM), (CLSM), (MCLSM) for Exmple 4 OLSM CLSM MCLSM Cse 2. I tis cse, te sigulr secod order iitil vlue differetil equtios re ivestigted. Tese equtios pper i some models of pysicl prolems d re ivestigted y my resercers (Wzwz, 2002; Kiymz d Mirsyedioglu, 2005; Aslov d Au-Alsik, 2008). Sice tese equtios ve sigulr poits, teir umericl solutios re of prmout importce. Te geerl form of tese equtios re s follows: p(t)y q(t)y r(t)y f(t), t [0, T], y(0) y 0, y (0) y 1, (15) fuctio p s some zeros i [0,T]. To pproximte te solutio i te sis t i } i0 wit lest squres metod, let y(t) i0 i t i (16) y usig te iitil vlues, te ukow prmeters 0 d 1 re determied s follows: 0 y 0, 1 y 1. By puttig (16) i (15) we ve i2 i L i (t) f (t) r (t), (17) r (t) is residul fuctio d L i (t) p(t)(i(i 1))t i2 q(t)it i1 r(t)t i, i 2,,, f (t) f(t) ( 1 q(t) ( 0 1 t)r(t)). For determiig ukow coefficiet i } i2 it is sufficiet to determie lest squres of f i te sis L i } i2. Similr to cse 1 we clculte ukow coefficiets i } i2 wit (OLSM), (CLSM), (MCLSM). We tke s 2 for MCLSM. Exmple 5. t2 y (1 t)y si(t)y f(t), t [0,1]. y(0) 1, y (0) 1

8 IJST (2015) 39A1: Exmple 6. (t 0.5)(t 0.7)y ty e t y f(t), t [0,1], y(0) 1, y (0) 0. Te rigt d side of te ove equtios re so cosidered suc tt e t, cos (t) is te solutios respectively. Te mximum solute errors of (OLSM), (CLSM), (MCLS) for exmple 5, 6 re reported i Tles 10, 11. Tle 10. Mximum solute errors of (OLSM), (CLSM), (MCLSM) for Exmple 5 OLSM CLSM MCLSM Coclusios Accordig to te give umericl results, it is cocluded tt te preseted coditiol lest squres d modified lest squres metods, eside mitiig ccurcy d stility, ve low computtiol cost d tis is vlule dvtge for te ew metods. It sould e metioed tt te prmeter s itroduced i modified ci lest squres is cose experimetlly i suc wy tt y icresig te ill-posedess of te prolem, tis prmeter is cose close to 1. Of course, te preseted metod of sectio 4 c e expressed i oter formts tt will e discussed more i te ext rticles. Tle 11. Mximum solute errors of (OLSM), (CLSM), (MCLSM) for Exmple 6 OLSM CLSM MCLSM Refereces Aks, E. N., Ozdes, A., & Ozis, T. (2006). A umericl solutio of Burgers equtio sed o lest squres pproximtio. Applied Mtemtics d Computtio, 176, Alexder, P., & George, F. P. (1990). A prllel lest squres colloctio cojugte grdiet pproc for te dvectio diffusio equtio. Advces i Wter Resources, 13, Aslov, A., & Au-Alsik, I. (2008). Furter developmets to te decompositio metod for solvig sigulr iitil-vlue prolems. Mtemticl d Computer Modellig, 48, Boli, E., Adolli, A., & Smord, S. (2014). Ci lest squres metod d ill-posed prolems. Iri Jourl of Sciece & Tecology, 38A2, Boli, E., & Delves, L. M. (1979). A Augmeted Glerki Metod for First Kid Fredolm Equtios. J. Ist. Mts. Applics., 24, Boli, E., Lotfi, T., & Pripour, M. (2007). Wvelet momet metod for solvig Fredolm itegrl equtios of te first kid. Applied Mtemtics d Computtio, 186, Blis, C. A. (1989). Advced Egieerig Electromgetics. Wiley, New York. Bitsdze, A. V. (1995). Itegrl Equtios of First Kid. World Scietific Pulisig Co. Pte. Ltd. Cig, L. C., & Su-Yu Y. (2002). Alysis of te L 2 lest squres fiite elemet metod for te velocity vorticity pressure Stokes equtios wit velocity oudry coditios. Applied Mtemtics d Computtio, 130, Dtt, B. N. (2010). Numericl Lier Alger d Applictios. Secod Editio, SIAM. Delves, L. M., & Momed, J. L. (1985). Computtiol Metods for Itegrl Equtios. Cmridge Uiversity Press. Groetsc, C. W. (1984). Te Teory of Tikoov Regulriztio for Fredolm Equtios of te First Kid. Reserc Notes i Mtemtics, vol. 105, Pitm, Bosto. Ito, K., & Kuisc, K. (2008). Lgrge multipliers pproc to vritiol prolems d pplictios. SIAM. Jglur-Mo, J., Feijo, G., & Oeri, A. (2013). A Glerki, lest squres metod for time rmoic Mxwell equtios usig Ne de lec elemets. Jourl of Computtiol Pysics, 235, Jike, S., & Hugo, A. J. (2012). Effects of Jcoi polyomils o te umericl solutio of te pellet equtio usig te ortogol colloctio, Glerki, tu d lest squres metods. Computers & Cemicl Egieerig, 39, Jimig, W. (2012). Lest squres metods for solvig prtil differetil equtios y usig Bzier cotrol poits. Applied Mtemtics d Computtio, 219, Kicid, D. R., & Wrd, C. E. (2002). Numericl Alysis: Mtemtics of Scietific Computig. Americ Mtemticl Society. Kig, B. B., & Krueger, D. A. (2003). Burgers equtio: Glerki lest-squres pproximtios d feedck cotrol. Mtemticl d Computer Modellig, 38,

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