Legendre-coefficients Comparison Methods for the Numerical Solution of a Class of Ordinary Differential Equations

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1 IOSR Joual of Mathematcs (IOSRJM) ISS: Volume, Issue (July-Aug 01), PP Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday Dffeetal Equatos Olaguju, A. S. ad Olaegu, D.G. (Depatmet of Mathematcs ad Statstcal Sceces, Kwaa state Uvesty, Malete, gea) Abstact : We eploe the use of Legede polyomals of the fst kd solvg costat coeffcets, ohomogeous dffeetal equatos. To acheve ths, tal soluto s fomulated wth the use of Legede polyomals as bass fuctos. We theeafte apply dect ad dect compaso techques to educe the ete poblem whethe tal o bouday value poblems to a system of algebac equatos. umecal eamples ae gve to llustate the effcecy ad good pefomace of these methods. Keywods: Algebac equatos, Dect compaso, Idect compaso, Legede polyomals. I. ITRODUCTIO Specal atteto has ove the yeas bee gve to applcatos of othogoal fuctos, such as Chebyshev polyomals[1], Laguee polyomals[], Legede polyomals[3], Foue sees[4], block-pulse fuctos to meto a few. These fuctos ad polyomal sees have eceved cosdeable atteto dealg wth vaous poblems Egeeg ad Scetfc applcatos [5]. The ma chaactestc that cuts acoss these applcatos s that they educe these poblems to those of solvg a system of algebac equatos, thus geatly smplfyg the poblem[1]. th The focus of ths pape howeve, volves solvg the ode dffeetal equato of the fom; 1 d y dy d y f, y,,, (1) 1 wth suffcet codtos attached to the physcal boudaes of the poblem. The poblem of ths sot fds elevat applcatos a good umbe of scetfc studes such as populato dyamcs, decomposto of adoactve substaces ad moto of a vbatg body e.t.c. (see[6]) The appoach ths pape bascally etals substtutg to equato (1) a tal soluto of the fom: whee s the ode of the tal soluto, y (, a) () a 0 a ae specalsed codates called Degee of Feedom (DOF), ae othogoal fuctos. As meas of obtag the umecal values of the appomats a, coeffcets compaso techques ae appled, fst tems of coeffcets of depedet vaable (Dect compaso) ad secodly, a bd to ehace esults poduced by ths fst techque, compaso s lkewse caed out tems of coeffcets of Legede polyomals (Idect compaso). P II. LEGEDRE POLYOMIAL Fo easy efeece, the deftos ad ceta popetes of the Legede polyomals of the fst kd ae peseted. These polyomals, whch ae specal cases of Legede fuctos have the fst kd P defed as; ( 1) P P 1 P 1( ) (3) 1 wth tal codtos P 1ad P 0 fom (3), the followg ae the fst few Legede polyomals; P 1 0 P Page

2 Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday 3 1 P P P 4 8 Oe of the techques establshed ths pape demads that a dffeetal equato be wtte puely Legede fom. To acheve ths, a ecuece elato of the fom; ' ( P 1 P ) P (4) (1 ) fo 1 s appled fo the poblem of ode 1 (see [7]). th Fo hghe ode equatos, a ecuece elato fo obtag ode devatves of Legede polyomals s deved as; P ( k)! k ( ) ( k ) ( 1) (5) k0 ( k)!( k!) Fo the sake of poblems that ests tevals othe tha the atual teval 1 1, a shfted veso s obtaed va the use of; b a b a t (6) whch tasfoms the teval [a,b] to the teval [-1,1]. (see[1] ad [8]) Owg to othogoalty popety of these polyomals (Legede), they play a vey mpotat ole the umecal soluto of dffeetal equatos[9]. III. COSTRUCTIO OF TRIAL SOLUTIO The costucto of a tal soluto cossts of costuctg epessos fo each of the tal fuctos equato (). I choosg epesso fo these fuctos, a mpotat pactcal cosdeato s the use of fuctos that ae algebacally as smple as possble ad easy to wok wth [10]. Owg to these easos ad ts othogoalty popetes, Legede polyomals llustated secto of ths pape s used fo the costucto of tal fuctos. Theefoe the appled tal soluto takes the fom; whee P ae Legede polyomals of degee. y (, a) a P (7) IV. SOLUTIO TECHIQUES The appoach ths pape volves substtutg (7) to equato (1), to yeld esdual equato of the fom; 1 Cd y C 1d y Cd y R(, a) C0 y f. (8) 1 The subsequet steps demads geeatg system of equatos fom whch coeffcets of epasos a could be deved. I achevg ths, seveal eseaches have appled a umbe of techques such as collocato method, least squaes method, method of momets to meto a few. (see[10] ad [11]) I ths pape, we apply compaso techques ad ths s caed out two foms amely Dect ad Idect compaso techques. A. Dect Compaso Techque. Ths techque etals compag the coeffcets of depedet vaable equato (8), such that coeffcets of vaable of dffeet degees o the ght had sde (RHS) ae equated coespodgly to 0 15 Page

3 Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday coeffcets of o the left had sde (LHS) of the same esdual equato. Though ths appoach, we geeate a system of algebac equatos of the fom; whee C ad j C11 C1 C13 C1 a0 f 0 C1 C C3 C a1 f1 C 31 C3 C33 C3 a f C 1 C C 3 C a f f ae kow costats ad a ae the DOFs. The mat (9) s the solved alodsde wth equatos deved fom mposto of bouday/tal codtos o (7), thus poducg umecal values of a. These ae theeafte substtuted back to tal soluto (7) theeby yeldg the umecal soluto of equatos (1). B. Idect Compaso Techque. Ths techque etals wtg the ete dffeetal equato lea combato of Legede polyomals P. Ths s acheved such a way that afte tal soluto (7) s substtuted to (1), thee ests a cluso of Legede polyomals ad ts devatves of dffeet degees. These devatves ae epessed tems of Legede polyomals wthout devatves by the use of equato (4) ad (5). At ths pot, equato (1) s beg coveted to lea combato of Legede polyomals of the fom; 0 D P f ( P, P (10) 0 1, ) whee D ae the coeffcets of coespodg Legede polyomals ad P s the Legede polyomal of ode. It should howeve be oted that the RHS of (10) s thus coveted to Legede polyomal fom by the use of techques dscussed [1]. The coeffcets D o the LHS ae the equated coespodg to the coeffcets of P o the RHS. By ths appoach, a system of lea algebac equatos s geeated. Ths system cojucto wth equatos deved fom the mposto of bouday codtos ae solved to obta the DOFs, whch ae theeafte substtuted to tal soluto (7) thus yeldg the umecal soluto of dffeetal equato (1). It s ecessay to ote that equatos deved fom the mposto of gve codtos wth selected equatos fom coeffcets compaso ae tayloed to yeld a system of 1equatos, ths s bascally to gude agast ove-detemed ad ude-detemed cases (see [1]). I cases whee RHS cotaed opolyomal fucto, the fucto s fst epaded though the use of taylo sees epaso afte whch subsequet steps ae appled. V. UMERICAL EXAMPLES The descbed methods of dect ad dect compasos ae ths secto appled to a umbe of eamples typcal of the class of cosdeed poblems (costat coeffcets, o-homogeous dffeetal equatos) ad cuttg acoss poblems of ode oe ad hghe odes. Ths pcple s to llustate the applcablty ad effcecy of these techques ad also wth the am of compag the pefomaces. The ete soluto techque ae automated va the use of symbolc algebac pogam MATLAB. Eample 5.1 Solve the tal value poblem; dy y y( 0) 1 the aalytcal soluto s e 1 (9) Eample 5. Solve the tal value poblem; dy y 3e y(0) 0 16 Page

4 Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday the aalytcal soluto s 3( e e ) Eample 5.3 Solve the bouday value poblem; d y y y(0) y(1) 0 sh wth aalytcal soluto sh1 Eample 5.4 Fd the umecal soluto of the bouday value poblem; 4 3 d y d y d y y e y(0) 1 y''(0) 1 y(1) e y''(1) e the aalytcal soluto s e 5.1 Estmato of umecal eos. I ths subsecto, we epot some epesetatve esults tems of eo accued fom the soluto of eamples The eact eos ae computed by the use of the fomula; Eo= y y Whee y s the eact soluto ad y s the appomate soluto of degee. Table 1: Table of eos fo Eample 5.1 X =4 =6 =8 = Dect Idect Dect e e e Idect e e e Dect e e e e-016 Idect e e e e Dect 4.615e e e e-014 Idect 1.980e e e e Dect 1.873e e e e-01 Idect.156e e e e Dect e e e e-011 Idect e e e e Dect e e e e-010 Idect e e e e Dect e e e e-009 Idect 6.547e e e e Dect 6.819e e e e-009 Idect e e e e Dect e e e e-008 Idect e e e e Dect e e e e-008 Idect 1.059e e e e Page

5 Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday Table : Table of eos fo Eample 5. X =4 =6 =8 = Dect Idect Dect 8.003e e e e-009 Idect e e e e Dect.658e e e e-007 Idect e e e e Dect.095e e e e-006 Idect e e e e Dect e e e e-005 Idect 1.608e e e e Dect.8994e e e e-005 Idect 1.751e e e e Dect e e e e-004 Idect e e e e Dect e e e e-004 Idect e e e e Dect 3.447e e-00.7e e-003 Idect e e e e Dect e e e e-003 Idect e e e e Dect e e e e-003 Idect.761e e e e-003 Table 3: Table of eos fo Eample 5.3 X =4 =6 =8 = Dect Idect Dect 6.344e e e e-009 Idect 7.045e e e e Dect 1.510e e e e-009 Idect 1.305e e e e Dect 1.878e e e e-009 Idect e e e e Dect.4834e e e e-009 Idect.0166e e e e Dect e e e e-009 Idect.003e e e e Dect 3.408e e e e-008 Idect e e e e Dect e e e e-008 Idect 1.739e e e e Dect e e e e-008 Idect e e e e Dect.0897e e e e-008 Idect.0044e e e e Dect e-017 Idect e e-017 Table 4: Table of eos fo Eample 5.4 X =4 =6 =8 = Dect Idect Dect e e e e-008 Idect.1368e e e e Dect e e e e Page

6 Legede-coeffcets Compaso Methods fo the umecal Soluto of a Class of Oday Idect e e e e Dect e e e e-007 Idect e e e e Dect 5.691e e e e-007 Idect e e e e Dect e e e e-007 Idect e e e e Dect e e e e-007 Idect 7.936e e e e Dect 5.946e e e e-007 Idect e e e e Dect e e e e-007 Idect e e e e Dect.1408e e e e-008 Idect.569e e e e Dect Idect VI. Cocluso I ths pape, we gave a umecal teatmet of a class of dffeetal equato both tal ad bouday value poblems though Legede polyomals appled va the use of two kds of compaso techques amely dect ad dect. Cosdeg the esult poduced, t s otced that these two techques ae hghly effectve fo the class of poblem cosdeed, that s, costat coeffcet, o-homogeeous lea dffeetal equatos. Also fom the acheved accuaces as depcted though the tabulated eos, t s futhe obseved that as the degee of tal soluto ceases, bette accuaces wee obtaed theeby mmzg the eos. Also obseved, s the fact that dect compaso techque facltates a evely dstbuto of eos acoss the teval of cosdeato, theeby sevg as a mpovemet to dect compaso method especally o the ote of a eve dstbuto of obtaed eos. Though ths pape, the two techques of dect o dect compaso have bee demostated to yeld esult close eough to the eact soluto as to be useful applcato, ad the computatoal cost ae lkewse mmal whe compaed to a good umbe of estg methods lke Galek weghted esdual method, collocato method, methods of momet, fte dffeece method to meto a few. We theefoe suggest a eteso to othe classes of dffeetal ad tegal equatos. Refeeces [1] Maso, J.C. ad Hadscomb, D.C. (003): Chebyshev Polyomals, Rhapma & Hall CRC, Roca Rato, Lodo, ew Yok, Washgto D.C [] Cauto, C. Ad Quateo, A., Hussa, M.Y. ad Zag, T.A.; Spectal Methods; Fudametals sgle domas (006): Spge- Velag Bel Hedelbeg [3] Gav, B., Stamats, K. ad Kuyag, W. O the postvty of some basc Legede polyomal sums. Joual of Lodo Mathematcs Socety, 59, pp , Cambdge Uvesty Pess (1999). [4] Pesses, R. Ad Bades, M. (199), o the computato of Foue Tasfoms of sgula fuctos, J.Comp. Appl. Math. 43, [5] Tawo, O.A. ad Olaguju, A. S.: Chebyshev Methods fo the umecal Soluto of 4th ode Dffeetal Equatos, Poee Joual of Mathematcs ad Mathematcal scece 3(1), (011) [6] Keyszg, E. Advaced Egeeg mathematcs, 8th ed. Wley, (1999). [7] Olaguju, A.S. Chebyshev- Collocato Appomato Methods fo umecal Soluto of Bouday Value Poblems. (01), a Doctoal thess, Uvesty of Ilo, gea. [8] Boyd, J.P (000): Chebyshev ad Foue Spectal Methods, d ed. Dove, ew Yok. [9] Afke, G., Legede fuctos of the secod kd, Mathematcal methods fo physcsts, 3d ed. Olado, FL: Academc pess, pp , (1985) [10] Davd, S.B. Fte Elemet Aalyss, fom cocepts to applcatos, AT&T Bell Laboatoy, Whppay, ew Jesey, (1987) [11] Hema, B. Collocato Methods fo Voltea tegal ad elated fuctoal Dffeetal Equatos, Cambdge Uvesty pess, ew Yok, (004) [1] Gewal, B.S. (005): umecal Methods Egeeg ad Scece, 7th ed. Kaa Publshes, Delh. 19 Page