APPROXIMATE SOLUTION OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS BY MEANS OF A NEW RATIONAL CHEBYSHEV COLLOCATION METHOD
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1 thetcl nd oputtonl Applctons ol. 5 o. pp Assocton for Scentfc eserch APPOXIAE SOLUIO OF HIGHE ODE LIEA DIFFEEIAL EQUAIOS BY EAS OF A EW AIOAL HEBYSHE OLLOAIO EHOD Slh Ylçınbş * esrn Özso ehet Sezer Deprtent of thetcs Fcult of Scence nd Arts ell Br Unerst urde ns ure. slcn@fef.sdu.edu.tr Deprtent of thetcs Educton Fcult of Educton Adnn enderes Unerst Adın ure. nesrnozso@hoo.co Deprtent of thetcs Fcult of Scence uğl Unerst uğl ure. sezer@u.edu.tr Abstrct- In ths pper new pprote ethod for solng hgher-order lner ordnr dfferentl equtons wth rble coeffcents under the ed condtons s presented. he ethod s bsed on the rtonl hebshe u hebshe nd lor collocton ethods. he soluton s obtned n ters of rtonl hebshe functons. Also llustrte eples re gen to deonstrte the ldt nd pplcblt of the ethod. ewords- tonl hebshe Functons Hgher-order Ordnr Dfferentl Equtons lor nd hebshe ollocton ethods.. IODUIO n probles rsng n scence nd engneerng re forulted n bounded nd unbounded dons. ecentl nuber of dfferent ethods ssocted wth orthogonl sstes for solng hgher-order dfferentl equtons whch re the Herte spectrl ethod [] the Lguerre ethod [] the Jcob polnols ethod [5] the ethods bsed on rtonl hebhshe functons [67] the Lguerre tu ethod [8] nd the rtonl hebshe tu ethod [9] he been studed. On the other hnd hebshe nd lor tr nd collocton ethods for the pprote soluton of hgh-order dfferentl nd dfference equtons he been presented n n pper b Sezer et.l. [-6]. In ths pper the hebshe tu [9] the lor collocton [5] ethods re th deeloped nd ppled to the -order lner nonhoogenous dfferentl equton wth the ed condtons P g <
2 6 S. Ylçınbş. Özso nd. Sezer J cj c j λ j c j < j... J... nd the soluton s epressed n ters of the rtonl hebshe functons [9] s follows: n n n < here P nd g re contnuous functons on [ n n... re the coeffcents to be deterned n n... s the rtonl hebshe functons c j c j nd λ re pproprte constnts.. POPEIES OF HE AIOAL HEBYSHE FUIOS [9] In cses when errors ner the ends of n nterl [ b] re prtculr portnce weghtng functon whch s the for / b s often useful. It s supposed gn tht lner chnge n rbles hs trnfored the gen nterl nto the nterl [ ] so tht the weghtng functon becoes w /. In other words gret ret of other tpes of lest-squre polnol pproton cn be forulted n ters of other weghtng functons. In prtculr for the weghtng functon α β w α > β > oer [ ] whch reduces to Legendre cse when α β nd to the hebshe cse when α β /. he well-nown hebshe polnols re orthogonl n the nterl [ ] wth respect to the weght functon w / nd cn be deterned wth the d of the recurrence forule n n n n. he functons re defned b n n or clerl n n n n. hese functons re orthogonl wth respect to the weght functon w / n the nterl [.
3 Approte Soluton of Hgher Order Lner Dfferentl Equtons 7. FUDAEAL AIX ELAIOS Let us frst ssue tht the soluton of Eq. cn be epressed n the for whch s truncted hebshe seres n ters of functons. hen nd ts derte cn be put n the tr fors A ] [ 5 nd... A ] [ so tht... ] [ [ ] A... where... re the functons defned n Eq.... re coeffcents defned n Eq.. If we use the epresson n the functon then the tr becoes or 6 so tht ] [... ] [... In ths cse we re gong to use the lst row for odd lues of nd otherse preous one s the lst row of tr.
4 S. Ylçınbş. Özso nd. Sezer 8 For eple n the cses nd the tr becoes nd 8 8 onsequentl the th derte of the tr defned n 5 fro Eq 6 cn be obtned s nd thereb fro the epresson 5 [ ] A 7 where [ ] nd.. AIX ELAIO BASED O OLLOAIO POIS ow let us defne the collocton ponts s... r r c r 8 so tht. < I c c r hen we substtute the collocton ponts 8 nto Eq. to obtn the sste.... r g P r r r 9 he sste 9 cn be wrtten n the tr for G Y P where p p p P Y g g g G.
5 Approte Soluton of Hgher Order Lner Dfferentl Equtons 9 B puttng the collocton ponts r r... n the relton 7 we he the tr sste [ r ] r A r... or brefl where Y A L L L onsequentl fro the tr fors nd we obtn the fundentl tr equton for Eq. s P A G. et we cn obtn the correspondng tr fors for the condtons s follows: Usng the relton 7 for c we he the fundeentl tr equton correspondng to the ed condtons : J j so tht c j c< j... J. c j j c A [ λ ]... j 5. EHOD OF SOLUIO he fundentl tr equton for Eq. corresponds to sste of lgebrc equtons for the unnown coeffcents o.... Brefl we cn wrte Eq. s WA G or [ W G] so tht [ W ] P p q.... W pq We cn obtn the tr for for the ed condtons b ens of Eq. brefl s where U A λ ] or [U λ... 5 U [ ] J cj j c j [ u u... u ].
6 5 S. Ylçınbş. Özso nd. Sezer o obtn the soluton of Eq. under the condtons b replcng the rows of trces 5 b the lst rows of the tr we he the requred ugented tr w w... w g w w... w g w w... w g W. 6 u u... u λ u u... u λ u u... u λ ~ ~ [ G] If rn ~ ~ W ~ rn [ W G] hus the coeffcents n then we cn wrte ~ ~ W G A. 7 n... re unquel deterned b Eq.6. Also we cn esl chec the ccurc of the obtned solutons s follows [5]: Snce the obtned rtonl hebshe functon epnson s n pprote soluton of Eq. the resultng equton ust be stsfed pprotel tht s for [ b]... E P g or E s n poste nteger. If s n poste nteger s prescrbed then the truncton lt s ncresed untl the dfference E t ech of the ponts becoes sller thn the prescrbed. 6. ILLUSAIE EXAPLE In ths secton seerl nuercl eples re gen to llustrte the ccurc nd effecteness of propertes of the ethod. All of the were perfored on the coputer usng progr wrtten n AHEAIA 5.. he bsolute errors n bles re the lues of t selected ponts. Eple.[9] Eple Let us consder the followng two pont boundr lue proble
7 Approte Soluton of Hgher Order Lner Dfferentl Equtons 5 ] [ 8 wth nd pprote the soluton b the rtonl hebshe functons n n n where g P P P. hen for the collocton ponts re nd the fundentl tr equton of proble s { } G A P P P where P P P re trces of order 5 5 defned b P P P
8 5 S. Ylçınbş. Özso nd. Sezer he ugented tr fors of the condtons for re [ ]. hen we obtn the ugented tr 6 s W [ ~ ~ G ] We obtn the soluton herefore we fnd the soluton or n the for A. whch s ect soluton of two-pont boundr lue proble [9]. Eple.[9] Eple onsder the dfferentl equton [ 9. π
9 Approte Soluton of Hgher Order Lner Dfferentl Equtons 5 We ppled the collocton ethod nd soled ths proble. In ble the resultng lues for nd 8 usng the present ethod together wth the rtonlzed Hr nd u ethod wth 8 nd lso the ect lues of.e ep t dt π re tbulted. he present ethod s lso er effecte nd conenent. he errors n nuercl soluton of Eple re seen n Fgure. he error decreses when the nteger s ncresed. ble. Approtes nd ect lues for Eple Ect tonlzed hebshe Present Present Soluton Hr8 u et8 ethod ethod Fgure. Ect nd other ethod solutons of the Eple
10 5 S. Ylçınbş. Özso nd. Sezer Eple.[7]p.5 onsder the frst order lner ntl lue proble [] Followng the procedures n the preous eples we obtn the ugented tr n the for: [ ~ ~ W G ] hs sste hs the soluton herefore we fnd the soluton or whch s the ect soluton of Eple. A. Eple. Our sple eple s the lner ntl lue proble s follows Usng 7 to deterne the nddul ters of the collocton ethod we fnd Usng leds edtel to the soluton of proble gen b
11 Approte Soluton of Hgher Order Lner Dfferentl Equtons 55 hs epnson s pprote soluton tht s the frst fe ters of the lor seres epnsons of the hebshe soluton gen b Fo nd Prer [7p.7 ]. In Fgure the results obtned b our ethod re copred wth the results of Fo nd Prer [7p.7]. he present ethod s lso er effecte nd conenent. he errors n nuercl soluton of Eple re seen n Fgure. Fgure. uercl nd Fo-Prer soluton of the Eple Fgure. Error nlss for Eple 7. OLUSIO he rtonl hebshe collocton ethod bsed on the rtonl hebshe u nd hebshe-lor collocton ethods re used to sole the hgher-order ordnr dfferentl equtons nuercll. A consderble dntge of the ethod s tht the rtonl hebshe coeffcents of the soluton re found er esl b usng coputer progrs. For ths reson ths process s uch fster thn the other ethods. In ddton n nterestng feture of ths ethod s to fnd the nltcl solutons f the equton hs n ect soluton tht s rtonl functons. Illustrte eples wth the stsfctor results re used to deonstrte the pplcton of ths ethod.
12 56 S. Ylçınbş. Özso nd. Sezer he ethod cn lso be etended to the sste of lner dfferentl equtons wth rble coeffcents but soe odfctons re requred. 8. EFEEES. D. Funro nd O. n Approton of soe dffuson eoluton equtons n unbounded dons b Herte functon th. op B.Y. Guo Erroe estton of Herte spectrl ethod for nonlner prtl dfferentl equton th. op B.Y. Guo nd J. Shen Lguerre-Glern ethod for nonlner prtl dfferentl equtons on se nfnte nterl uer. th J. Shen Stble nd effcent spectrl ethods n unbounded dons usng Lguerre functons SIA J. uer. Anl B.Y. Guo Jcob spectrl pproton nd ts pplctons to dfferentl equtons on hlf lne J. oput. th J.P. Bod Orthogonl rtonl functons on se-nfnte nterl J. oput. Phs J.P. Bod Spectrl ethods usng rtonl bss functons on n nfnte nterl J. oput. Phs H.I. S Lguerre tu ethods for solng hgher-order ordnr dfferentl equtons J. oput. Anl. Appl Prnd nd. zzgh tonl hebshe tu ethod for solng hgher-order ordnr dfferentl equtons Inter. J. oput. th Sezer nd. n hebshe polnol solutons of lner dfferentl equtons Int. J. th. Educ. Sc. echnol A. Auz nd. Sezer hebshe polnol solutons of sstes of hgh-order lner dfferentl equtons wth rble coefcents Appl. th. oput Gulsu nd. Sezer he pprote soluton of hgh-order lner dfference equton wth rble coeffcents n ters of lor polnols Appl. th.oput Sezer nd. Gulsu A new polnol pproch for solng dfference nd Fredhol ntegro-dfference equtons wth ed rguent Appl. th. oput A. rete nd. Sezer A lor collocton ethod for the soluton of lner ntegro-dfferentl equtons Intern. J. oputer th Sezer A. rete nd. Gulsu lor polnol solutons of sstes of lner dfferentl equtons wth rble coeffcents Intern. J. oput. th A. Auz nd. Sezer A hebshe collocton ethod for the soluton of lner dfferentl equtons Intern. J. oput. th L. Fo nd I.B. Prer hebshe polnols n uercl Anlss Oford Unerst press El House London F.B. Hldebrnd Introducton to uercl Anlss cgrw-hll Boo opn Inc. ew Yor 956.
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