Collocation Method for Ninth order Boundary Value Problems Using Quintic B-Splines

Transcription

1 Iteatoal Joual of Egeeg Ivetos e-issn: , p-issn: Volume 5, Issue 7 [Aug. 16] PP: 8-47 Collocato Metod fo Nt ode Bouday Value Poblems Usg Qutc B-Sples S. M. Reddy Depatmet of Scece ad Humates, Seed Isttute of Scece ad Tecology,Hydeabd-Ida-51 1 Abstact: A fte elemet metod volvg collocato metod wt septc B-sples as bass fuctos as bee developed to solve t ode bouday value poblems. Te t, egt ad sevet ode devatves fo te depedet vaable s appoxmated by te fte dffeeces. Te bass fuctos ae edefed to a ew set of bass fuctos wc umbe matc wt te umbe of collocated pots selected te space vaable doma. Te poposed metod s tested o two lea ad oe olea bouday value poblems. Te soluto to a olea poblem as bee obtaed as te lmt of a sequece of solutos of lea poblems geeated by te quasleazato tecque. Numecal esults obtaed by te peset metod ae good ageemet wt te exact solutos avalable te lteatue. Keywods: Collocato metod, Septc B-sple, Nt ode bouday value poblem, Absolute eo. I. INTRODUCTION I ts pape, we cosde a geeal t ode bouday value poblem ( 9 ) ( 8 ) ( 7 ) ( 6 ) ( 5 ) y ( x ) y ( x ) y ( x ) y ( x ) y ( x ) 1 4 (4) y ( x ) y ( x ) y ( x ) y ( x ) y ( x ) b ( x ), c x d subect to bouday codtos y ( c ) A, y ( d ) C, y ( c ) A, y ( d ) C, y ( c ) A, y ( d ) C, y ( c ) A, y ( d ) C, y ( c ) A wee A, A 1, A, A, A 4,C, C 1, C, C, ae fte eal costats ad a (x), a 1 (x), a (x), a (x), a 4 (x), a 5 (x), a 6 (x), a 7 (x), a 8 (x), a 9 (x) ad b(x) ae all cotuous fuctos defed o te teval [c, d]. Te t-ode bouday value poblems ae kow to ase te study of astopyscs, ydodyamc ad ydo magetc stablty [5]. A class of caactestc-value poblems of ge ode (as ge as twety fou) s kow to ase ydodyamc ad ydomagetc stablty [5]. Te exstece ad uqueess of te soluto fo tese types of poblems ave bee dscussed Agawal []. Fdg te aalytcal solutos of suc type of bouday value geeal s ot possble. Ove te yeas, may eseaces ave woked o t-ode bouday value poblems by usg dffeet metods fo umecal solutos. Cawla ad Katt [6] developed a fte dffeece sceme fo te soluto of a specal case of olea ge ode two pot bouday value poblems. Wazwaz [16] developed te soluto of a specal type of ge ode bouday value poblems by usg te modfed Adoma decomposto metod. Hassa ad Etuk [1] povded soluto of dffeet types of lea ad olea ge ode bouday value poblems by usg Dffeetal tasfomato metod. Tauseef ad Amet [14] peseted te soluto of t ad tet ode bouday value poblems by usg omotopy petubato metod wtout ay dscetzato, leazato o estctve assumptos. Tauseef ad Amet [15] developed modfed vaatoal metod fo solvg t ad tet ode bouday value poblems toducg He s polyomals te coecto fuctoal. Jafa ad S [8] peseted omotopy petubato metod fo solvg te bouday value poblems of ge ode by efomulatg tem as a equvalet system of tegal equatos. Tawfq ad Yasse [9] developed semaalytc tecque fo te soluto of ge ode bouday value poblems usg two-pot oscllatoy tepolato to costuct polyomal soluto. Hossa ad Islam [] peseted te Galek metod wt Legede polyomals as bass fuctos fo te soluto of odd ge ode bouday value poblems. Sam [1] developed spectal collocato metod fo te soluto of m t ode bouday value poblems wt elp of Tcebysev polyomals by covetg te gve dffeetal equato to a system of fst ode bouday value poblems. So fa, t ode bouday value poblems ave ot bee solved by usg collocato metod wt septc B-sples as bass fuctos. I ts pape, we ty to peset a smple fte elemet metod wc volves collocato appoac wt septc B-sples as bass fuctos to solve te t ode bouday value poblem of te type (1)-(). Ts pape s ogazed as follows. I secto II of ts pape, te ustfcato fo usg te collocato metod as bee metoed. I secto III, te defto of septc B-sples as bee descbed. I secto IV, descpto of Page 8 (1) ()

2 te collocato metod wt septc B-sples as bass fuctos as bee peseted ad secto V, soluto pocedue to fd te odal paametes s peseted. I secto VI, umecal examples of bot lea ad olea bouday value poblems ae peseted. Te soluto to a olea poblem as bee obtaed as te lmt of a sequece of soluto of lea poblems geeated by te quasleazato tecque [4]. Fally, te last secto s dealt wt coclusos of te pape. II. JUSTIFICATION FOR USING COLLOCATION METHOD I fte elemet metod (FEM) te appoxmate soluto ca be wtte as a lea combato of bass fuctos wc costtute a bass fo te appoxmato space ude cosdeato. FEM volves vaatoal metods lke Rtzs appoac, Galeks appoac, least squaes metod ad collocato metod etc. Te collocato metod seeks a appoxmate soluto by equg te esdual of te dffeetal equato to be detcally zeo at N selected pots te gve space vaable doma wee N s te umbe of bass fuctos te bass [1]. Tat meas, to get a accuate soluto by te collocato metod oe eeds a set of bass fuctos wc umbe matc wt te umbe of collocato pots selected te gve space vaable doma. Fute, te collocato metod s te easest to mplemet amog te vaatoal metods of FEM. We a dffeetal equato s appoxmated by m t ode B-sples, t yelds (m + 1) t ode accuate esults [11]. Hece ts motvated us to solve a t ode bouday value poblem of type (1)-() by collocato metod wt septc B-sples as bass fuctos. III. DEFINITION OF SEPTIC B-SPLINES Te septc B-sples ae defed [7, 11, 1]. Te exstece of septc sple tepolate s(x) to a fucto a closed teval [c, d] fo spaced kots (eed ot be evely spaced) of a patto c = x < x 1 < < x -1 < x = d s establsed by costuctg t. Te costucto of s(x) s doe wt te elp of te septc B-sples. Itoduce foutee addtoal kots x -7, x -6, x -5, x -4, x -, x -, x -1, x +1, x +, x +, x +4, x +5, x +6 ad x +7 suc a way tat x -7 <x -6 <x -5 <x -4 <x - <x - <x -1 <x ad x <x +1 <x + <x + <x +4 < x +5 < x +6 < x +7. Now te septc B-sples B ( x )' s ae defed by 4 ( x x) B x x ( ) 4 ( ), 7, x [ x, x ] 4 4 o t e w se wee ( ), 7 x x f x x ( x x), f x x 7 ad 4 ( x ) ( x x ) 4 wee {B - (x), B - (x), B -1 (x), B (x), B 1 (x),,b (x), B +1 (x), B + (x), B + (x)} foms a bass fo te space s 7 ( ) of septc polyomal sples. Scoebeg [1] as poved tat septc B-sples ae te uque ozeo sples of smallest compact suppot wt te kots at x -7 <x -6 <x -5 <x -4 <x - <x - <x -1 <x <x 1 < <x -1 <x <x +1 <x + <x + <x +4 <x +5 <x +6 <x +7. IV. DESCRIPTION OF THE METHOD To solve te bouday value poblem (1) subect to bouday codtos () by te collocato metod wt septc B-sples as bass fuctos, we defe te appoxmato fo y( x ) as y ( x ) B ( x ) () Page 9

3 Wee ' s ae te odal paametes to be detemed ad B ( x ) ' s ae septc B-sple bass fuctos. I te peset metod, te mes pots x, x,..., x, x ae selected as te collocato pots. I collocato metod, 1 te umbe of bass fuctos te appoxmato sould matc wt te umbe of collocato pots [6]. Hee te umbe of bass fuctos te appoxmato () s +7, wee as te umbe of selected collocato pots s -. So, tee s a eed to edefe te bass fuctos to a ew set of bass fuctos wc umbe matc wt te umbe of selected collocato pots. Te pocedue fo edefg te bass fuctos s as follows: Usg te defto of septc B-sples, te Dclet, Neuma, secod ode devatves bouday codtos, td ode devatve bouday codtos ad fout ode devatve bouday codto of (), we get te appoxmate soluto at te bouday pots as y ( c ) y ( x ) ( ) B x A (4) (5) y ( d ) y ( x ) B ( x ) C y ( c ) y ( x ) B ( x ) A 1 (6) 1 (7) y ( d ) y ( x ) B ( x ) C y ( c ) y ( x ) B ( x ) A (8) (9) y ( d ) y ( x ) B ( x ) C (1) y ( c ) y ( x ) B ( x ) A y ( d ) y ( x ) B ( x ) C (11) ( ) ( ) ( ) 4 (1) y c y x B x A Elmatg,,,,,,, ad fom te equatos () to (1), we get te appoxmato fo y(x) as 1 y ( x ) w ( x ) T ( x ) (1) Wee 1 A w ( x ) 4 w ( x ) w ( x ) S ( x ) 4 ( 4) 1 S ( x ) Page 4

4 A w ( x ) C w ( x ) w ( x ) w ( x ) R ( x ) R ( x ) 4 R( x ) R( x ) A w ( x ) C w ( x ) w ( x ) w ( x ) Q ( x ) Q ( x ) 1 1 Q ( x ) Q ( x ) 1 1 A w ( x ) C w ( x ) w ( x ) w ( x ) P ( x ) P ( x ) 1 P ( x ) P ( x ) A C w ( x ) B ( x ) B ( x ) 1 B ( x ) B ( x ) S ( x ) S ( x ) S ( x ), 1 T ( x) ( ) S x 1, S ( x ), 4, 5,..., 1 (14) R( x ) R ( x ) R ( x ), R( x ) 1,, S ( x ) ( ), R x 4, 5,..., 4 R( x ) R ( x ) R ( x ), R( x ),, 1 Q ( x ) Q ( x ) Q ( x ),,1,, 1 Q ( x ) 1 R ( x ) ( ), 4, 5,..., 4 Q x Q ( x ) Q ( x ) Q ( x ),,, 1, 1 Q ( x ) 1 P( x ) P ( x ) P ( x ), 1,,1,, P ( x ) Q ( x ) ( ), 4, 5,..., 4 P x P( x ) P ( x ) P ( x ),,, 1,, 1 P ( x ) B ( x ) B ( x ) B ( x ),, 1,,1,, B ( x ) P ( x ) ( ), 4, 5,..., 4 B x B ( x ) B ( x ) B ( x ),,, 1,, 1, B ( x ) Page 41

5 Now te ew bass fuctos fo te appoxmato y(x) ae {T (x), =,,, -1} ad tey ae umbe matc wt te umbe of selected collocated pots. Sce te appoxmato fo y(x) (1) s a septc appoxmato, let us appoxmate y (7), y (8) ad y (9) at te selected collocato pots wt fte dffeeces as ( 6 ) ( 6 ) ( 7 ) y y 1 1 y fo,,..., 1 (15) ( 7 ) ( 7 ) ( 7 ) ( 8 ) y y y 1 1 y fo,,..., 1 (16) ( 7 ) y y y y 1 1 y fo,,..., (17) ( 9 ) y y y y 1 y fo 1 (18) wee 1 y y ( x ) w ( x ) T ( x ) (19) Now applyg te collocato metod to (1), we get ( 9 ) ( 8 ) ( 7 ) ( 6 ) ( 5 ) y y y y y y y y y y b ( x ) fo,,, Substtutg (15) to (19) (), we get () ( 6 ) ( 6 ) ( 6 ) w ( x ) S ( x ) w ( x ) S ( x ) w ( x ) S ( x ) w ( x ) ( 6 ) S ( x ) ( 6 ) ( 6 ) ( 6 ) ( 6 ) ( 6 ) ( 6 ) w ( x ) T ( x ) w ( x 1 ) ( ) ( ) ( ) 1 T x w x T x w ( x ) T ( x ) w ( x ) T ( x ) ( 6 ) ( 6 ) ( 5 ) ( 5 ) w ( x ) T ( x ) w ( x ) T 4 ( x ) ( ) ( ) ( ) a x 5 w x T x 1 1 w ( x ) T ( x ) w ( x ) T ( x ) ( ) ( ) ( ) ( ) ( ) 8 w x T x a x w x T x 9 fo =,,, -. (1) Page 4

6 1 1 1 ( 6 ) ( 6 ) ( 6 ) w ( x ) ( ) ( ) ( ) ( ) ( ) ( ) S x w x S x w x S x w x ( 6 ) S ( x ) 1 1 ( 6 ) ( 6 ) ( 6 ) w ( x ) T ( x ) w ( x ) 1 1 ( 6 ) ( 6 ) ( 6 ) T ( x ) w ( x ) T ( x ) w ( x ) T ( x ) w ( x ) T ( x ) ( 6 ) ( 6 ) ( 5 ) ( 5 ) w ( x ) T ( x ) w ( x ) T 4 ( x ) ( ) ( ) ( ) a x 5 w x T x 1 1 w ( x ) T ( x ) w ( x ) T ( x ) w ( x ) T ( x ) w ( x ) T ( x ) 8 9 fo =-1. () Reaagg te tems ad wtg te system of equatos (1) ad () matx fom, we get A B () wee A [ ]; a 1 ( 6 ) ( 6 ) ( 6 ) a T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) ( 6 ) ( 6 ) ( 6 ) ( 5 ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T x a x T x a x T x a x T x fo =,., -; =,,, -1. (4) 1 ( 6 ) ( 6 ) ( 6 ) a T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) 1 1 ( 6 ) ( 6 ) ( 6 ) ( 5 ) T ( x ) T ( x ) T ( x ) T ( x ) T ( x ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T x a x T x a x T x a x T x fo = -1; =,,, -1. (5) B [ b ]; 1 ( 6 ) ( 6 ) ( 6 ) b w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) ( 6 ) ( 6 ) ( 6 ) ( 5 ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) ( ) ( ) ( ) ( ) ( ) ( ) w ( x ) a 6 7 x w x a 8 x w x a 9 x w x fo =,,, -. (6) Page 4

7 1 ( 6 ) ( 6 ) ( 6 ) b w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) ( 6 ) ( 6 ) ( 6 ) ( 5 ) w ( x ) w ( x ) w ( x ) w ( x ) w ( x ) ) ( ) ( ) ( ) ( ) ( ) w ( x ) a ( x w x a x w x a x w x fo = -1. (7) ad [,,, ] T. 1 V. SOLUTION PROCEDURE TO FIND THE NODAL PARAMETERS Te bass fucto S ( x ) s defed oly te teval [ x, x ] ad outsde of ts teval t s zeo. 4 4 Also at te ed pots of te teval [ x, x ] te bass fucto T ( x ) vases. Teefoe, T ( x ) s 4 4 avg o-vasg values at te mes pots x, x, x, x, x ad zeo at te ote mes pots. Te 1 1 fst fou devatves of T ( x ) also ave te same atue at te mes pots as te case of T ( x ). Usg tese facts, we ca say tat te Tus te stff matx A s a twelve dagoal bad matx. Teefoe, te system of equatos () s a tweleve bad system ' s. Te odal paametes ' s ca be obtaed by usg bad matx soluto package. We ave used te FORTRAN-9 pogammg to solve te bouday value poblem (1)-() by te poposed metod VI. NUMERICAL RESULTS To demostate te applcablty of te poposed metod fo solvg te t ode bouday value poblems of te type (1) ad (), we cosdeed two lea ad oe olea bouday value poblems. Te obtaed umecal esults fo eac poblem ae peseted tabula foms ad compaed wt te exact solutos avalable te lteatue. Example 1: Cosde te lea bouday value poblem ( 9 ) y y 9 e x, x 1 (8) subect to y ( ) 1, y (1), y ( ), y (1) e, y ( ) 1, y (1) e, y ( ), y (1) e, y ( ). x Te exact soluto fo te above poblem s y x (1 x ) e. Te poposed metod s tested o ts poblem wee te doma [, 1] s dvded to 1 equal subtevals. Te obtaed umecal esults fo ts poblem ae gve Table 1. Te maxmum absolute eo obtaed by 5 te poposed metod s Table 1: Numecal esults fo Example 1 Absolute eo by x te poposed metod E E E E-5 Page 44

8 Example : Cosde te E E E E E-5 lea bouday value poblem ( 9 ) ( 7 ) y y xy y s x y y 5 xs x co sx x co sx xs x s xco sx xco sx, x 1 subect to y ( ), y (1) co s1, y ( ) 1, y (1) co s1 s1, y ( ), y (1) s1 co s1, (9) y ( ), y (1) co s1 s1, y ( ). Te exact soluto fo te above poblem s y x c o s x. Te poposed metod s tested o ts poblem wee te doma [, 1] s dvded to 1 equal subtevals. Te obtaed umecal esults fo ts poblem ae gve Table. Te maxmum absolute eo obtaed by 5 te poposed metod s Table : Numecal esults fo Example Absolute eo by x te poposed metod E E E E E E E E E-6 Example : Cosde te olea bouday value poblem ( 9 ) y y y co s x, x 1 () subect to y ( ), y (1) s1, y ( ) 1, y (1) co s1, y ( ), y (1) s1, y ( ) 1, y (1) co s1, y ( ). Te exact soluto fo te above poblem s y s x. Te olea bouday value poblem () s coveted to a sequece of lea bouday value poblems geeated by quasleazato tecque [] as ( 9 ) y y y y y y c o s x y y,1,,... (1) ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) Subect to y ( ), y (1) s1, y ( ) 1, y (1) c o s1, y ( ), y (1) s1, ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) y ( ) 1, y (1) c o s1, y ( ). ( 1) ( 1) ( 1) Page 45

9 Hee y s te ( 1) t appoxmato fo y( x ). Te doma [, 1] s dvded to 1 equal subtevals ( 1 ) ad te poposed metod s appled to te sequece of lea poblems (1). Te obtaed umecal esults fo ts poblem ae peseted Table. Te maxmum absolute eo obtaed by te poposed metod s.8616x1-5. Table : Numecal esults fo Example Absolute eo by x te poposed metod E E E E E E E E E-5 VII. CONCLUSIONS I ts pape, we ave developed a collocato metod wt septc B-sples as bass fuctos to solve t ode bouday value poblems. Hee we ave take mes pots x, x,..., x, x as te collocato 1 pots. Te septc B-sple bass set as bee edefed to a ew set of bass fuctos wc umbe matc wt te umbe of selected collocato pots. Te poposed metod s appled to solve seveal umbe of lea ad o-lea poblems to test te effcecy of te metod. Te umecal esults obtaed by te poposed metod ae good ageemet wt te exact solutos avalable te lteatue. Te obectve of ts pape s to peset a smple metod to solve a t ode bouday value poblem ad ts easess fo mplemetato. REFERENCES [1] Abdel-Halm Hassa, I.H. Vedat Suat Etuk. Solutos of dffeet types of te lea ad olea ge ode bouday value poblems by Dffeetal tasfomato metod, Euopea Joual of Pue ad Appled Matematcs, 9, (), pp [] Agawal, R.P. Bouday Value Poblems fo Hge Ode Dffeetal Equatos, 1986, Wold Scetfc, Sgapoe. [] Bellal Hossa, M.d. Safqul Islam, M.d. A Novel Numecal appoac fo odd ge ode bouday value poblems, Matematcal Teoy ad Modelg, 14, 4(5) (14), pp [4] Bellma, R.E.; Kalaba, R.E. Quasleazato ad Nolea Bouday Value Poblems, 1965, Ameca Elseve, New Yok. [5] Cadaseka, S. Hydodyamc ad Hydomagetc Stablty, 1981, Dove, New Yok. [6] Cawla, M.M., Katt, C.P. Fte dffeece metods fo two-pot bouday value poblems volvg g ode dffeetal equatos, BIT, 1979, 19, pp. 7-. [7] Cal de-boo. A Patcal Gude to Sples, 1978, Spge-Velag. [8] Jafa Sabe-Nadaf, S Zamatkes. Homotopy petubato metod fo solvg ge ode bouday value poblems, Appled Matematcal ad Computatoal Sceces, 1, 1(), pp [9] Luma N.M.Tawfq, Samae M.Yasse, Soluto of ge ode bouday value poblems usg Sem- Aalytc tecque, Ib Al-Hatam Joual fo Pue ad Appled Scece, 1, 6(1), pp [1] Reddy, J.N. A Itoducto to te Fte Elemet Metod, d Edto, 5, Tata Mc-GawHll Publsg Compay Ltd., New Del. [11] Pete, P.M. Sples ad Vaatoal Metods, 1989, Jo-Wley ad Sos, New Yok. [1] Sam Kuma Bowmk. Tcebycev polyomal appoxmatos fo mt ode bouday value poblems, Iteatoal Joual of Pue ad Appled Matematcs, 15, 98(1), pp [1] Scoebeg, I.J. O Sple Fuctos, MRC Repot 65, 1966, Uvesty of Wscos. Page 46

10 [14] Syed Tauseef Moyud-D, Amet Yldm. Soluto of tet ad t ode bouday value poblems by omotopy petubato metod, J.KSIAM, 1, 14(1), pp [15] Syed Tauseef Moyud-D, Amet Yldm. Solutos of tet ad t ode bouday value poblems by modfed vaatoal teato metod metod, Applcatos ad Appled Matematcs: A Iteatoal Joual (AAM), 1, 5(1), pp [16] Wazwaz, A.M. Appoxmate solutos to bouday value poblems of ge ode by te modfed decomposto metod, Computes ad Matematcs wt Applcatos,, 4, pp Page 47