1. INTRODUCTION In this paper, we consider a general ninth order linear boundary value problem (1) subject to boundary conditions

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1 NUMERICAL SOLUTION OF NINTH ORDER BOUNDARY VALUE PROBLEMS BY PETROV-GALERKIN METHOD WITH QUINTIC B-SPLINES AS BASIS FUNCTIONS AND SEXTIC B-SPLINES AS WEIGHT FUNCTIONS K. N. S. Kas Vswaaham a S. V. Kamay Ch. Depame of Mahemacs, Naoal Isue of Techology, Waagal, Ia E-Mal: kas_w@yahoo.co. ABSTRACT I hs pape a fe eleme meho volvg Peov-Galek meho wh quc B-sples as bass fucos a sexc B-sples as wegh fucos has bee evelope o solve a geeal h oe bouay value poblem wh a pacula case of bouay coos. The bass fucos ae eefe o a ew se of bass fucos whch vash o he bouay he Dchle, he Neuma, seco oe a h oe evave ype of bouay coos ae pescbe. The wegh fucos ae also eefe o a ew se of wegh fucos whch umbe mach wh he umbe of eefe bass fucos. The popose meho was apple o solve seveal examples of h oe lea a olea bouay value poblems. The obae umecal esuls wee fou o be goo ageeme wh he exac soluos avalable he leaue. Keywos: peov-galek meho, quc B-sple, sexc B-sple, h oe bouay value poblem, absolue eo.. INTRODUCTION I hs pape, we cose a geeal h oe lea bouay value poblem (9 (8 (7 (6 p( u ( p( u ( p( u ( p3( u ( (5 p( u ( p5( u ( p6( u( p7( u( p ( u( p ( u( b(, c 8 9 subec o bouay coos uc ( A, u ( C, u( c A, u( C u( c A, u( C, u( c A, u ( C, u ( c A, 3 3 A, C, A, C, A, C, A 3, C 3, A ae fe eal cosas a p (, p (, p (, p 3(, p (, p 5(, p 6(, p 7(, p 8(, p 9( a b( ae all couous fucos efe o he eval [c, ]. The h-oe bouay value poblems ae kow o ase he suy of asophyscs, hyoyamc a hyo magec sably []. A class of chaacescvalue poblems of hghe oe (as hghe as wey fou s kow o ase hyoyamc a hyo magec sably []. The exsece a uqueess of he soluo fo hese ypes of poblems have bee scusse Agawal []. Fg he aalycal soluos of such ype of bouay value poblems geeal s o possble. Ove he yeas, may eseaches have woke o hoe bouay value poblems by usg ffee mehos fo umecal soluos. Chawla a Ka [3] evelope a fe ffeece scheme fo he soluo of a specal case of olea hghe oe wo po bouay value poblems. Wazwaz [] evelope he soluo of a specal ype of hghe oe bouay value poblems by usg he ( ( mofe Aoma ecomposo meho a he pove he soluo he fom of a aply covege sees. Hassa a Euk [5] pove soluo of ffee ypes of lea a olea hghe oe bouay value poblems by usg ffeeal asfomao meho. Tauseef a Ahme [6] pesee he soluo of h a eh oe bouay value poblems by usg homoopy peubao meho whou ay scezao, leazao o escve assumpos. Tauseef a Ahme [7] evelope mofe vaaoal meho fo solvg h a eh oe bouay value poblems oucg He's polyomals he coeco fucoal. Jafa a Sh [8] pesee homoopy peubao meho fo solvg he bouay value poblems of hghe oe by efomulag hem as a equvale sysem of egal equaos. Tawfq a Yasse [9] evelope Sem-Aalyc echque fo he soluo of hghe oe bouay value poblems usg wo-po oscllaoy epolao o cosuc polyomal soluo. Hossa a Islam [] pesee he Galek meho wh Legee polyomals as bass fucos fo he soluo of o hghe oe bouay value poblems. Sam [] evelope specal collocao meho fo he soluo of m h oe bouay value poblems wh help of Tchebychev polyomals by coveg he gve ffeeal equao o sysem of fs oe bouay value poblems. Kas Vswaaham a Rey [] solve h oe bouay value poblems by Peov- Galek meho wh quc B-sples as bass fucos a sepc B-sples as wegh fucos. So fa, h oe bouay value poblems have o bee solve by usg Peov-Galek meho wh quc B-sples as bass fucos a sexc B-sples as wegh fucos. Ths movae us o solve a h oe bouay value poblem by Peov-Galek meho wh quc B- sples as bass fucos a sexc B-sples as wegh fucos. 63

2 I hs pape, we y o pese a smple fe eleme meho whch volves Peov-Galek appoach wh quc B-sples as bass fucos a sexc B- sples as wegh fucos o solve a geeal h oe bouay value poblem of he ype (-(. Ths pape s ogaze as follows. Seco eals wh he usfcao fo usg Peov-Galek meho. I Seco 3, a escpo of Peov-Galek meho wh quc B- sples as bass fucos a sexc B-sples as wegh fucos s explae. I pacula we fs ouce he cocep of quc B-sples, sexc B-sples a followe by he popose meho wh he specfe bouay coos. I Seco, he poceue o solve he oal paamees has bee pesee. I seco 5, he popose meho s ese o seveal lea a olea bouay value poblems. The soluo o a olea poblem has bee obae as he lm of a sequece of soluo of lea poblems geeae by he quasleazao echque [3]. Fally, he las seco, he coclusos ae pesee.. JUSTIFICATION FOR USING PETROV- GALERKIN METHOD I Fe Eleme Meho (FEM he appoxmae soluo ca be we as a lea combao of bass fucos whch cosue a bass fo he appoxmao space ue coseao. FEM volves vaaoal mehos lke Raylegh Rz meho, Galek meho, Leas Squaes meho, Peov-Galek meho a Collocao meho ec. I Peov-Galek meho, he esual of appoxmao s mae ohogoal o he wegh fucos. Whe we use Peov-Galek meho, a weak fom of appoxmao soluo fo a gve ffeeal equao exss a s uque ue appopae coos [, 5] especve of popees of a gve ffeeal opeao. Fuhe, a weak soluo also es o a classcal soluo of gve ffeeal equao, pove suffce aeo s gve o he bouay coos [6]. Tha meas he bass fucos shoul vash o he bouay he Dchle ype of bouay coos ae pescbe a also he umbe of wegh fucos shoul mach wh he umbe of bass fucos. Hece hs pape we employe he use of Peov-Galek meho wh quc B-sples as bass fucos a sexc B-sples as wegh fucos o appoxmae he soluo of h oe bouay value poblem. 3. DESCRIPTION OF THE METHOD Defo of quc B-sples a sexc B- sples: The quc B-sples a sexc B-sples ae efe [7-9]. The exsece of quc sple epolae s( o a fuco a close eval [c, ] fo space kos (ee o be evely space of a pao c = < < < - < = s esablshe by cosucg. The cosuco of s( s oe wh he help of he quc B-sples. Iouce e aoal kos -5, -, -3, -, -, +, +, +3, + a +5 such a way ha -5< -< -3< - < -< a < +< +< +3< +< +5. by B, Now he quc B-sples 3 5 ( 3 ( ( a (, [, ] f 5 5 (,, 3 3 f ( ( 3 3 ohewse B (' s ae efe {B -(, B -(, B (, B (,,B -(, B (, B +(, B +(} foms a bass fo he space S 5 ( of quc polyomal sples. Schoebeg [9] has pove ha quc B-sples ae he uque ozeo sples of smalles compac suppo wh he kos a -5< -< -3< < < < <...< -< < +< +< +3< +.< +5. I a smla aalogue sexc B-sples S ('s ae efe by 6 (, [ 3, ] S( 3 (, ohewse a ( f 6 6 (, 3, ( ( f {S -3(, S -(, S -(, S (, S (,,S -(, S (, S +(, S +(} foms a bass fo he space S 6 ( of sexc polyomal sples wh he ouco of wo moe aoal kos -6 a +6 o he aleay exsg kos -5 o +5. Schoebeg [9] has pove ha sexc B- sples ae he uque ozeo sples of smalles compac suppo wh he kos a -6< -5< -< -3< -< -< < < < -< < +< +< +3< +< +5< +6. To solve he bouay value poblem ( subec o bouay coos ( by he Peov-Galek meho wh quc B-sples as bass fucos a sexc B- sples as wegh fucos, we efe he appoxmao fo u( as u ( B( (3 α s ae he oal paamees o be eeme a B ( s ae quc B-sple bass fucos. I Peov- Galek meho, he bass fucos shoul vash o he bouay he Dchle ype of bouay coos ae specfe. I he se of quc B-sples {B -(, B -(, B (, B (,,B -(, B (, B +(, B +(}, he bass 63

3 fucos B -(, B -(, B (, B (, B (, B -(, B -(, B (, B +( a B +( o o vash a oe of he bouay pos. So, hee s a ecessy of eefg he bass fucos o a ew se of bass fucos whch vash o he bouay he Dchle ypes of bouay coos ae specfe. Whe he chose appoxmao sasfes he pescbe bouay coos o mos of he bouay coos, gves bee appoxmao esuls. I vew of hs, he bass fucos ae eefe o a ew se of bass fucos whch vash o he bouay he Dchle, he Neuma, seco a h oe evave ypes of bouay coos ae pescbe. The poceue fo eefg of he bass fucos s as follows. Usg he efo of quc B-sples, he Dchle, he Neuma, seco a h oe evave bouay coos of (, we ge he appoxmae soluo a he bouay pos as A uc ( u ( B( C u( u( B ( (5 A u( c u( B( (6 C u( u( B( (7 A u( c u( B( (8 C u( u( B( (9 A u( c u( B( 3 3 ( C u( u( B ( ( Elmag α -, α -, α, α, α -, α, α + a α + fom he equaos (3 o (, we ge u ( w ( R ( ( A w ( C w ( w ( w3 ( R( R ( (3 R ( R ( A w( C w( w3( w( Q ( Q ( ( Q ( Q ( A w C w w ( w( P ( P ( (5 ( ( P ( P ( A C w( B ( B ( (6 B ( B( R( ( (, R R R ( R ( R (, 3,,..., 3 R ( R( R (, R ( Q ( ( (,, Q Q Q ( R( Q(, 3,,..., 3 Q ( Q( Q(,, Q ( P( ( (,,, P P P ( Q( P(, 3,,..., 3 P ( P( P (,,, P ( (7 (8 (9 B ( B ( B (,,,, B ( ( P( B(, 3,,..., 3 B( B ( B (,,,, B ( The ew se of bass fucos he appoxmao u( s { R, =,,-}. Hee w( akes cae of gve se of Dchle, Neuma, seco a h oe evave ype of bouay coos a R 's a s fs, seco a h oe evaves vash o he bouay. I Peov-Galek meho, he umbe of bass fucos he appoxmao shoul mach wh he umbe of wegh fucos. Hee he umbe of bass 63

4 fucos he appoxmao fo u( efe ( s -3, as he umbe of wegh fucos s +6. So, hee s a ee o eefe he wegh fucos o a ew se of wegh fucos whch umbe mach wh he umbe of bass fucos. The poceue fo eefg he wegh fucos s as follows: Le us we he appoxmao fo v( as v ( S( ( 3 S ( 's ae sexc B-sples a hee we assume ha above appoxmao v( sasfes he coos vc (, v (, v( c, v(, v( c, v(, v( c, v(, v (. ( Applyg he bouay coos ( o (, we ge he appoxmae soluo a he bouay pos as 3 vc ( v ( S( (3 3 v ( v ( S( ( 3 V (3 v ( ( V (, =,,..., - V ( V ( (33 V( V (, 3. V ( U ( U ( U (,, U ( (3 V( U (, 3,,..., U ( U ( U (, 3,. U ( U ( U ( U (,,, U ( (35 U ( U (, 3,,..., U ( U ( U(, 3,,. U ( 3 vc ( v ( S( (5 3 v( v( S( (6 3 v( c v( S ( (7 3 v( v( S ( (8 3 v( c v( S ( (9 3 v( v( S ( (3 ( ( ( (3 3 v v S Elmag β -3, β -, β -, β, β -, β -, β, β + a β + fom he equaos ( a (3 o (3, we ge he appoxmao fo v( as U ( U ( U (,,,, U ( (36 U ( U (, 3,,..., U ( U ( U (, 3,,,. U ( S ( S ( S 3(,,,,, S3( (37 U ( S(, 3,..., S( S( S(, 3,,,,. S( Now he ew se of wegh fucos fo he appoxmao v( s { V V, =,,,-3}. Hee 's a s fs, seco a h oe evaves vash o he bouay. Also fouh oe evave of V s a gh bouay also vash. Applyg he Peov-Galek meho o ( wh he ew se of bass fucos { R, =,,-} a wh he ew se of wegh fucos { V 3}, we ge, =,,,- 633

5 [ p ( u ( p( u p( u ( p( u ( (9 (8 (7 (6 3 p ( u ( p( u ( p( u( p( u( ( pu ( ( p( uv (] ( bv ( ( 8 9 fo,,, -3. (38 Iegag by pas he fs fve ems o he lef ha se of (38 a afe applyg he bouay coos pescbe (, we ge (9 p( u ( V ( [ p ( V ( ] A (39 5 [ p 5 ( V( ] u ( ( 8 p ( u ( V ( [ p ( V ( ] u ( ( (7 ( ( ( p u V [ p V( ] u( ( 3 3 (6 ( ( ( p u V [ p V( ] u( ( (5 ( ( ( p u V [ p V( ] u( (3 Subsug (39 o (3 (38 a usg he appoxmao fo u( gve (, a afe eaagg he ems fo esulg equaos, we ge a sysem of equaos he max fom as A B ( A [ ]; a 5 a {[ p 5 ( V( p ( V( p5 ( V( ] R ( [ p ( V( p6( V( ] R ( [ p 3( V( p7( V( ] R ( [ p ( V( p8 ( V( ] R ( p ( V( R ( } 9 fo =,,, -3; =,, -. (5 B [ b ]; b { bv ( ( {[ p( V ( p( V ( p ( V(] w ( [ p ( V( p ( V(] w( [ p 3( V( p7( V( ] w( [ p ( V( p8( V( ] w ( p 9( V( w(}} p ( V( A fo =,,..., -3. (6 a [ ] T. 3. PROCEDURE TO FIND THE SOLUTION FOR NODAL PARAMETERS A ypcal egal eleme he max A s I m m I m m v ( ( Z(, ( ae he m quc B-sple bass fucos o he evaves, v ( ae he sexc B-sple wegh fucos o he evaves. I may be oe ha I m = f (, ( 3, 3 ( m, m. To evaluae each I m, we employe 7-po Gauss-Legee quaaue fomula. Thus he sffess max A s a welve agoal ba max. The oal paamee veco has bee obae fom he sysem A B usg he ba max soluo package. We have use he FORTRAN-9 pogam o solve he bouay value poblems ( - ( by he popose meho. 63

6 5. NUMERICAL RESULTS To emosae he applcably of he popose meho fo solvg he h oe bouay value poblems of he ype ( a (, we cosee hee lea a wo olea bouay value poblems. The obae umecal esuls fo each poblem ae pesee abula foms a compae wh he exac soluos avalable he leaue. Example : Cose he lea bouay value poblem (9 u u 9 e, (7 subec o u(, u(, u(, u( e, u(, u( e, u(, u( 3 e, u ( 3. The exac soluo fo he above poblem s u ( e. The popose meho s ese o hs poblem he oma [, ] s ve o equal subevals. The obae umecal esuls fo hs poblem ae gve Table-. The maxmum absolue eo obae by he popose meho s x -5. Table-. Numecal esuls fo Example. Absolue eo by he popose meho e e e E E E E E E-6 The popose meho s ese o hs poblem he oma [, ] s ve o equal subevals. The obae umecal esuls fo hs poblem ae gve Table-. The maxmum absolue eo obae by he popose meho s.7697x -5. Table-. Numecal esuls fo Example. Absolue eo by he popose meho..597e E E E E E E E E-5 Example 3: Cose he lea bouay value poblem (9 (7 u u u u s u u 5s cos coss (9 s coscos, subec o u(, u( cos, u(, u( coss, u(, u( scos, u( 3, u( 3coss, u (. The exac soluo fo he above poblem s u cos. The popose meho s ese o hs poblem he oma [, ] s ve o equal subevals. The obae umecal esuls fo hs poblem ae gve Table-3. The maxmum absolue eo obae by he popose meho s.6868x -5. Example : Cose he lea bouay value poblem u s u u ( s e, (8 (9 subec o u(, u( e, u(, u( e, u(, u( e, u(, u( e, u (. u e. The exac soluo fo he above poblem s 635

7 Table-3. Numecal esuls fo Example 3. Absolue eo by he popose meho..9576e-7..96e e E E E E E E-8 Example : Cose he olea bouay value poblem (, (9 u e u e u uu e e e (5 subec o The exac soluo fo he above poblem s u e. The olea bouay value poblem (5 s covee o a sequece of lea bouay value poblems geeae by quasleazao echque [3] as u u u u u e (9 u( ( e ( e u( ( ( ( ( u( u( u( ( ( ( ( e e e ( u e u u f o,,,... subec o u( (, u( ( e, u ( (, u ( ( e, u ( (, u ( ( e, u ( (, u ( ( e, u (. ( Hee ( (5 u s he( h appoxmao fo u (. The oma [, ] s ve o equal subevals a he popose meho s apple o he sequece of lea poblems (5. The obae umecal esuls fo hs poblem ae pesee Table-. The maxmum absolue eo obae by he popose meho s 9.799x -5. Table-. Numecal esuls fo Example. Absolue eo by he popose meho..3886e e E E E E E E E-6 Example 5: Cose he olea bouay value poblem cos, (5 (9 u uu 3 subec o u(, u( s, u(, u( cos, u(, u( s, u(, u( cos, u (. The exac soluo fo he above poblem s u s. The olea bouay value poblem (5 s covee o a sequece of lea bouay value poblems geeae by quasleazao echque [3] as u u u u u u (9 ( ( ( ( ( ( 3 u( u ( cos,,,,... subec o u( (, u( ( s, u ( (, u ( ( cos, u ( (, u ( ( s, u (, u ( cos, u (. ( ( ( Hee ( (53 u s he( h appoxmao fo (. u The oma [, ] s ve o equal subevals a he popose meho s apple o he sequece of lea poblems (53. The obae umecal esuls fo hs poblem ae pesee Table-5. The maxmum absolue eo obae by he popose meho s x

8 Table-5. Numecal esuls fo Example 5. Absolue eo by he popose meho E-7..56E E E E E E E E-6 6. CONCLUSIONS I hs pape, we have employe a Peov- Galek meho wh quc B-sples as bass fucos a sexc B-sples as wegh fucos o solve h oe bouay value poblems wh specal case of bouay coos. The quc B-sple bass se has bee eefe o a ew se of bass fucos whch vash o he bouay he Dchle, he Neuma, seco a h oe evave ypes of bouay coos ae pescbe. The sexc B-sples ae eefe o a ew se of wegh fucos whch umbe mach he umbe of eefe se of bass fucos. The soluo o a olea poblem has bee obae as he lm of a sequece of soluo of lea poblems geeae by he quasleazao echque [3]. The popose meho has bee ese o hee lea a wo olea h oe bouay value poblems. The umecal esuls obae by he popose meho ae goo ageeme wh he exac soluos avalable he leaue. The segh of he popose meho les s easy applcably, accuae a effce o solve h oe bouay value poblems. REFERENCES [] S. Chaashekha. 98. Hyoyamc a Hyomagec sably, Dove, New Yok. [] R. P. Agawal Bouay Value Poblems fo Hghe Oe Dffeeal Equaos, Wol Scefc, Sgapoe. [3] M. M. Chawla a C.P. Ka Fe ffeece mehos fo wo-po bouay value poblems volvg hgh oe ffeeal equaos. BIT. 9: [] A.M Wazwaz.. The appoxmae soluos o bouay value poblems of hghe oe by he mofe ecomposo meho. Compues a Mahemacs wh Applcaos. : [5] I. H. Abel-Halm Hassa a Vea Sua Euk. 9. Soluos of ffee ypes of he lea a olea hghe oe bouay value poblems by ffeeal asfomao meho. Euopea Joual of Pue a Apple Mahemacs. (3: 6-7. [6] Sye Tauseef Mohyu-D a Ahme Ylm.. Soluo of eh a h oe bouay value poblems by homoopy peubao meho, J.KSIAM. (: 7-7. [7] Sye Tauseef Mohyu-D a Ahme Ylm.. Soluos of eh a h oe bouay value poblems by mofe vaaoal eao meho. Applcaos a Apple Mahemacs: A Ieaoal Joual (AAM. 5(: -5. [8] Jafa Sabe-Naaf a Sh Zahmakesh.. Homoopy peubao meho fo solvg hghe oe bouay value poblems. Apple Mahemacal a Compuaoal Sceces. (: 99-. [9] Luma N.M.Tawfq a Samahe M.Yasse. 3. Soluo of hghe oe bouay value poblems usg sem-aalyc echque. Ib Al-Haham Joual fo Pue a Apple Scece. 6(: 8-9. [] M. Bellal Hossa a M. Shafqul Islam.. A ovel umecal appoach fo o hghe oe bouay value poblems, Mahemacal Theoy a Moelg. (5: -. [] Sam Kuma Bhowmk. 5. Tchebychev polyomal appoxmaos fo m h oe bouay value poblems. Ieaoal Joual of Pue a Apple Mahemacs. 98(: [] K.N.S. Kas Vswaaham a S.M. Rey. 5. Numecal Soluo of h oe bouay value poblems by Peov-Galek meho wh quc B- sples as bass fucos a sexc B-sples as wegh fucos. Pocea Egeeg. 7: 7-3. [3] R.E. Bellma a R.E. Kalaba Quasleazao a Nolea Bouay Value Poblems, Ameca Elseve, New Yok. [] L. Bes, F. Joh a M. Schechee. 96. Paal Dffeeal Equaos. Joh Wley Ie scece, New Yok. 637

9 [5] J. L. Los a E. Magees. 97. No-Homogeeous Bouay Value Poblem a Applcaos. Spge- Velag, Bel. [6] A. R. Mchel a R. wa The Fe Eleme Meho Paal Dffeeal Equaos. Joh Wley a Sos, Loo. [7] P. M. Pee Sples a Vaaoal Mehos. Joh-Wley a Sos, New Yok, USA. [8] Cal e-boo A Pacal Gue o Sples. Spge-Velag. [9] I. J. Schoebeg O Sple Fucos, MRC Repo 65, Uvesy of Wscos. 638