International Journal of Computer Science and Electronics Engineering (IJCSEE) Volume 3, Issue 1 (2015) ISSN (Online)

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1 Numecal Soluto of Ffth Oe Bouay Value Poblems by Petov-Galek Metho wth Cubc B-sples as Bass Fuctos Qutc B-sples as Weght Fuctos K.N.S.Kas Vswaaham, S.M.Rey Abstact Ths pape eals wth a fte elemet metho volvg Petov-Galek metho wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos to solve a geeal ffth oe bouay value poblem wth a patcula case of bouay cotos. The bass fuctos ae eefe to a ew set of bass fuctos whch vash o the bouay the Dchlet type of bouay cotos ae pescbe. The weght fuctos ae also eefe to a ew set of weght fuctos whch umbe match wth the umbe of eefe bass fuctos. The popose metho was apple to solve seveal eamples of ffth oe lea olea bouay value poblems. The obtae umecal esults wee fou to be goo ageemet wth the eact solutos avalable the lteatue. Keywos Cubc B-sple, Ffth oe bouay value poblem, Petov-Galek metho, Qutc B-sple. I I. INTRODUCTION N ths pape, we cose a geeal ffth oe lea bouay value poblem ( ( a ( y ( a ( y ( a ( y( a ( y( 1 a ( y( a ( y( b(, c (1 subect to bouay cotos y( c A, y( C, y( c A, y( C, y( c A ( 1 1 A, A 1, A, C, C 1 ae fte eal costats a (, a 1 (, a (, a (, a (, a ( b( ae all cotuous fuctos efe o the teval [c,. The ffth oe bouay value poblems occu the mathematcal moellg of the vscoelastc flows othe baches of mathematcal, physcal egeeg sceces [1,. The estece uqueess of the soluto fo these types of poblems have bee scusse Agawal [. Fg the aalytcal solutos of such type of bouay F. A. K.N.S.Kas Vswaaham, Pofesso, Depatmet of Mathematcs, Natoal Isttute of Techology, Waasgal 6, Ia (Phoe: , e-mal: kas_tw@yahoo.co. S. B. S.M.Rey, Reseach Schola, Depatmet of Mathematcs, Natoal Isttute of Techology, Waasgal 6, Ia (e-mal: smmtw@gmal.com. value poblems geeal s ot possble. Ove the yeas, may eseaches have woke o bouay value poblems by usg ffeet methos fo umecal solutos [-[17. The vaous umecal methos ae cosee the lteatue such as Decomposto metho [, Sple techques [, Nopolyomal sple techques [6-[8, Homotopy petubato metho [9, Local polyomal egesso [1, Sc-Galek metho [11, Vaatoal teato metho [1-[1, Collocato metho [1-[1 Galek metho [ So fa, ffth oe bouay value poblems have ot bee solve by usg Petov-Galek metho wth cubc B-sples as bass fuctos qutc B- sples as weght fuctos. Ths motvate us to solve a ffth oe bouay value poblem by Petov-Galek metho wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos. I ths pape, we ty to peset a smple fte elemet metho whch volves Petov-Gelek appoach wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos to solve a geeal ffth oe bouay value poblem of the type (1-(. Ths pape s ogaze as follows. Secto, eals wth the ustfcato fo usg Petov-Galek Metho, I Secto, a escpto of Petov-Galek metho wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos s eplae. I patcula we fst touce the cocept of cubc B-sples, qutc B-sples followe by the popose metho wth the specfe bouay cotos. I Secto, the poceue to solve the oal paametes has bee pesete. I secto, the popose metho s teste o seveal lea olea bouay value poblems. The soluto to a olea poblem has bee obtae as the lmt of a sequece of soluto of lea poblems geeate by the quasleazato techque [18. Fally, the last secto, the coclusos ae pesete. II. JUSTIFICATION FOR USING PETROV-GALERKIN METHOD I Fte Elemet Metho(FEM the appomate soluto ca be wtte as a lea combato of bass fuctos whch costtute a bass fo the appomato space ue coseato. FEM volves vaatoal methos lke 87

2 Raylegh Rtz metho, Galek metho, Least Squaes metho, Petov-Galek metho Collocato metho etc. I Petov-Galek metho, the esual of appomato s mae othogoal to the weght fuctos. Whe we use Petov-Galek metho, a weak fom of appomato soluto fo a gve ffeetal equato ests s uque ue appopate cotos [19-[ espectve of popetes of a gve ffeetal opeato. Futhe, a weak soluto also tes to a classcal soluto of gve ffeetal equato, pove suffcet atteto s gve to the bouay cotos [1. That meas the bass fuctos shoul vash o the bouay the Dchlet type of bouay cotos ae pescbe also the umbe of weght fuctos shoul match wth the umbe of bass fuctos. Hece ths pape we employe the use of Petov- Galek metho wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos to appomate the soluto of a ffth oe bouay value poblem. III. DESCRIPTION OF THE METHOD Defto of cubc B-sples qutc B-sples: The cubc B-sples qutc B-sples ae efe [-[. The estece of cubc sple tepolate s( to a fucto a close teval [c, fo space kots (ee ot be evely space of a patto c s establshe by costuctg t. The costucto of s( s oe wth the help of the cubc B-sples. Itouce s atoal kots -, -, -1, +1, + + such a way that - < - < -1 < < +1 < + < +. Now the cubc B-sples B ( (, B ( ' s ae efe by (, [, ( (,, ( ( othewse f f the set {B (, = -1,,1,,,+1} foms a bass fo the space S ( of cubc polyomal sples. Schoebeg [ has pove that cubc B-sples ae the uque ozeo sples of smallest compact suppot wth the kots at - < - < -1 < < 1 < < -1 < < +1 < + < +. I a smla aalogue qutc B-sples R ('s ae efe by (, [, R( (, othewse ( f (,, ( ( f the set {R (, = -,-1,,1,,,+1,+} foms a bass fo the space S ( of qutc polyomal sples wth the toucto of fou moe atoal kots,,, to the aleay estg kots - to. Schoebeg [ has pove that qutc B-sples ae the uque ozeo sples of smallest compact suppot wth the kots at - < - < - < - < -1 < < 1 < < -1 < < +1 < + < + < + < +. To solve the bouay value poblem (1 subect to bouay cotos ( by the Petov-Galek metho wth cubc B-sples as bass fuctos qutc B-sples as weght fuctos, we efe the appomato fo y( as 1 y( B ( ( 1 α s ae the oal paametes to be eteme B s ae cubc B-sple bass fuctos. I Petov-Galek metho, the bass fuctos shoul vash o the bouay the Dchlet type of bouay cotos ae specfe. I the set of cubc B-sples {B (, = - 1,,1,,,+1}, the bass fuctos B -1 (, B (, B 1 (, B - 1(, B ( B +1 ( o ot vash at oe of the bouay pots. So, thee s a ecessty of eefg the bass fuctos to a ew set of bass fuctos whch vash o the bouay the Dchlet type of bouay cotos ae specfe. The poceue fo eefg of the bass fuctos s as follows. Usg the efto of cubc B-sples the Dchlet bouay cotos of (, we get the appomate soluto at the bouay pots as A y( c y( B ( B ( B ( ( C y( y( B ( B ( B ( ( Elmatg 1 1 fom the equatos (, ( (, we get y( w( P ( (6 A C w( B 1( B 1( (7 B ( B ( 1 1 B ( B ( B 1(,,1 B 1( (8 P( B(,,,..., B( B( B 1(, 1, B 1( The ew set of bass fuctos the appomato y( s {P (, =,1,,}. Hee w( takes cae of gve set of 88

3 Dchlet bouay cotos P s vash o the bouay. I Petov-Galek metho, the umbe of bass fuctos the appomato shoul match wth the umbe of weght fuctos. Hee the umbe of bass fuctos the appomato s + 1, as the umbe of weght fuctos s +. So, thee s a ee to eefe the weght fuctos to a ew set of weght fuctos whch umbe match wth the umbe of bass fuctos. The poceue fo eefg the weght fuctos s as follows: Let us wte the fucto v( as v( R ( (9 R s ae qutc B-sples hee we assume that above fucto v( satsfes coespog homogeeous bouay cotos of Dchlet Neuma bouay cotos of (. That meas v( efe (9 satsfes the cotos v(c=, v(=, vʹ(c=, vʹ(= (1 Applyg the bouay cotos (1 to (9, we get the appomate soluto at the bouay pots as v( c v( R( (11 v( v( R ( (1 v( c v( R( (1 v( v( R( (1 Elmatg β -, β -1, β +1 β + fom the equatos (9 (11 to (1, we get the fucto v( as v( T ( (1 S ( S ( S 1(,,1, S 1( T( S (,,,..., S( S ( S 1(,, 1, S 1( (16 R ( R ( R (, 1,,1, R ( (17 S ( R(,,,..., R( R( R (,, 1,, 1 R ( Let us take {T (, =,1,,} as the set of weght fuctos fo the appomato y( efe (6. Hee T ( s the evatves vash o the bouay. Applyg the Petov-Galek metho to (1 wth the ew set of bass fuctos {P (, =,1,,} wth the ew set of weght fuctos{t (, =,1,,}, we get [ a ( y ( a ( y ( a ( y( a ( y( ( ( 1 a ( y( a ( y( T ( b( T ( (18 fo =,1,, Itegatg by pats the fst thee tems o the left h se of (18 afte applyg the bouay cotos pescbe (, we get ( a( T ( y ( a( T ( y( ( ( ( ( 1 a T A a T C a ( T ( A a ( T ( y( 1 ( 1 1 [ [ a ( T ( y ( a ( T ( y( a ( T ( y ( [ ( ( ( a T y (19 ( (1 Substtutg (19, ( (1 (18 usg the appomato fo y( gve (6, afte eaagg the tems fo esultg equatos, we get a system of equatos the mat fom as A B ( A = [a ; a {[ a1 ( T ( a ( T ( a ( T ( P ( ( [ a ( T ( a ( T ( P ( a ( T ( P ( } a ( T ( P ( fo =, 1,, ; =, 1,, B = [b ; b { b( T ( {[ a1( T ( a( T ( a( T ( w( [ a( T ( a( T ( w( ( a( T ( w( }} a( T ( w( a( T ( A a( T ( C 1 a ( T ( 1 A fo =, 1,..., α = [α α 1 α. 89

4 IV. PROCEDURE TO FIND THE SOLUTION FOR NODAL PARAMETERS A typcal tegal elemet the mat A s 1 m I m m1 Im v ( ( Z( ( ae the cubc B- m sple bass fuctos o the evatves v ( ae the qutc B-sple weght fuctos o the evatves. It may be ote that I = f m (, (, (, m m. To 1 evaluate each I m, we employe -pot Gauss-Legee quaatue fomula. Thus the stffess mat A s a e agoal b mat. The oal paamete vecto has bee obtae fom the system Aα = B usg the b mat soluto package. We have use the FORTRAN-9 pogam to solve the bouay value poblems (1 - ( by the popose metho. V. NUMERICAL RESULTS To emostate the applcablty of the popose metho fo solvg the ffth oe bouay value poblems of the type (1 (, we cosee two lea two olea bouay value poblems. The obtae umecal esults fo each poblem ae pesete tabula foms compae wth the eact solutos avalable the lteatue. EXAMPLE 1: Cose the lea bouay value poblem ( ( y y e y e [ e ( cos ( (1 e ( s, 1 subect to y( y(1, y( 1, y(1 es1, y (. The eact soluto fo the above poblem s y e ( 1 s. The popose metho s teste o ths poblem the oma [, 1 s ve to 1 equal subtevals. The obtae umecal esults fo ths poblem ae gve Table I. The mamum absolute eo obtae by the popose metho s TABLE I NUMERICAL RESULTS FOR EXAMPLE 1 Absolute eo by the popose metho.1.98e e E E E E E-7.8.8E E-6 EXAMPLE : Cose the lea bouay value poblem ( ( y ( y y ( 1 y ( y (6 y e cos 6, 1 subect to y(, y(, y(1 1es1, y(1 e( s1 cos1, y( 6. The eact soluto fo the above poblem s y = e s +. The popose metho s teste o ths poblem the oma [, 1 s ve to 1 equal subtevals. The obtae umecal esults fo ths poblem ae gve Table II. The mamum absolute eo obtae by the popose metho s TABLE II NUMERICAL RESULTS FOR EXAMPLE Absolute eo by the popose metho E E-6..79E E-..199E E E E E- EXAMPLE : Cose the olea bouay value poblem ( y 8 y e, 1 (7 (1 subect to y(, y(1 l, y( 1, y(1., y( 1. The eact soluto fo the above poblem s y=l(1+. The olea bouay value poblem (7 s covete to a sequece of lea bouay value poblems geeate by quasleazato techque [18 as ( y( ( 1 1 ( 1 y e y (8 8 y( y( 1y ( e e,,1,,... (1 subect to y (, y (1 l, y ( 1, y (1., y ( 1. ( 1 ( 1 ( 1 ( 1 ( 1 Hee y (+1 s the (+1 th appomato fo y(. The oma [, 1 s ve to 1 equal subtevals the popose metho s apple to the sequece of lea poblems (8. The obtae umecal esults fo ths poblem ae pesete Table III. The mamum absolute eo obtae by the popose metho s TABLE III NUMERICAL RESULTS FOR EXAMPLE Absolute eo by the popose metho E E-7..78E E E E E E-6.9.E+ EXAMPLE : Cose the olea bouay value poblem ( y y y [ y e y e e [ y e, 1 (9 9

5 subect to y( 1, y(1 e, y(, y(1 e, y(. The eact soluto fo the above poblem s y=e -. The olea bouay value poblem (9 s covete to a sequece of lea bouay value poblems geeate by quasleazato techque [18 as ( y( y e yy [ y e y ( 1 ( ( 1 ( ( 1 y( y( y( ( ( 1 ( ( ( 1 y e y [[ y e 8y e y y( y( [ ( ( (1 ( [ ( e y y e y y e y( y( ( ( e ([ y e 8y e y subect to y ( 1, y (1 e, y (, ( 1 ( 1 ( 1 y (1 e, y (. ( 1 ( 1, ( =,1,, ( Hee y (+1 s the (+1 th appomato fo y(. The oma [, 1 s ve to 1 equal subtevals the popose metho s apple to the sequece of lea poblems (. The obtae umecal esults fo ths poblem ae pesete Table IV. The mamum absolute eo obtae by the popose metho s TABLE IV NUMERICAL RESULTS FOR EXAMPLE Absolute eo by the popose metho.1.769e E E E E E E-6.8.1E E-6 VI. CONCLUSION The umecal esults obtae by the popose metho ae goo ageemet wth the eact solutos avalable the lteatue. The stegth of the popose metho les ts easy applcablty, accuate effcet to solve ffth oe bouay value poblems. [6 Muhamma Azam Kha, Sa-ul-Islam, Ikam A.Tmz, E.H.Twzell Saaat Ashaf, A class of methos base o o-polyomal sple fuctos fo the soluto of a specal ffth oe bouay value poblems, Joual of Mathematcal Aalyss Applcatos, vol. 1, pp , 6. [7 Shah S.Sq Ghazala Akam, Soluto of ffth oe bouay value poblems usg o-polyomla sple techque, Apple Mathematcs Computato, vol. 17, pp , 6. [8 Shah S. Sq, Ghazala Akam Salma A.Malk, Noployomal setc sple metho fo the soluto alog wth covegece of Lea specal case ffth oe two-pot bouay value poblems, Apple Mathematcs Computato, vol. 19, pp. -1, 7. [9 Muhamma Aslam Noo Sye Tauseef Mohyu-D, A e_cet algothm fo solvg ffth-oe bouay value poblems, Mathematcal Compute Moellg, vol., pp. 9-96, 7. [1 Hkmet Cagla Naza Cagla, Soluto of ffth-oe bouay value poblems by usg Local polyomal egesso, Apple Mathematcs Computato, vol. 186, pp. 9-96, 7. [11 Mohame El-Gamel, Sc the umecal soluto of ffth oe bouay value poblems, Apple Mathematcs Computato, vol. 187, pp , 7. [1 Muhamma Aslam Noo Sye Tauseef Mohyu-D, Vaatoal teato metho fo ffth oe bouay value poblems usg He s ployomals, Mathematcal Poblems Egeeg, vol. 8, Acle I [1 Zhao-Chuwu, Appomate aalytcal solutos of ffth oe bouay value poblems by the Vaatoal teato metho, Compute Mathematcs wth Applcatos, vol. 8, pp. 1-17, 9. [1 A.Lam, H.Maou, D.Sbbh A.T, Setc sple soluto of ffth oe bouay value poblems, Mathematcs Computes Smulato, vol. 77, pp. 7-6, 8. [1 K.N.S.Kas Vswaaham Y.Showy Rau, Quatc B-sple collocato metho fo ffth oe bouay value poblems, Iteatoal Joual of Compute Applcatos, vol. (1, pp. 1-6, 1. [16 K.N.S.Kas Vswaaham P.Mual Ksha, Qutc B-sple Galek metho fo ffth-oe bouay value poblems, ARPN Joumal of Egeeg Apple Sceces, vol. (, pp. 7-77, 1. [17 K.N.S.Kas Vswaaham Seevasulu Ballem, Numecal soluto of ffth-oe bouay value poblems by Galek metho wth quatc B- sples, Iteatoal Joual of Compute Applcatos, vol. 77(17, pp. 7-1, 1. [18 R.E.Bellma R.E. Kalaba, Quasleazato Nolea Bouay Value Poblems, Ameca Elseve, New Yok, 196. [19 L.Bes, F.Joh M.Schechete, Patal D_eetal Equatos, Joh Wley Ite scece, New Yok, (196. [ J.L.Los E.Magees, No-Homogeeous Bouay Value Poblem Applcatos. Spge-Velag, Bel, 197. [1 A.R.Mtchel R.wat, The Fte Elemet Metho Patal D_eetal Equatos, Joh Wley Sos, Loo, [ P.M. Pete, Sples Vaatoal Methos, Joh-Wley Sos, New Yok, [ Cal e-boo, A Patcal Gue to Sples, Spge-Velag, [ I.J. Schoebeg, O Sple Fuctos, MRC Repot 6, Uvesty of Wscos, REFERENCES [1 A.R.Daves, A.Kaageoghs T.N.Phllps, Spectal Galek methos fo the pmay two-pot bouay value poblem moellg vscoelastc flows, Iteatoal Joual fo Numecal Methos Egeeg, vol.6, pp , [ H.N.Cagla, S.H.Cagla E.H.Twzell, The umecal soluto of ffth oe bouay value poblems wth sth egee B-sple fuctos, Apple Mathematcs Lettes, vol. 1, pp. -, [ R.P. Agawal, Bouay Value Poblems fo Hghe Oe D_eetal Equatos, Wol Scetfc, Sgapoe, [ Abul-Ma Wazwaz, The umecal soluto of ffth oe bouay value poblems by the Decomposto metho, Joual of Computatoal Apple Mathematcs, vol. 16, pp. 9-7, 1. [ Shah S. Sq Ghazala Akam, Setc sple solutos of ffth oe bouay value poblems, Apple Mathematcs Lettes, vol., pp , 7. 91