SEPTIC B-SPLINE COLLOCATION METHOD FOR SIXTH ORDER BOUNDARY VALUE PROBLEMS

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1 VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com SETIC -SLINE COLLOCTION METHOD FOR SIXTH ORDER OUNDRY VLUE ROLEMS K.N.S. Ks Vsdhm d. Mul Ksh Deptmet of Mthemtcs Ntol Isttute of Techolog Wgl Id E-Ml: ks_t@hoo.co. STRCT I ths ppe sth ode oud vlue polems s solved umecll collocto method. The soluto s ppomted s le comto of septc -sple fuctos. The septc -sples costtute ss fo the spce of septc poloml sples. I the method the ss fuctos e edefed to e set of ss fuctos hch ume mtch th the ume of selected collocto pots. To test the effcec of the method sevel umecl emples of sth ode le d ole oud vlue polems e solved the poposed method. Numecl esults oted the poposed method e good geemet th the ect solutos vlle the ltetue. Keods: sth ode oud vlue polems collocto method septc -sples d mt solute eo.. INTRODUCTION Geell sth ode oud vlue polems se sevel ches of ppled mthemtcs d phscs. I the ook tte Chdsekh [] e c fd tht he fte hozotl le of flud s heted fom elo d s ude the cto of otto stlt sets. Whe ths stlt s od covecto the od dffeetl equto s sth ode od dffeetl equto. I ths ppe e cosdeed sth ode oud vlue polem of the tpe f g < < suect to Whee f d g e cotuous fuctos o [ ] d d e fte el costts. The estece d uqueess of soluto of such tpe of oud vlue polems c e foud the ook tte gl []. El-Gmel et l. [] used Sc- Glek method to solve sth ode oud vlue polems. km d Sddq [] solved the oud vlue polem of tpe d th o-poloml sple techque. Sddq et l. [5] solved the sme oud vlue polems usg qutc sples. lso Sddq d km [] used septc sples to solve the oud vlue polems of tpe d. Lm et l. [] used sple collocto method to solve the sth ode oud vlue polems. I ths ppe e t to peset smple collocto method usg septc -sples s ss fuctos to solve the sth ode oud vlue polem of tpe d. I fte elemet method FEM the ppomte soluto c e tte s le comto of ss fuctos hch costtute ss fo the ppomto spce ude cosdeto. FEM volves vtol methods lke Rlegh Rtz Glek Lest Sques d Collocto etc. The collocto method seeks ppomte soluto equg the esdul of the dffeetl equto to e detcll zeo t N selected pots collocto pots the gve spce vle dom hee N s the ume of ss fuctos the ss. Tht mes to get ccute soluto the collocto method oe eeds set of ss fuctos hch ume mtch th the ume of collocto pots selected the gve spce vle dom [8] d lso the collocto method s the esest to mplemet mog the vtol methods of FEM. Tht s h e ted to use the collocto method to solve the sth ode oud vlue polem of tpe d. I secto of ths ppe the defto of the septc -sples hs ee desced. I secto the descpto of the method s peseted. I secto the cosstec of the sstem s dscussed. Numecl esults oted the poposed method e peseted secto 5. The poposed method s tested o thee le d to ole sth ode oud vlue polems. The soluto of ole oud vlue polem s oted s the lmt of sequece of solutos of le polems geeted the quslezto techque [9]. Fll the lst the coclusos of the ppe e peseted.. DEFINITION OF SETIC -SLINES The cuc -sples d qutc -sples e defed [ ]. I sml logue the estece of the seveth degee sple tepolte s to fucto closed tevl [] fo spced kots < <...< < s estlshed costuctg t. The costucto of s s doe th the help of the septc -sples. Itoduce foutee ddtol kots 5 5 d such tht < < 5 < < < < < d < < < < < 5 < <. No the seveth degee -sples s e defed

2 VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com othe se ] [ π hee f f d π Hee the set {... } foms ss fo the spce S π of seveth degee poloml sples []. The septc - sples e the uque o zeo sples of smllest compct suppot th kots t < < 5 < < < < < <... < < < < < < 5 < <.. DESCRITION OF THE METHOD To solve the sth ode specl cse oud vlue polem d the collocto method th septc -sples s ss fuctos e defe the ppomto fo s α hee α s e the odl pmetes to e detemed. To ppl the collocto method oe hs to select the collocto pots the gve spce vle dom. These collocto pots ume should mtch th the ume of ss fuctos the ppomto. Hee e hve tke the mesh pots s the selected collocto pots. I the ppomto e c oseve tht the ume of ss fuctos s. ut the ume of mesh pots collocto pots the spce vle dom s. So thee s ecesst to edefe the ss fuctos to e set hch should cot ss fuctos. Fo ths e poceed the follog me. Usg the defto of sth ode -sples desced secto d the oud codtos e get the ppomto fo d t the oud pots s α α 5 α α α 8 α 9 Elmtg α α α α α d α fom the equtos to 9 e get the ppomto fo s α hee ' ' No the e set of ss fuctos s { } d the ume of ss fuctos mtch th the ume of selected collocto pots.

3 VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com pplg the collocto method th the edefed set of ss fuctos... to the polem e get d d α d d f α g fo Retg the ove sstem of equtos the mt fom e get α 8 hee d [ ]; f d fo d 9 d [ ]; g f d fo α ] T L d α [ α α α. CONSISTENCY OF THE SYSTEM Theoem. The mt the sstem 8 s osgul. oof. The ss fucto s defed ol the tevl [ ] d outsde of ths tevl t s zeo. lso t the ed pots of the tevl [ ] the ss fucto vshes. Theefoe s hvg o-vshg vlues t the mesh pots d t othe mesh pots the vlue of s zeo. It s cle tht fom the defto of septc - sples defed secto the vlue of t s domtg fo he comped th the vlues of fo. The devtves of upto sth ode lso hve the sme tue t the mesh pots s the cse of. Usg these fcts e c s tht the mt defed 9 s seve dgol d mt th ozeo etes d domt pcpl dgol elemets. Hece the mt s osgul. Sce the mt s osgul the sstem 8 s cosstet d thus the odl pmetes c e oted usg d mt soluto pckge. We hve used FORTRN pogmmg to develop the pckge d hece oted the esults the poposed method. 5. NUMERICL EXMLES To demostte the pplclt of the poposed method fo solvg the sth ode specl cse oud vlue polems of tpe d e cosdeed thee le d to o le oud vlue polems of such tpe. These emples hve ee chose ecuse ethe ltcl o ppomte solutos e vlle the ltetue d the solutos oted the poposed method e comped th the ect solutos. Emple Cosde the follog sth ode le oud vlue polem e < < suect to e e The ltcl soluto of the ove polem s - e. solute eos oted the poposed method e peseted Tle-. The mmum solute eo oted the poposed method fo ths polem s Tle-. Numecl esults fo the emple th step leg th h.. Ect soluto solute eo poposed method E- 5.E-..95E-.E-5..8E-.9E-..589E-.9E-.5.8E- 8.E-..85E-.8E E-.E E-.858E-5.9.E-.5E-5 Emple Cosde the follog sth ode le oud vlue polem cos < < suect to s cos The ltcl soluto of the ove polem s - s. 8

4 VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com solute eos oted the poposed method e peseted Tle-. The mmum solute eo oted the poposed method fo ths polem s. 5. Tle-. Numecl esults fo the emple th step legth h.. Ect soluto solute eo poposed method E-.9588E E-.59E E- 5.9E-. -.5E-.8E E E E-.898E E-.8E-.8 -.E-.59E E-.85E- Emple Cosde the follog oud vlue polem e < < suect to e e sth ode le The ltcl soluto of the ove polem s - e. solute eos oted the poposed method e peseted Tle-. The mmum solute eo oted the poposed method fo ths polem s. 5. Tle-. Numecl esults fo the emple th step legth h.. Ect soluto solute eo poposed method E-.595E-5. 9.E-.8E E-.5E E- 5.8E-.5 8.E-.595E-..885E-.E-5..59E-.88E E-.99E E-.99E-5 Emple Cosde the follog sth ode ole oud vlue polem e suect to < < e e 5 e The ltcl soluto of the ove polem s e. pplg the quslezto techque [9] to the ove ole polem 5 e get sequece of le polems s [e [ ] e ] fo... Hee epesets th ppomto to. The soluto of ole polem 5 s oted the lmt of sequece of solutos of the le polems. solute eos oted the poposed method e peseted Tle-. The mmum solute eo oted the poposed method fo ths polem s.9. Tle-. Numecl esults fo the emple th step legth h.. Ec t soluto solute eo poposed method..5.e e e e e e-..5.9e e e-5 Emple 5 Cosde the follog sth ode ole oud vlue polem e suect to < < -/e - /e /e The ltcl soluto of the ove polem s e -. pplg the quslezto techque [8] to the ove ole polem e get sequece of le polems s [e [ ] e ] fo

5 VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com Hee epeset Th ppomto to. The soluto of ole polem s oted the lmt of sequece of solutos of the le polems 8. solute eos oted the poposed method e peseted Tle-. The mmum solute eo oted the poposed method fo ths polem s. 5. Tle-5. Numecl esults fo the emple 5 th step legth h.. Ect soluto solute eo poposed method. 9.8E-.995E E-.9E-5..88E-.9E-..E-.E-5.5.5E-.E E-.9E E E E- 8.88E-.9.59E- 5.59E-. CONCLUSIONS I ths ppe e hve developed collocto method th septc -sples s ss fuctos to solve sth ode specl cse oud vlue polem. I the collocto method e hve selected the mesh pots s collocto pots. The septc -sple ss set hs ee edefed to e set hch the ume of ss fuctos s equl to the ume of collocto pots. The poposed method s ppled to solve thee le polems d to o-le polems to test the effcec of the poposed method. The umecl esults oted the poposed method e good geemet th the ect solutos vlle the ltetue. The oectve of ths ppe s to peset smple method to solve sth ode specl cse oud vlue polem. [] Ghzl km Shhd S. Sddq.. Soluto of sth ode oud vlue polem usg opoloml sple techque. ppled Mthemtcs d Computto. 8: 8-. [5] Shhd S. Sddq Ghzl km d Sm Nzee.. utc sple soluto of le sth ode oud vlue polems. ppled Mthemtcs d Computto. 89: [] Shhd S. Sddq d Ghzl km. 8. Septc sple solutos of sth ode oud vlue polems. Joul of computtol d ppled Mthemtcs. 5: 88-. [] dellh Lm Hmd Mou Dss Sh hmed T d hmed Zd. 8. Sple collocto method fo solvg le sth ode oud-vlue polems. Itetol Joul of Compute Mthemtcs. 85: -8. [8] J. N. Redd. 5. toducto to the fte elemet method. d Edto. Tt Mcg Hll ulshg Comp ltd. Ne Delh. [9] R.E ellm d R.E. Kl. 95. uslezto d Nole oud vlue polems. mec Elseve Ne Yok. [] C. de oo. 98. ctcl Gude to Sple Spge-Velg. [] I.J. Schoeeg. 9. O Sple Fuctos. MRC Repot 5 Uvest of Wscos. []. M. ete Sples d Vtol Methods. Joh-Wle d Sos Ne Yok. REFERENCES [] S. Chdsekh. 9. Hdodmc d Hdomgetc stlt. Cledo ess Ofod Repted: Dove ooks Ne Yok. 98. [] R.. gl 98. oud vlue polems fo hghe-ode dffeetl equtos. Wold Scetfc Sgpoe. [] Mohmed El-Gmel Joh R. Co d hmed I zed.. Sc-Glek method fo solvg le sth ode oud vlue polems. Mthemtcs of computto. : 5-.