Comparison of B-spline Method and Finite Difference Method to Solve BVP of Linear ODEs
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1 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER Compriso of -splie Metod d Fiite Differece Metod to Solve VP of Lier ODEs Jici Cg Qili Yg Log Zo College of ScieceHebei Uited Uiversit Tgs Hebei 6Ci {ici@eut.edu.c} Abstrct -splie fuctios pl importt roles i bot mtemtics d egieerig. To describe umericl metod for solvig te boudr vlue problem of lier ODE wit secod-order b usig -splie. First te cubic -splie bsis fuctios re itroduced te we use te lier combitio of cubic -splie bsis to pproimte te solutio. Fill we obti te umericl solutio b solvig tri-digol eutios. Te results re compred wit fiite differece metod troug two emples wic sows tt te -splie metod is fesible d efficiet. Ide Terms -splie fuctio oudr-vlue problem Fiite differece metod I. INTRODUCTION Ordir Differetil Eutios (ODE s log istor d widel pplied i m fields. Te umericl solutio of ODE s mde gret developmet i te t cetur. Tere ve bee emerged m ew ides s well s m comple metods for solvig ODE so tt te umericl metods for solvig ODE s bee deepeed. Sstems of ordir differetil eutio ve bee pplied to m problems i psics egieerig biolog d so o. Te teor of splie fuctios is ver ctive field of pproimtio teor d boudr vlue problems ( VPs we umericl spects re cosidered. I tis pper we discuss direct metod bsed o -splie for two-poit boudr vlue problems of secod-order ordir differetil eutio. Tere re m publictios delig wit tis problem wit some metods. Referece []For istce -splie pplied to delig wit te o-lier problems; Referece []A fiite differece metod s bee proposed; Referece [-6]I series of pper b Cglr VPs of tird fift were solved usig fourt d sit-degree splies; - splie metod for solvig lier sstem of secod-order boudr vlue d sigulr boudr vlue problems etc. I te preset pper cubic -splie is used to solve two-poit boudr vlue problems s te followig lier sstems wic re ssumed to ve uiue solutio i te itervl []. m f.( ( were m d f d m re cotiuous. re give fuctios I sectio we ve give te defiitio of te - splie metod. Te splie teciue presets to pproimte te solutio of two-poit boudr vlue problems i sectio. I sectio we ve solved two problems usig te metod d te m-bsolute errors d grps ve lso bee sow. Sectio 5reports te mor coclusio d furter developmets. II. THE CUIC -SPLINE A. Te defiitio of te -splie fuctio Referece [7] Let Ω { } be set of prtitio of[ ] te zero degree -splie is defied s follows: [ i i i oterwise d for positive p it is defied i te followig recursive form: i i p p i p i p i p i p i i p i We ppl tis recursio to get te cubic -splie it is defied s follows: [ [ 6 [ [ oterwise. Te properties of -splie fuctios ( Trsltio Ivrice: i i ( i p p ( Compct Supported: ( Derivtio formul: [ i p i i p ACADEMY PULISHER doi:./cp
2 5 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER Were ( i p p! p! i p i i p i i p i i p Ⅲ. -SPLINE SOLUTIONS FOR LINEAR OUNDARY VALUE Let PROLEMS c. ( be pproimte solutio of E.(were uow rel coefficiet d ci is re cubic -splie fuctios. Let re grid poits i te so tt i i i itervl [ b] b ( b.it is reuired tt te pproimte solutio(stisfies te differetil eutio t te poits. Puttig ( i (it follow tt i [ ( m( ( ( ( ] c i i i i i. ( f ( i i d boudr coditio c be writte s ( c for. ( ( c for. (5 Te splie solutio of ( is obtied b solvig te followig mtri eutio. Te sstems of lier eutios i te uows c c c re obtied usig(c obti te umericl solutio. Tis sstems c be writte i te mtri-vector form s follows: A F. (6 Were c c [ c ] T [ f ( f ( f ] T F d A is ( ( -dimesiol tri-digol mtri give b b c. (7 b c A b c lso te coefficiets i te mtri A ve te followig form 6 i( i m( i ( i ( i bi( i ( i ( i 6 ci( i m ( i ( i ( i Te sstem of lier eutios c be build s sow below: c b c c b c b c c c f 6 f ( IV. NUMERICAL RESULTS Referece [8]I tis sectio two umericl emples re studied b -splie fiite differece metod. Te results obtied b te metod re compred wit te lticl solutio so tt we get te mimum bsolute errors te demostrte te ccurc of te -splie metod. We c fid tt our metod i compriso wit te metod of fiite differece is muc better wit view to ccurc d utiliztio. Moreover te mimum bsolute errors re give i Tble d.te umericl results re illustrted i Fig. d Fig.. Emple : Solve te followig boudr vlue problem e. (8 ( Te lticl solutio: e ACADEMY PULISHER
3 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER 5 Respectivel te observed mimum bsolute errors for vrious vlue of re give i Tble.Te umericl results re illustrted i Fig. d. Metod : we c get te coefficiet mtri A b usig(7 for A ( 6 ( 6 ( 6 ( 6 ( 6 ( 6 Ad F [ ] T Te if. we c fid s follows: Ad we c get te fuctio [ ] T c for emple [. ; [.. ; [.. ; [.. ; [..5 ; [.5.6 ; [.6.7 ; [ ; ; 6 7 [ [. ; Terefore umericl solutios re obtied b te - splie metod te follow tt te lticl solutios re give b.5. (.5 (.85 5( ( ( 7.8 8( 8.5 (.856 d Tus te m-bsolute error is give b δ.55 Referece []Metod Fiite differece metod: At first te itervl of solutio is divided ito m smll regios d get te set of iterl ode. I tese odes we use differece coefficiet isted of differetil. We reect te tructio error d estblis te differetil eutios. Te we c obti te umericl solutio b combiig te boudr coditios. Cosider te lier boudr vlue problem. ( [ b] ACADEMY PULISHER
4 β b. ( Colloctio poits re ot verges i itervl[ ] b let re grid poits i te itervl [ ] b so tt b we use first-order d secod-order cetered differece isted of te first d secod derivtive t te iterl ots d substitute ito Ο Ο Te we get differetil eutio wic tructio error is Ο it follows tt [ ] [ ] Also combied wit te boudr vlue problem β b Ad te lier eutios s follow [ ] [ ] [ ] β Tt is AY Were A Y β From (8 d ( we ve e. ( Hece umericl solutios re obtied b te Fiite differece metod te follow tt lso te m-bsolute error is give b. δ Emple : we solve te followig eutios were Wic s te ect solutio is e e e e e e e e Respectivel te observed mimum bsolute errors for vrious vlue of re give i Tble.Te umericl results re illustrted i Fig. d.as is evidet from te umericl results te preset metod pproimtes te ect solutio ver well. Metod : we c get te coefficiet mtri A b usig(7 for 5 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER ACADEMY PULISHER
5 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER A Ad F [ ] T Te if. we c fid s follows: [ ] T Ad we c get te fuctio c emple for ; [ [.. ; [.. ; [.. ; ; [ [.5.6 ; [.6.7 ; [.7.8 ; [.8. ; [. ; 7 Terefore umericl solutios re obtied b te - splie metod te follow tt te lticl solutios re give b.57.6 (.6 (.758 5( 5. 6( 6.8 7( ( 8.5 (.8 d.6.8 (. (.57 5( ( ( ( 8 8. (.86 Tus te m-bsolute error is give b δ.86 Metod : Te umericl solutios re obtied b te Fiite differece metod te follow tt ACADEMY PULISHER
6 5 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER lso te m-bsolute error is give b δ. From te results we will see te differece betwee tem d coclude tt te -splie metod is te better to iterpolte smoot fuctios t oters. Te umericl results for our emple re sow i Tble d wic sow tt tere is big differece for te errors betwee -splie metod d te Fiite differece metod uless tere is o remrble differece mog te ccurc of te oter metod i te cse were f is sufficietl smoot. TALEⅠ Sows te m-bsolute errors for te two metods wit respect to te true solutio Metods H M-bsolute errors Fiitedifferece.. metod -splie..55 metod TALEⅡ Sows te m-bsolute errors for te two metods wit respect to te true solutio Metods H M-bsolute errors Fiite differece.. metod -splie metod..86 problems of differetil eutios. Severl refereces give i tis pper re of gret prcticl importce but spce costrits did ot llow teir discussio ere. Fill it c be observed from tis rticle tt sigifict mout of wor s bee doe d tere is lrge scope of wor to be doe i tis field. Referece [-]Te bove two emples re te deformtio of sigulr perturbtio problem. Te sigulrl-perturbed differetil eutio is tt ε m f Fig. -splie Fig. Fiite Differece V. CONCLUSION AND OUTLOOK A fmil of -splie metod s bee cosidered for te umericl solutio of boudr vlue problems of lier ordir differetil eutios. Te cubic -splie s bee tested o problem. From te test emples we c s tt te ccurc is better t te fiite differece metod. Te umericl results sowed tt te preset metod is pplicble teciue d pproimtes te solutio ver well. Te implemettio of te preset metod is ver es cceptble d vlid sceme. Tis metod gives comprble results d is es to compute.also tis metod produces splie fuctio wic m be used to obti te solutio t poit i te rge weres te fiite differece metod gives te solutio ol t te cose ots. Tis metod is esil trctble d c redil be pplied to oter Fig. -splie ACADEMY PULISHER
7 JOURNAL OF COMPUTERS VOL. 6 NO. OCTOER 55 Fig. Fiite Differece subect to ( A d ( positive prmeter m d were < ε ε is re sufficietl smoot rel vlued fuctios. It is so ttrctive to mtemticis due to te fct tt te solutio eibits multi-scle crcter tt is regios of rpid cge i te solutio er te ed poits or te solutio eperieces te globl peomeo of rpid oscilltio trougout te etire itervl. Tpicll tese problems rise ver freuetl i fluid dmics elsticit utum mecics cemicl rector teor d m oter llied res. I recet ers tere re wide clss of specil purpose metods vilble for solvig te bove tpe problems. ut tis field will be oe of our future reserc wors. ACKNOWLEDGMENT Tis wor ws supported b Eductiol Commissio of Hebei Provice of Ci (No.8 Z6 Nturl Sciece Foudtio of Hebei Provice of Ci (No.A75 d Nturl Sciece Foudtio of Hebei Provice of Ci (No.A8;. REFERENCES [] Himet Cglr Nz Cglr d Memet Ozer -splie solutio of o-lier sigulr boudr vlue problems risig i psiolog Cos Solutios d Frctls pp.-7. [] M. M. cwl. C. P. tti A fiite differece metod for clss of sigulr two poit boudr vlue problems IMA. J. Number. Al pp [] Nz Cglr Himet Cglr -splie metod for solvig lier sstem of secod-order boudr vlue problems Computers d Mtemtics wit Applictios pp [] Nz Cglr Himet Cglr -splie solutio of sigulr boudr vlue problems Applied Mtemtics d Computtio pp [5] H.N.CglrS.H.Cglr d E.H.Twizell Te umericl solutio of tird-order boudr vlue problems wit fourt-degree -splie futios It. J. Comput. Mt pp.7-8. [6] H. N. Cglr S. H. Cglr d E. H. Twizell Te umericl solutio of fift-order boudr vlue problems wit sit-degree -splie futios Appl.Mt.Lett pp.5-. [7] Wg Re-og Li Cog-u d Zu Cu-gg Computtiol Geometr. eijig: Sciece Press 8. [8] Himet Cglr Nz Cglr d Kled EIfituri splie iterpoltio compred wit fiite differece fiite elemet d fiite volume metods wit pplied to twopoit boudr vlue problems Applied Mtemtics d Computtio pp [] Re Yu-ie Numericl Alsis d MATLA Implemettio. eijig: Higer Eductio Press 8. [] i Li Kiti Li Zegig Ceg -splie solutio of sigulrl perturbed boudr vlue problems risig i biolog Cos Solutios d Frctlspp.-8. [] M. K. Kdlboo Pueet Aror -splie colloctio metod for te sigulr- perturbtio problem usig rtificil viscosit Computers d Mtemtics wit Applictios pp [] SisiG.I.Grid Approimtio of sigulrl perturbed boudr vlue problems wit covective terms Sov.J.Numer.Ad.Mt.Modeligpp [] Mo Kumr A differece metod for sigulr twopoit boudr vlue problems Applied Mtemtics d Computtio pp [] J. Li A Robust Fiite Elemet Metod for Sigulrl Perturbed Elliptic Problem wit Two Smll Prmeters Computtiol d Applied Mtemtics pp.- 8. [5] Mo Kumr A differece metod for sigulr twopoit boudr vlue problems Applied Mtemtics d Computtio pp [6] i Li Kiti Li d Zegig Ceg -splie solutio of sigulrl perturbed boudr vlue problem risig i biolog Cos Solitos d Frctls pp. 8. [7] M. K. Kdlboo Pueet Aror -splie colloctio metod for te sigulr- perturbtio problem usig rtificil viscosit Computers d Mtemtics wit Applictio pp [8] S. Cdr Ser Ro Mues Kumr Epoetil - splie colloctio metod for self-doit sigulrl perturbed boudr vlue problems Applied Numericl Mtemtics pp [] R.K. Mot Nvit J A clss of vrible mes splie i compressio metods for sigulrl perturbed two poit sigulr boudr vlue problems Applied Mtemtics d Computtio pp [] R. K. AWA A Computtiol Metod for Self-Adoit Sigulr Perturbtio Problems Usig Quitic Splie Computers d Mtemtics wit Applictios pp [] R.K. MotUrvsi Aror A fmil of o-uiform mes tesio splie metods for sigulrl perturbed twopoit sigulr boudr vlue problems wit sigifict first derivtives Applied Mtemtics d Computtio pp [] R.H Wg J. C. Cg. A Kid of ivrite Splie Spce Over Rectgulr Prtitio d Pure edig of Ti Plte Applied Mtemtics d Mecics pp [] R.H Wg J. C. Cg. Te Mecicl cgroud of ivrite Splie Spce S Jourl of Iformtio d Computtiol Sciece pp ACADEMY PULISHER
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