Development of a New Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations
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1 IOS Journl o Memics IOSJM ISSN: Volume Issue July-Aug PP -9 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil Equions Ogunrinde. B. dugb S. E. Deprmen o Memicl Sciences Ekii Se Universiy Ado Ekii Nigeri. Absrc: Tis pper is on e developmen o new sceme or e soluion o iniil vlue problems in ordinry dierenil equions. We presen some bsic conceps nd undmenl eories wic re very vil o e developmen o e sceme. Te new sceme is o order seven nd is derivion is bsed on e represenion o e eoreicl soluion by perurbed polynomil uncion o degree our. Tis sceme is suible or problems ssocied wi e sysems o ordinry dierenil equions ving oscillory or eponenil soluion. Key words: Convergence Eponenil Soluion Ordinry Dierenil Equion I. Inroducion In e yers ps lrge number o meods suible or solving ordinry dierenil equions ve been proposed. A mjor impeus o developing numericl procedures ws e invenion o e clculus by Newon nd Leibniz s is led o ccure memicl models or pysicl reliy suc s Sciences Engineering Medicine nd Buess. Tese memicl models cnno be usully solved eplicily nd numericl meod o obin pproime soluions is needed. Anoer imporn spec o e developmen o numericl meods ws e creion o logrims by Npier nd oers giving muc simpler mnner o crrying ou e rimeic operions o muliplicion division nd eponeniion. Up o e le 8 s i ppers mos memicins were quie brod in eir ineres. Generlly e eiciency o ny o ese meods depends on e sbiliy nd ccurcy properies. Te ccurcy properies o dieren meods re usully compred by considering e order o convergence s well s e runcion error coeicien o e vrious meods. Te sources o moivion or is work re ose o wo proposed numericl inegrion sceme o order our nd considered o order ive suied or e soluion o iniil vlue problem in ordinry dierenil equion o e orm b rele ving eponenil soluion were nd b re rel undeermined coeiciens nd re comple prmeers nd inroduced new sceme o order si wic is n improvemen over. Tere re mny numericl inegring scemes o genere e numericl soluions developed by severl uors suc s nd 7 jus o menion ew. In is pper we sll consider e iniil vlue problem o e orm y y y nd develop n lgorim o order seven wic cn eecively solve iniil vlue problem in ordinry dierenil equion. II. Some Bsic Conceps We sll consider e ollowing bsic conceps wic re very vil o e developmen o e new sceme.. Sbiliy A numericl meod is sid o be sble i e dierence beween e numericl soluion nd e ec soluion cn be mde s smll s possible is i ere eiss wo posiive numbers e nd K suc e ollowing olds. y y K e n n. Consisency A numericl sceme wi n incremen uncion n y is sid o be consisen wi e iniil vlue problem under considerion i y y n. Convergence A numericl meod is sid o be convergen i or ll iniil vlue problem sisying e ypoesis o Lipsciz condiion given by Pge
2 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil y y L y y Were e Lipsciz condiion is denoed by L.Te necessry nd suicien condiions or convergence re e sbiliy nd consisency.. ound o Error Tis cn be deined s e error due o compuing device. Tey rise becuse i is possible o represen ll rel numbers ecly on inie-se mcine. I cn be represened memiclly s y p n n n Were yn is e pproime soluion nd pn is e compuer oupu. Te mgniude depends in e sorge nd e rimeic operion doped. In suc cses double precision re employed o gurnee n deque pproimion. III. Developmen o New Sceme o Order Seven Tis secion presens e derivion o e new sceme o order seven or e soluion o iniil vlue problems in ordinry dierenil equions.. Te Bsic Inerpoln Le us ssume e eoreicl soluion y o e IVP problem y y y cn be represened loclly in e inervl by e polynomil inerpoling uncion. b rele Were I we pu nd b re rel undeermined coeiciens nd re comple prmeers. i i i And in we obin e ollowing inerpoling uncion be I we move urer o deine e uncions nd s ollows e 8 rom 7 nd 8 we obin 9 We sll ssume yis e numericl soluion o e eoreicl soluion y nd y. We deine mes poin s ollows:... We en move orwrd o impose e ollowing consrins on e inerpoling uncion 9.. T e inerpoling uncion mus coincide wi e eoreicl soluion nd. In order words we required y i.e. nd y. i.e. b. We lso require e irs second ird our nd i derivives wi respec o o e inerpoling uncion respecively coincide wi e dierenil equion s well s is irs second ird our nd i derivives wi respec o. In e oer words we require : 7 Pge
3 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil Pge Tis implies Noe were i denoes e i derivive o y wi respec o by seing nd. We obin direcly rom 8 nd e 9 rom 8 we ve b rom 7 en rom were
4 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil 7 Pge rom Ten rom Tereore I we subrc rom nd pply 8 we obin e ollowing relions y y e Were = 7 8 To obin e new sceme puing e vlues o 7 nd 8 in we ve
5 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil 8 Pge y y e 9 Equion 9 is e new sceme o order seven.
6 Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil IV. Conclusion Tis pper considers e developmen o new numericl sceme or e soluion o iniil vlue problem in ordinry dierenil equions.aer successul derivion o e lgorim i is obvious e sceme will be useul in solving iniil vlue problem mos especilly problems ving oscillory or eponenil soluions. Compring e sceme wi e eig sndrd scemes especilly o Ibijol 997 wic is o order ive Ogunrinde wic is o order Si in ours is o order seven wic sows i is o iger order.owing o ime nd compuer ime llocion we ve no considered e nlysis o sbiliy propery o is sceme in gre deils. eerences unl S. O. 98: Numericl Inegrors or Si nd Higly Oscillory Dierenil Equions Memics o Compuion 7-9. Gusci W. 9: Numericl Inegrion o Ordinry Dierenil Equions bsed on Trigonomeric Polynomils Numerisce Memik Ibijol E. A. 997: New Scemes or Numericl Inegrion o Specil Iniil Vlue Problems in Ordinry Dierenil Equions P.D. Tesis Universiy o Benin Nigeri. Ibijol E. A nd Ogunrinde. B. : On New Numericl Sceme or e Soluion o Iniil Vlue Problems Acceped Ausrlin Journl o Bsic nd Applied Sciences. Josep-Loius Lgrnge. 8: On oo inding nd Polynomil Inerpolion in Ordinry Dierenil Equions. Ogunrinde. B. 9: On New Numericl Sceme or e Soluion o Iniil Vlue Problem in Ordinry Dierenil Equions. 7 Wllce C. S nd Gup G. K. 97: Generl Liner Mulisep Meods o Solve Ordinry Dierenil Equions. Te Ausrlin Compuer Journl Vol Pge
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