The Derivation of Implicit Second Derivative Method for Solving Second-Order Stiff Ordinary Differential Equations Odes.
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1 IOSR Jourl o Mtetcs (IOSR-JM) e-issn: - p-issn: 9-6X. Volue Issue Ver. I (Mr. - Apr. ) PP - Te Dervto o Iplct Secod Dervtve Metod or Solvg Secod-Order St Ordr Deretl Equtos Odes. Y. Skwe Deprtet o Mtetc Adw Stte Uverst Mub-Nger. Correspodg Autor: Y. Skwe Abstrct: A sgle-step brd block etod or tl vlue probles o geerl secod order Ordr Deretl Equtos s bee studed ts pper. I te dervto o te etod power seres s dopted s bss ucto to obt te cotuous scee troug collocto d terpoltos pproc. Tlor etod s lso used togeter wt ew etod to geerte te o-overlppg uercl results. Te ewl costructed etod s te ppled to solve te sste o secod-order st ordr deretl equtos d te ccurc s better we copred wt te estg etods ters o error. Kewords: Power Seres Collocto d Iterpolto Metod Hbrd Block Metod St ODEs Sste o Secod Order ODEs Dte o Subsso: -- Dte o cceptce: I. Itroducto Most rel le probles tt rse vrous elds o stud be t egeerg or scece re odeled s tetcl odels beore te re solved. Tese odels ote led to deretl equtos. Nuerous probles suc s cecl ketcs orbtl dcs crcut d cotrol teor d Newto s secod lw pplctos volve secod-order ODEs [-]. Ordr deretl equtos (ODEs) re cool used or tetcl odelg dverse elds suc s egeerg operto reserc dustrl tetcs bevorl sceces rtcl tellgece geet d socolog. Ts tetcl odelg s te rt o trsltgproble ro pplcto re to trctble tetcl orultos wose teoretcl d uercl lss provde sgt swers d gudce useul or te orgtg pplcto []. Ts tpe o proble c be orulted eter ters o rst-order or ger order ODEs. I ts rtcle te sste o secod-order ODEs o te ollowg or s cosdered. " " " b b b Te etod o solvg ger-order ODEs b reducg te to sste o rst-order pproc volves ore uctos to evlute te d te leds to coputtol burde s etoed []-[]. Te ultstep etods or solvg ger-order ODEs drectl ve bee developed b scolrs suc s [6]- []. Te o ts pper s to develop ew uercl etod or solvg sstes o secod-order st ODEs. II. Dervto O Te Metod I ts secto oe-step brd block etod wt two o-step pots or solvg Equto () s derved. Let te power seres o te or d () v.... DOI:.99/- Pge ()
2 Te Dervto o Iplct Secod dervtve etod or solvg Secod-order St Ordr Deretl DOI:.99/- Pge be te pprote soluto to Equto () or were s N re te rel coecets to be detered v s te uber o collocto pots s te uber o terpolto pots d s costt step sze o te prtto o tervl b wc s gve b b N. Derettg Equto () oce d twce elds:. v (). " v () Iterpoltg Equto () t te selected tervls.e. d colloctg Equto () d () t ll pots te selected tervl.e. d gves te ollowg equtos wc c be wrtte tr or: () Applg te Guss elto etod o Equto () gves te coecet or s. Tese vlues re te substtuted to Equto () to gve te plct cotuous brd etod o te or:. (6) Derettg Equto (6) oce elds:
3 Te Dervto o Iplct Secod dervtve etod or solvg Secod-order St Ordr Deretl DOI:.99/- Pge d d d d. () Were te cotuous scees re
4 I(z) Te Dervto o Iplct Secod dervtve etod or solvg Secod-order St Ordr Deretl III. Covergecelss. Order d error Costts o te Metods Accordg to [] te order o te ew etod Equto () s obted b usg te Tlor seres d t s oud tt te developed etod s uorl order Te wt error costts vector o: C T. Cosstec Deto.:Te brd block etod () s sd to be cosstet t s order ore t or equl to oe.e. P. Tereore te etod s cosstet.. Zero Stblt Deto.: Te brd block etod ()sd to be zero stble te rst crcterstc polol r vg roots suc tt r z d rz te te ultplct o rz ust ot greter t two. I order to d te zero-stblt o brd block etod () we ol cosder te rst crcterstc polol o te etod ccordg to deto (.) s ollows r r B r r z Wc ples r.hece te etod s zero-stble sce r z d rz.. Covergece Teore (.): Cosstec d zero stblt re sucet codto or ler ultstep etod to be coverget.sce te etod () s cosstet d zero stble t ples te etod s coverget or ll pot.. Regos o Absolute Stblt (RAS) Usg te MATLAB pckge we were ble to plot te stblt regos o te block etod (see g. below). Ts s doe b reorultg te block etod s geerl ler etod to obt te vlues o te trces ccordg to [] []. Te trces re substtuted to te stblt tr d usg MATLAB sotwre te bsolute stblt regos o te ew etods re plotted s sow g. below Re(z) Fgure: Rego o Absolute Stblt. DOI:.99/- Pge
5 Te Dervto o Iplct Secod dervtve etod or solvg Secod-order St Ordr Deretl.6 Nuercl Ipleetto To stud te ecec o te block brd etod or K we preset soe uercl eples b Skwe et-l []. I ts secto te perorce o te ew sgle-step brd block etod s eed usg te ollowg two sstes o secod-order tl vlue probles o ordr deretl equtos wt K d K. Tbles d sow te coprso o te uercl results o te ew etod wt te estg etods Skwe et-l []or solvg Eple d. Eple Wt Ect Soluto e e 999 e e. (See Skwe et-l []) Tble : Coprso o bsolute errors or eple. Eple Wt Ect Soluto e e (See Skwe et-l []). DOI:.99/- Pge
6 Te Dervto o Iplct Secod dervtve etod or solvg Secod-order St Ordr Deretl Tble : Coprso o bsolute errors or eple. IV. Coclusos I ts rtcle uorl order egt plct sgle-step block etod wt two o-step pots s derved v te terpolto d collocto pproc. Te bsolute errors rsg ro eples d usg te ew etod were copred wt te estg etod []. Skwe et-l solves eples d t s evdet ro te tbles preseted bove tt te ewl proposed etod perors better t Skwe et-l []. Te etod s lso desrble b vrtue o possessg g order o ccurc. Te developed etod s cosstet A stble coverget wt rego o bsolute stblt d s uorl order egt. Reereces []. Mud R.; Y.Y.A A st order plct brd bckwrd deretto orule (HBDF) or block soluto o ordr deretl equtos. Aer. J. Mt. Sttstcs. p []. M. Alksssbe; Zur O. Iplct oe-step block brd trd-dervtve etod or te drect soluto o tl vlue probles o secod order ordr deretl equtos. J. ppl.t. p. []. Or Z.; Sul M. Prllel r-pot plct block etod or solvg ger order ordr deretl equtos drectl. J. ICT 66. []. Y. Skwe; J. Sbo; P. Tub; T. Y. kg.order Te Iplct Oe-Step Hbrd Block Metod or Te Soluto o St Secodorder Ordr Deretl Equtos.IJEAS []. Jes A.; Ades A.; Josu S. Cotuous block etod or te soluto o secod order tl vlue probles o ordr deretl equto. It. J. Pure Appl. Mt. 6. [6]. Or Z.; Sule M.B. Prllel two-pot eplct block etod or solvg g-order ordr deretl equtos. It. J. Sul. Process Model.6.. J. Coput. Appl. []. Vgo-Agur J.; Ros H. Vrble stepsze pleetto o ultstep etods or Mt []. Ades A.O.; Ake T.A.; Udo O. Iproved cotuous etod or drect soluto o geerl secod order ordr deretl equtos. J. Nger. Assoc. Mt. Ps [9]. Awoe D.O. A P-stble ler ultstep etod or solvg geerl trd order o ordr deretl equtos. It. J. Coput. Mt []. Herc P. Soe Applctos o te Quotet-Derece Algort. Proc. Sp. Appl. Mt []. Lbert J.D. Coputtol Metods ODEs; Jo Wle d Sos: New York NY USA 9. []. Y. Skwe.; J. Sbo.; T. Y. Kg. Te costructo o plct oe-step block brd etods wt ultple o-grd pots or te soluto o st ODEs. JSRR -. []. Y. Skwe; G. M. Kuleg; I. A. Bkr.Secod dervtve Hbrd Block Bckwrd Deretto Forule or Nuercl Soluto o St Sstes. JAMCS p. (): -. Y. Skwe. " Te Dervto o Iplct Secod Dervtve Metod or Solvg Secod-Order St Ordr Deretl Equtos Odes." IOSR Jourl o Mtetcs (IOSR-JM). (): -. DOI:.99/- Pge
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