AN A(α)-STABLE METHOD FOR SOLVING INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

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1 Advances n Derenal Eqaons and Conrol Processes 4 Pshpa Pblshng Hose, Allahabad, Inda Avalable onlne a hp://pphm.com/ornals/adecp.hm Volme, Nmber, 4, Pages AN A(α)-STABLE METHOD FOR SOLVING INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS Tmoh A. Anake, Shela A. Bshop, Adeola O. Adesana and Mchael C. Agarana Deparmen o Mahemacs Covenan Unvers Oa, Ogn Sae, Ngera e-mal: moh.anake@covenannvers.ed.ng e-mal: shela.bshop@covenannvers.ed.ng e-mal: mchael.agarana@covenannvers.ed.ng Deparmen o Mahemacs and Comper Scence Modbbo Adamma Unvers o Scence and Technolog Yola, Adamawa Sae, Ngera e-mal: zorlar@ahoo.com Absrac An A(α)-sable mplc one sep hbrd mehod or he nmercal approxmaon o solons o nal vale problems o general second order ordnar derenal eqaons s proposed. The mehod s developed b nerpolaon and collocaon o a power seres approxmae solon and mplemened as smlaneos negraors va block mehod. The sabl and convergence o he mehods are deermned. Nmercal expermens are condced on sample problems and he absole error esmaes o he resls are presened. Receved: November, 3; Revsed: Febrar, 4; Acceped: March, 4 Mahemacs Sbec Classcaon: L, L, L. Kewords and phrases: mplc, collocaon, nerpolaon, approxmaon, sabl, convergence.

2 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana. Inrodcon De o sophscaon n compng, mahemacal modelng o real le ssems hrow p complex mahemacal eqaons whch pose he challenge o obanng close orm solons. Mosl, hese complex mahemacal eqaons are eher n he orm o paral derenal eqaons (PDE) or ordnar derenal eqaons (ODE). O neres o s however, s he laer, where he models pose nal vale problems (IVP). For hese knds o problems, developng ecen and accrae nmercal mehods has ncreasngl been o mch neres o researchers n he area o nmercal mehods and analss over he ears. Indeed, hrogh a vare o approaches, several nmercal mehods have been proposed; rangng rom he sngle sep Rnge-Ka pe mehods [, ], hrogh Adam pe mlsep mehods [4,, 9, 8, ] o he now ver poplar block mehods [,, 3, 8, 3, 9]. Besdes hese mehods, are her hbrd varans [,,,, 4, ]. These mehods respecvel, have her sebacks whch mpacs on her ecenc and accrac. Thereore, he overrdng obecve n developng new mehods has alwas been o mprove on he ecenc and convergence wh he lmae am o redcng he error o approxmaon. Ths, s or nenon n hs paper o develop a more ecen and accrae mplc one sep hbrd mehod or he drec solon o general second order IVP o ODE o he orm: ( x,, ), x [ a, b] ( a) ζ, ( a) ζ. Ths class o problems oen arses n areas sch as conrol heor, chemcal knecs, crc heor, mechancs and bolog. Unqe solons have been shown o exs or problems o hs class n []. The lao o hs paper s as ollows: he nex secon descrbes he dervaon o he proposed nmercal mehod, hs s ollowed b he analss o he mehod or sabl and convergence n secon hree. In secon or, ()

3 An A(α)-sable Mehod or Solvng Inal Vale Problems 3 he assocaed block ormlaon or he mplemenaon o he mehod s presened, hs s ollowed b nmercal expermens on some seleced problems n secon ve. Fnall, conclson s gven n secon sx and reerences hereaer.. Dervaon o he Mehod In hs secon, a connos represenaon o an mplc one-sep hbrd mehod s derved. Le πn : a x < x < < xn < xn n be a paron o he negraon nerval [ a, b], no N sbnervals, [ x, x ], wh consan sep sze gven b h x x ;,,..., N. Also, le he bass polnomal be a power seres polnomal o he orm m Y ( x ) a x () compleel deermned b m nknown parameers a,,,,..., m. Inrodcng n osep pons, μ,,,..., n, n, n he n one sep srcre, (see []), a connos mplc one sep hbrd mehod s obaned. Ths s accomplshed, b nerpolang () a he pons x μ( n ) and x μ n a Sormer-Cowell ashon, (see [3]), and collocang () a he n pons x, ( μ ). A combnaon o hese procedres gve rse o a ssem o m eqaons o degree a mos m n he orm: m a x μs μs, s n, n (3a) m ( ) a x r r, r, μ,, (3b) where s and r represen he nerpolaon and collocaon pons respecvel.

4 4 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana The ssem o eqaons (3) s solved or he vales o he nknown parameers a,,,..., m whch are hen sbsed no (). B sng he ransormaon, x x μn n he reslng algebrac ssem, we h obaned he proposed connos mplc one-sep hbrd mehod and s rs dervave n he orm: () α ( ) μn μn αμ ( ) ( ) n μ( n) h β () βμ () μ, (4a) n () α μ ( ) μ α μ( ) ( ) n n n μ( n) () (), h β β μ μ (4b) n where or arbrar d R, αd ( ) and β d ( ) are connos coecens n, ( x ) s he nmercal approxmaon o he analcal solon a he pon x x h and ( x,, ). Obanng vales o b evalang (4a) x x,, μ, ;,,..., n and (4b) a x x,, μ, ;,,..., n, respecvel, he desred dscree nmercal mehods and her rs dervaves are obaned. In parclar, we se n, ha s, sx osep pons are nrodced beween x and x. Then, a power seres approxmae solon () o degree m 9 elds a ssem o eqaons, each compleel deermned b he coecens a,,,..., 9. Followng he procedre descrbed earler we obaned he connos mplc one sep hbrd mehods:

5 An A(α)-sable Mehod or Solvng Inal Vale Problems () () ( ) α α () (), β β μ μ h (a) () ( ) ( ) α α () (). β β μ μ h (b) Obanng vales or b evalang (a) a he pons, x x,, 4, 3,,, elds specc dscree mehods expressed n erms o her coecens n Table. In a smlar manner, when (b) s evalaed a, x x or vales o, specc dervave mehods expressed b her coecens n Table are obaned. Table. The coecens o he mehod (a) or,,, 4, 3,

6 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana Table. The coecens or he mehod (b) or,,, 4, 3,,, In parclar, he man mehod and s rs dervave are obaned as ollows: h, n () h n () 3. Analss o he Mehods In hs secon, he order, local error consan, zero sabl, conssenc, convergence and absole sabl and A(α)-sabl o he mehod () s deermned.

7 An A(α)-sable Mehod or Solvng Inal Vale Problems 3.. Order and error consan To oban he order and error consans or he new mehods, rewre () n he orm o he lnear derence operaor L [ ( x) ; h] ( x h) α ( x μ h) α ( x μ h) μn n μn d β ( x h) βμ ( x μh), (8) h where ( x) C [ a, b] s an arbrar es ncon. Then expand ( x h) and ( x h),, μ, or all respecvel n Talor seres abo x and collec erms n powers o sch ha (8) becomes: L [ ( x) ; h] C ( x) C h ( ) ( x) C h ( ) ( x) C h ( ) ( x) n p ( p C p h ) ( x), (9) where he consan coecens C q, q,,,... are dened as ollows: p p n p C C k k α α C q k k k q ( ) ( ) q q α q q β μ β! μ q. Denon 3... The derence operaor L and he assocaed mehod s sad o be o order p C C C and C. p p. The erm C p s called he error consan and mples ha he

8 8 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana local rncaon error (l..e) s dened b: p p p p 3 l.. e. C h O( h ). () We have esablshed rom or compaon ha mehod () has order p 8 and error consans C p 3.. Zero sabl, conssenc and convergence Denon 3.. The rs and second characersc polnomals o he algorhm () are dened respecvel as k ρ( z ) α, (a) k z σ( z ) β, (b) z where z s he prncpal roo, α k and α β. Denon 3.3. The mehod () s sad o be zero sable as h no roo o (a), ρ ( z) has modls greaer han one, and ever roo o modls one has mlplc no greaer han one. For or mehod (), we obaned (a) as ollows; ρ ( z ) z z z. () Clearl, he condons n Denon 3.3 are sased hence, he algorhm s zero sable. The conssenc o he mehod s esablshed b he ac ha he order o he algorhm s greaer han one, (see []). Followng [], convergence s esablshed b he zero sabl and conssenc o mehod ().

9 3.3. Sabl An A(α)-sable Mehod or Solvng Inal Vale Problems 9 Absole sabl or he algorhm s deermned b means o he bondar locs mehod. Consder he sabl polnomal Π( z, h ) ρ( z) hσ( z), (3) where h h ω d and ω are assmed consan. d The sabl polnomal (3) s obaned b applng he connos mplc one sep hbrd mehods () o he scalar es problem; The ollowng denons shall gde or conclsons. ω. (4) Denon 3.4 (Absole sabl). The algorhm, () s sad o be absolel sable or a gven h all he roos z φ o (3) sas z φ <, φ,,..., ( r ). Denon 3. (Regon o absole sabl). The regon R o he complex h -plane sch ha he roos o he polnomal Π ( z, h ) le whn he n crcle whenever h les n he neror o he regon s called he regon o absole sabl. Denon 3. (A(α)-sabl). A lnear mlsep mehod s A(α)-sable, π α, he regon o absole sabl ncldes he nne wedge S { h : π arg( h ) < α}. () α We esablshed rom or compaon ha mehod (), s absolel sable and ndeed A(α)-sable. The A(α)-sabl proper s shown n Fgre.

10 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana Fgre. A(α)-sabl o mehod (). 4. Implemenaon The mehods obaned rom Tables and are combned o orm a block mehod gven n vecor noaon b A Y E h [ DF( ) BF( )], m m γλ m m () where A s a sqare den marx o order 4; T μ,, μ, E, D, B are consan coecen marces, Y ( ), (, ), T m F ( Ym ) ( μ, ), F( m ) ( ), λ s he power o dervave n (4) and γ s he order o he problem. The consan coecen marces are obaned as ollows: T E 3 4 T D T

11 An A(α)-sable Mehod or Solvng Inal Vale Problems The block ormlaon or he mplemenaon o hese schemes s accordng o [8]. A sngle applcaon o he revsed block ormla generaes smlaneosl, approxmae solons and rs dervave solons a he sep pons x, x and all he osep pons: x μ,,..., n. The procedre s a block b block procedre where nal condons are obaned explcl a x,,,..., N sng he comped vales. The sarng vales or sbseqen block s hen comped rom he prevos block or he mplemenaon o he mehod over he sbnervals: [ x ], [ x, x ],..., [ x N, x ]. x, N. Nmercal Expermens In hs secon, nmercal expermens are perormed sng some sample problems o es he ecenc and accrac o he hbrd mehods. The resls are compared wh resls obaned rom exsng mehods n Tables 3, 4 and respecvel, sng n each case a xed sep sze h.. Problem.. Theorecal solon: x( ), ( ), ( ) ln x. x

12 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana Table 3. Comparson o absole errors and CPU me beween mehod (), [3] and [8] or Problem. Problem.. ( ), π, π 4 3 Theorecal solon: sn x. Table 4. Comparson o absole errors and CPU me beween mehod () and [] or Problem. Problem.3. ψ, ( ), ( ), ψ Theorecal solon: ( x) cos x sn x.

13 An A(α)-sable Mehod or Solvng Inal Vale Problems 3 Table. Comparson o absole errors and CPU me beween mehod () and [] or Problem.3. Conclson An A(α)-sable connos mplc one sep hbrd mehod whch s boh ecen, accrae and economcal has been developed n hs paper. I has been esablshed ha he order p 8. mehod obaned converges ver as or xed sep szes as shown n he me akes o oban solons a he respecve grd pons. I s worh nong ha apar rom servng as sarng vales, he smlaneos block solons can hemselves be sed as negraors. I s also evden ha dervave solons can be obaned a ndvdal grd pons as well. Nmercal expermens perormed on sample problems elded he resls repored n Tables 3, 4 and respecvel. In vew o he comparson made wh solons obaned rom block mehod [8], block predcor-correcor mehod [] and block hbrd predcor-correcor mehod [3], we observed ha mehod () gave beer resl; eldng ver low error o approxmaon and sed lesser CPU me (n seconds) han hese mehods. The mehod developed s recommended or he drec solon o hgher order nal vale problems o ordnar derenal eqaons, even or s problems. Reerences [] A. O. Adesana, M. R. Odeknle and A. O. Adeee, Connos block-hbrd-

14 4 T. A. Anake, S. A. Bshop, A. O. Adesana and M. C. Agarana predcor-correcor mehod or he solon o ( x,, ), Inernaonal Jornal o Mahemacs and So Compng () (), 3-4. [] A. O. Adesana, M. R. Odeknle and M. A. Alkal, Three seps block predcorcorrecor mehod or he solon o general second order ordnar derenal eqaons, Inernaonal Jornal o Engneerng Research and Applcaons (4) (), 9-3. [3] A. O. Adesana, M. R. Odeknle and M. A. Alkal, Order sx block predcorcorrecor mehod or he solon o ( x,, ), Canadan Jornal on Scence and Engneerng Mahemacs 3(4) (), 8-8. [4] A. O. Adesana, T. A. Anake and G. J. Oghonon, Connos mplc mehod or he solon o general second order ordnar derenal eqaons, Jornal o Ngeran Assocaon o Mahemacal Phscs (9), -8. [] A. O. Adesana, T. A. Anake and M. O. Udoh, Improved connos mehod or drec solon o general second order ordnar derenal eqaons, Jornal o Ngeran Assocaon o Mahemacal Phscs 3 (8), 9-. [] T. A. Anake, Connos mplc hbrd one-sep mehods or he solons o general second order nal vale problems o ordnar derenal eqaons, Docoral Thess, Covenan Unvers, Oa, Ngera,. [] T. A. Anake, D. O. Awoem and A. O. Adesana, A one sep mehod or he solon o general second order ordnar derenal Eqaons, Inernaonal Jornal o Scence and Technolog (4) (), 9-3. [8] D. O. Awoem, E. A. Adeble, A. O. Adesana, T. A. Anake, Moded block mehod or he drec solon o second order ordnar derenal eqaons, Inernaonal Jornal o Appled Mahemacs and Compaon 3(3) (), [9] D. O. Awoem, A new Sxh-order algorhm or general second order ordnar derenal eqaon, In. J. Comp. Mah. (), -4. [] D. O. Awoem and O. M. Idow, A class o hbrd collocaon mehod or hrd order ordnar derenal eqaon, Inernaonal Jornal o Comper Mahemacs 8() (), [] J. C. Bcher, Nmercal Mehods or Ordnar Derenal Eqaons, John Wle and Sons Ld., Wes Sssex, 3. [] R. D Ambroso, M. Ferro and B. Paernoser, Two seps hbrd collocaon mehod or ( x, ), Appl. Mah. Le. (9), -8.

15 An A(α)-sable Mehod or Solvng Inal Vale Problems [3] S.O. Fanla, Block mehods or second order IVPs, Inern. J. Comp. Mah. 4(9) (99), -3. [4] C. W. Gear, Hbrd mehods or nal vale problems n ordnar derenal eqaons, SIAM J. Nmer. Anal. (94), 9-8. [] P. Henrc, Dscree Varable Mehods n ODE, John Wle and Sons, New York, 9. [] S. N. Jaor, On a class o hbrd mehods or ( x,, ), In. J. Pre Appl Mah. 9(4) (), [] J. D. Lamber, Compaonal Mehods n Ordnar Derenal Eqaons, John Wle, New York, 93. [8] I. Le, and S. P. Norse, Sper convergence or mlsep collocaon, Mah. Comp. (989), -9. [9] L. F. Shampne and H.A. Was, Block mplc one sep mehods, Mah. Comp. 3(8) (99), 3-4. [] J. Vgo-Aglar and H. Ramos, Varable sepsze mplemenaon o mlsep mehods or ( x,, ), J. Comp. Appl. Mah. 9 (), 4-3. [] D. V. V. Wend, Exsence and nqeness o solons o ordnar derenal eqaons, Proc. Amer. Mah. Soc. 3() (99), -33.

16 Paper # PPH-3-DE Kndl rern he proo aer correcon o: The Pblcaon Manager Pshpa Pblshng Hose Vaa Nwas 98, Mmordgan Allahabad- (Inda) along wh he prn charges* b he ases mal *Invoce aached Proo read b:. Coprgh ranserred o he Pshpa Pblshng Hose Sgnare: Dae:... Tel:... Fax:.. e-mal:.... Nmber o addonal reprns reqred. Cos o a se o copes o addonal Ero. per page. ( copes o reprns are provded o he correspondng ahor ex-gras)