μ are complex parameters. Other

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1 A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria. Departmet of Mathematics ad Computer Sciece, Modibbo Adama Uiversity of Techology, Yola, Nigeria. joshuasuday000@yahoo.com * ABSTRACT This research paper presets the developmet, aalysis, ad implemetatio of a ew umerical itegrator capable of solvig first order iitial value problems i ordiary differetial equatios. The algorithm developed is based o a local represetatio of the theoretical solutio y ( ) to the iitial value problem by a oliear iterpolatig fuctio (comprisig of the combiatio of polyomial, epoetial ad cyclometric fuctios). The itegrator is further applied o sampled problems to geerate umerical results. From the results obtaied, the ew umerical itegrator ca be said to be computatioally reliable ad igeious. (Keywords: umerical itegrator, iterpolatig fuctio, oliear, iitial value problem, approimatios) INTRODUCTION Most pheomea that occur owadays i the fields of physical, chemical, biological, ad maagemet scieces, or i egieerig ad ecoomics, ca be modeled i the form of differetial equatios. It is also iterestig to ote that solutios to most differetial equatios that arise from these models caot be easily obtaied by aalytical meas. Therefore, approimate solutios are eeded which are geerated by meas of umerical techiques. As reported i Mickes (994), for arbitrary values of the system parameters at the preset time, oly umerical itegratio techique ca provide accurate solutio to the origial differetial equatio. Also, for ay umerical method to be coverget, it has to be a sufficietly accurate represetatio of the differetial system (Lambert, 99). I this paper, we develop a ew umerical itegrator capable of solvig equatios of the form, y = f(, y), y( ) = y () 0 0 May umerical itegratio schemes to geerate the umerical solutios to problems of the form () have bee proposed ad developed by several authors. Most of these itegrators were developed by represetig the theoretical solutio y ( ) to Equatio () by a iterpolatig fuctio (liear or oliear) f ( ). This type of costructio was first reported i Fatula (976). He proposed a umerical itegrator which is particularly well suited to solve problems of the form () havig oscillatory or epoetial solutios. This method was based o the local represetatio of the theoretical solutio y ( ) to the IVP () i the iterval [, ] by a oliear polyomial iterpolatig ( ) fuctio F( ) = a a breale ρ μ, where 0 a0, aadb are real udetermied coefficiets, while ρad μ are comple parameters. Other schemes iclude those developed by Ademiluyi (987), Ibijola (997), Kama ad Ibijola (000), Wazwaz (000), Ibijola ad Oguride (00), Ibijola ad Suday (00), ad Ibijola, Bamisile ad Suday (0), to metio a few. Studies have show that the Iterpolats used by the authors above basically cosist of the combiatio of a polyomial ad epoetial fuctio. Havig see the performace of these schemes, we are motivated ad challeged to ivestigate what happes if a oliear Iterpolat that cosists of the combiatio of The Pacific Joural of Sciece ad Techology Volume. Number. May 0 (Sprig)

2 polyomial, epoetial ad cyclometric (trigoometric) fuctios is used to derive a ew umerical itegrator. We shall state without proof, the theorem that guaratees the eistece ad uiqueess of solutio of the IVP (). Theorem (Lambert, 97; Fatula, 988) Let f (, y) be defied ad cotiuous for all poits (, y) i the regio D defied by {(, y) : a b, < y < } where a ad b fiite, ad let there eist a costat L such that for every, y, y such that (, y) ad (, y ) are both i D : f (, y) f (, y ) L y y () The if η is ay give umber, there eist a uique solutio y () of the iitial value problem (), where y ( ) is cotiuous ad differetiable for all (, yid. ) The iequality () is kow as a Lipschitz coditio ad the costat L as a Lipschitz costat. Defiitio (Lambert 99) A Numerical Scheme or Numerical Method or Numerical Itegrator (sometimes shorteed to scheme or method or itegrator ) is a differece equatio ivolvig a umber of cosecutive approimatios y j, j = 0,,,..., k from which it will be possible to compute sequetially the sequece{ y = 0,,,..., N}. The iteger k is called the step umber of the scheme. If, k =, the method is called oe-step, while if k >, the method is called a multi-step or k step. THE NEW INTERPOLANT We develop a ew Iterpolat by combiig polyomial, epoetial ad cyclometric (trigoometric) fuctios. Let us assume that the theoretical solutio y ( ) to the iitial value problem () ca be locally represeted i the iterval,[, ], 0 by the o-liear polyomial iterpolatig fuctio, F( ) = a a a a e bsi () 0 where a0, a, a, a, b, ad are udetermied coefficiets. Let y be the umerical estimate to the theoretical solutio y( ) at the poit = ad that f = f(, y). Let also, = 0 ( ) h, = 0,,,... (4) We the move forward to impose the followig coditios o the ew iterpolatig fuctio (); a) That the iterpolatig fuctio must coicide with the theoretical solutio at = ad =. I other words, we require that, F( ) = a0 a a ae bsi (5) ad F( ) = a a a a e bsi (6) 0 It implies that F( ) = y( ) ad F( ) = y( ) from Equatios (5) ad (6), respectively. b) We further require that the first, secod, third ad fourth derivatives with respect to of the iterpolatig Fuctio (), respectively coicide with the differetial equatio as well as its first, secod, third ad fourth derivatives with respect to at. I other words, we require that, F ( ) = f F ( ) f F ( ) f = = ad, 4 F ( ) = f (7) The Pacific Joural of Sciece ad Techology Volume. Number. May 0 (Sprig)

3 DERIVATION OF THE NEW NUMERICAL INTEGRATOR If F( ) ad F( ) coicide with yad y respectively ad that () i F ( ) deotes the ith total derivative of f ( y, ) with respect to, ad also adopt the Maclauri series epasio for the epoetial fuctio, we have: e = =... r r 0 r!!! (8) We ow substitute the first few terms of Equatio (8) ito (5), this gives: F( ) = y = a a a 0 a ( ) bsi!! (9) Differetiatig Equatio (9) with respect to at gives: F( ) = a a a ( ) bcos f (0) = F ( ) = a a ( ) bsi = f () () () F ( ) = a bcos = f () F ( ) = bsi = f () (4) From Equatio (), b f si = (4) = ( cot( )) a f f (6) Puttig Equatios (4) ad (6) ito Equatio (), we have: a ( cot( ))( f f (7) f ) si = f si a = f ( f f cot( )) f (8) Substitutig Equatios (4), (6) ad (8) ito Equatio (0), we have: a f ( f f cot( )) f ( cot( )) f f f cos = f si (9) a = f f ( f f cot( )) f ( f f cot ( )) f cot ( ) (0) Usig the assumptio that F( ) = y ad F( ) = y, we subtract Equatio (5) from (6), this gives us: Substitute Equatio (4) i (), we have, f a cos f = si (5) y y = a ( ) a ( ) a ( e e ) b(si si ) Usig the fact that, () The Pacific Joural of Sciece ad Techology Volume. Number. May 0 (Sprig)

4 = h = h h si( ) = si( h) = si cos( h) cos si( h) So that Equatio () becomes: () ( ) h y y = ah a h h a e ( e ) [(si cos( ) cos si( )) si ] b h h Substitutig the values of Equatio (), we have: () a, a, a, adbi f f ( f f cot( )) f y = y h ( f f cot ( )) f cot ( ) f ( f f cot( )) f ( h h ) (si cos( ) ( f f cot( )) h h f e ( e ) si cos si( h)) si (4) Note that, f si [(si cos( h) cos si( h)) si ] [ cos( ) cot ( )si( ) ] = f h h (5) We shall ow substitute the first few terms of the series epasio for cos( h),si( h), ad e h i Equatio (6) ito (4): r r ( h) cos( h) = ( ) ( r)! 0 r r ( h) si( h) = ( ) 0 (r )! r h ( h) e = 0 ()! r (6) The Pacific Joural of Sciece ad Techology 4 Volume. Number. May 0 (Sprig)

5 f f ( f f cot( )) f y = y h ( f f cot ( )) f cot ( ) h hf ( f f cot( )) f ( ) h h ( f f cot( )) e h!! h h h h h h f cot( ) h! 4! 6!! 5! 7! (7) Fially, we have our ew umerical itegrator as; f f ( f f cot( )) f ( f f cot ( )) f cot ( ) h y f ( cot( )) ( ) y h f f f = h h ( f f cot( )) e (!)( )! h h h f cot ( ) h h h! 4! 6!! 5! 7! (8) NUMERICAL IMPLEMENTATION OF THE NEW INTEGRATOR We shall ow proceed to implemet the ew umerical itegrator (8) o problems of the form (). The implemetatio is carried out usig FORCE.0 programmig applicatio laguage. Problem (Logistic Model): The logistic model fids applicatios i various fields, amog which are; eural etworks, statistics, medicie, physics ad so o. I Neural etworks for eample, the logistic model is used to itroduce oliearity i the model ad/or to clamp sigals to withi a specific rage. I Statistics, they are used to model how the probability p of a evet may be affected by oe or more eplaatory variables. I Medicie, they are beig used to model the growth of tumors. I Chemistry, the cocetratio of reactats ad products i autocatalytic reactios follows the logistic fuctio. Whe special iitial coditios are applied to the logistic model, we obtai the logistic IVP: y y y y = ( ), (0) = 0.5 (9) The Pacific Joural of Sciece ad Techology 5 Volume. Number. May 0 (Sprig)

6 with the theoretical solutio, 0.5 yt () = ( e t ) (0) O the applicatio of the ew umerical itegrator (8), we obtai the result show below with step legth h=0.. Table : Performace of the New Itegrator (8) o y = y( y), y(0) = 0.5, h=0. h Numerical Solutio Eact Solutio Error Problem (Growth Model) Let us cosider the differetial equatio of the form: dy = yy, (0) = 000, t [ 0,] () d Equatio () represets the rate of growth of bacteria i a coloy. We shall assume that the model grows cotiuously ad without restrictio. Oe may ask how may bacterial are i the coloy after some hours if a idividual produces a average of 0. offsprig every hour? We assume that yt () is the populatio size at time t. This therefore implies that () may be writte as: [ ] y = 0. y, y(0) = 000, t 0, () with the eact solutio, y( t) t = e () O the applicatio of the ew umerical itegrator (8), we obtai the result show below with step legth h=0.. Table : Performace of the New Itegrator (8) y = 0. y, y(0) = 000, t 0,, h=0. o [ ] h Numerical Solutio Eact Solutio Error Problem Cosider the iitial value problem, y = 4 y,(0) = with the theoretical solutio, (4) y ( ) = e (5) O the applicatio of the ew umerical itegrator (8), we obtai the result show below with step legth h=0.. Table : Performace of the New Itegrator (8) o y = 4 y,(0) =, h=0. h Numerical Solutio Eact Solutio Error CONCLUSION We coclude that the umerical itegrator (8) is computatioally reliable goig by the results obtaied above, we therefore recommed it as a umerical scheme for estimatig the solutio to equatios of the form (). The Pacific Joural of Sciece ad Techology 6 Volume. Number. May 0 (Sprig)

7 REFERENCES. Ademiluyi, R.A New Hybrid Methods for Systems of Ordiary Differetial Equatios. Ph.D. Thesis, Uiversity of Bei, Nigeria.. Fatula, S.O A New Algorithm for Numerical Solutio of Ordiary Differetial Equatios. Computer ad Mathematics with Applicatios. : Fatula, S.O Numerical Methods for Iitial Value Problems i Ordiary Differetial Equatios. Academic Press: New York, NY. 4. Ibijola, E.A New Scheme for Numerical Itegratio of Special Iitial Value Problems i Ordiary Differetial Equatios. Ph.D. Thesis Submitted to the Departmet of Mathematics, Uiversity of Bei, Nigeria. 5. Ibijola, E.A. ad Suday, J. 00. A Comparative Study of Stadard ad Eact Fiite-Differece Schemes for Numerical Solutio of Ordiary Differetial Equatios Emaatig from the Radioactive Decay of Substaces. Australia Joural of Basic ad Applied Scieces. 4(4): Ibijola, E.A. ad Oguride, R.B. 00. O a New Numerical Scheme for the Solutio of IVPs i ODEs. Australia Joural of Basic ad Applied Scieces. 4(0): Wazwaz, A.M A New Algorithm for Calculatig Adomia Polyomials for Noliear Operatio. Appl. Math. Comput. :5-69. ABOUT THE AUTHORS J. Suday, is a Lecturer at the Adamawa State Uiversity, Mubi-Nigeria. He holds a Master of Sciece (M.Sc) degree i Computatioal Mathematics. M. R. Odekule, is a Professor of Computatioal Mathematics at the Modibbo Adama Uiversity of Techology, Yola-Nigeria. His research iterests are i computatioal mathematics. SUGGESTED CITATION Suday, J., ad M.R. Odekule. 0. A New umerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. Pacific Joural of Sciece ad Techology. ():-7. Pacific Joural of Sciece ad Techology 7. Ibijola, E.A. ad Suday, J. 0. O the Covergece, Cosistecy ad Stability of a Stadard Fiite Differece Scheme. America Joural of Scietific ad Idustrial Research, (): Ibijola, E.A., Bamisile, O.O., ad Suday, J. 0. O the Derivatio ad Applicatios of a New Oe- Step Method Based o the Combiatio of Two Iterpolatig Fuctios. America Joural of Scietific ad Idustrial Research. (): Kama, P. ad Ibijola, E.A O a New Oe- Step Method for Numerical Solutio of Ordiary Differetial Equatios. Iteratioal Joural of Computer Mathematics. 78(4). 0. Lambert, J.D. 97. Computatioal Methods i Ordiary Differetial Equatios. Joh Willey ad Sos: New York, NY.. Lambert, J.D. 99. Numerical Methods for Ordiary Differetial Systems: The Iitial Value Problem. Joh Wiley ad Sos: New York, NY.. Mickes, R.E. 994.No-Stadard Fiite Differece Models of Differetial Equatios. World Scietific: Sigapore. The Pacific Joural of Sciece ad Techology 7 Volume. Number. May 0 (Sprig)