On the convergence, consistence and stability of a standard finite difference scheme

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1 AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 2, Sciece Huβ, ttp:// ISSN: X, doi:.525/ajsir O te covergece, cosistece ad stabilit of a stadard fiite differece sceme * Prof. Emmauel Adeolu Ibijola ad 2 Josua Suda * Correspodig Autor: Departmet of Matematical scieces, Uiversit of Ado-Ekiti, P. M. B. 5363, Ekiti State, Nigeria ( , ibjemm@aoo.com) 2 Departmet of Matematical scieces, Adamawa State Uiversit, P.M. B. 25, Mubi-Nigeria ABSTRACT For a umerical metod to be efficiet, igeious ad computatioall reliable, it is epected tat it be coverget, cosistet ad stable. I tis paper, we ivestigate te covergece, cosistec ad stabilit of a Stadard Fiite-Differece Sceme developed b Ibijola ad Suda (2). Tis sceme as also bee sow to be applicable i solvig Ordiar Differetial Equatios. Kewords: Stadard Fiite-Differece Sceme, Covergece, Cosistec, Stabilit ad Iitial Value Problems INTRODUCTION It is a kow ad documeted fact tat a give liear or o-liear equatio does ot ave a complete solutio tat ca be epressed i terms of a fiite umber of elemetar fuctios. It is also a kow fact tat oe of te was to solve suc problem is to seek a approimate solutio b meas of various perturbatio metods (Ross 964, Humi ad Miller 989). It must be stated ere tat te above procedure will ol old for limited rages of te sstem parameters ad te idepedet variable (Mickes, 994). As reported i Mickes(994), for arbitrar values of te sstem parameters at te preset time, ol umerical itegratio tecique ca provide accurate solutios to te origial differetial equatio. It is because of te above facts tat Ibijola ad Suda (2) developed a ew Stadard Fiite Differece Sceme, capable of solvig equatios of te form; [ ] f(,, ( ), a, b = = () Trougout, we sall cosider equatios of te form () wic occur i psical ad biological scieces, maagemet scieces ad egieerig. Ifact, te importace of solvig equatios of te form () caot be over empasized. Tis is because of te fact tat ma processes i te fields metioed above are govered b te equatios of te form (). Let us cosider teorem below wic guaratees te eistece of a uique solutio of a iitial value problem of te ature (). Teorem (Lambert 973, Fatula 988) Let f (, be defied ad cotiuous for all poits (, i te regio D defied b a b, < <, a ad b fiite, ad let tere eist a costat L suc tat for ever,, suc tat (, ad (, ) are bot i D ; f (, f (, ) L (2) Te if η is a give umber, tere eist a uique solutio () of te iitial value problem (). Te iequalit (2) is kow as a Lipscitz coditio ad te costat L as a Lipscitz costat. Defiitio (Herici 962) A metod for solvig a differetial equatio i wic te approimatio to te solutio at te poit ca be calculated if ol,, ad is

2 Am. J. Sci. Id. Res., 2, 2(2): kow as a ONE-STEP METHOD. It is a commo fact to write te fuctioal depedece o te quatities, ad i te form;, φ( (3) were φ(, ; is te icremet fuctio. Defiitio 2 (Covergece) A umerical sceme is said to be coverget if for all iitial value problem () satisfig te potesis of Lipscitz coditio, Ma ( N ) arbitrar poit [ a, b] as or if for a, te global error fulfills te followig relatiosip; lim MaE, provided is alwas a mes poit. Defiitio 3 (Cosistece) A umerical sceme wit a icremet fuctio φ( is said to be cosistet wit te iitial value problem (), if φ( ) = f (, ). Note tat if te sceme developed is cosistet wit te iitial value problem (), te, =. Te cocept of cosistec of oe-step metod is ver crucial i te sese tat it cotrols te magitude of te local trucatio error. STABILITY (Herici 962) A umerical solutio of te class of sstem () is said to be stable if te differece betwee te umerical solutio ad te teoretical solutio ca be made as small as possible, i.e. if tere eists two positive umbers e ad K suc tat te followig olds; ( ) K e (4) Teorem 2 Let = ( ) ad l = l( ) deote two differet solutios of our differetial equatio uder cosideratio, wit te iitial coditio specified as ( ) = η ad l ) = η respectivel, suc tat ( l l < ε, ε >. If te two umerical estimates are geerated b; l Te coditio tat; φ( = l φ (, l ; (5) (6) l K η η (7) is te ecessar ad sufficiet coditio tat a umerical sceme be stable ad coverget. We sall ow proceed to cosider te derivatio, covergece, cosistec, stabilit ad applicatios of te ew Stadard Fiite Differece Sceme. Derivatio of te ew stadard fiite-differece sceme (Ibijola ad Suda 2) Let us assume tat te teoretical solutio ( ) to te iitial value problem () ca be locall represeted i te iterval [, ], b te o-polomial iterpolatig fuctio; F( ) = a ae, were a, a, ad are costats (8) At te poit =,we ave F( ) = a ae (9) Wile at te poit =, equatio (8) becomes, ( ) F = a ae () Differetiatig (8) gives; F ( ) = ae () ad; F ( ) = ae (2) We furter assume tat (2) coicides wit () ad ece we ave; a e = f (, ) f (, ) a = (3) e To proceed furter, we assume tat te teoretical ad umerical solutio iitiall coicide, ece F ( ) = (4) F (5) ( ) = 75

3 Am. J. Sci. Id. Res., 2, 2(2): Usig (9), (), (4) ad (5), we ave ( = a e e ) (6) ( e e ) f (, ) e (7) Usig te fact tat = ad = ( ), equatio (7) becomes; f = ( e ) (8) Equatio (8) is te required umerical sceme. We must metio ere tat tis umerical metod is strog eoug to solve problems of te form (). Equatio (8) is te ew Stadard Fiite-Differece Sceme ad it is a oe-step metod. Covergece of te ew stadard fiite differece sceme: Let us start b establisig te fact tat our ew umerical sceme i equatio (8) ca be epressed as a oe-step metod i te form of equatio (3), were φ( ; ), is te icremet fuctio. We sall ow proceed to derive te icremet fuctio. At = ad =, equatio (8) becomes; F( ) = a ae (9) ad F( ) a ae = (2) If we assume tat F( ) ad F( ), coicides wit ad respectivel ad F () i deote te it total derivatives of f (, wit respectivel to, we ave F ( ) = ae = f (2) Solvig for a i (2) gives f a = (22) e Cosiderig te assumptios i equatios (9) ad (2) as F( ) ad F( ) = respectivel we ave our first umerical itegrator geerated as f = ( e ) (23) Epadig e i series form we ave; r 3 3 ( e = =... ( r!) (24) r= Substitutig equatio (24) ito (23) ad epadig it will give us te followig equatio; 3 3 f = (... ) 2! 3! (25) Let 3 = f(...) 2! 3! (26) = f(...) 2! 3! (27) A =... (28) Te our umerical sceme ca be writte compactl as were = fa (29) φ (, ; = f A (3) is te icremet fuctio. We sall ow proceed to sow tat our metod (8) is coverget. Tis is oe of te fidigs of tis work. Teorem 3 Let te icremet fuctio of te sceme (8) defied b (3) be te fuctio of argumets i te regio D defied b [ ab, ], (, ),, were> ad let tere eists a costat L, te, φ(, ; φ(, ; L (3) 76

4 Am. J. Sci. Id. Res., 2, 2(2): for all ( ad ( i te regio just defied. Tis is te ecessar ad sufficiet coditio for te covergece of te metod (8). Proof Recall tat from (3), te icremet fuctio φ (, ; = f A, were A =... (32) Cosiderig equatio (3), we ca also write ad φ (, ; = Af(, ) (33) φ(, ; = Af(, ) (34) φ( φ( = A f(, ) f(, ) (35) Let be defied as a poit i te iterior of te iterval wose ed poits are ad, applig Mea Value Teorem, we ave f(, f (, ) f(, ) = ( ) (36) If we defie M = Sup (, ) D f (, ) (37) f (, ) φ(, ;) φ(, ;) = A ( ) (38) φ(, ; φ(, ; = AM( ) (39) Takig te absolute values of bot sides give; φ(, ; φ(, ; AM (4) If we let L = AM, te equatio (4) turs to φ(, ; φ(, ; L (4) wic is te coditio for covergece. Cosistec of te ew stadard fiite-differece sceme Recall Fatula (988), tat sas a umerical sceme wit a icremet fuctio of φ ( is cosistece wit respect to te iitial value problem () if φ ( ) = f(, ) (42) We sa tat our ew umerical sceme is cosistet sice (8) reduces to (42) we = i te equatio below. φ ( = f(...) (43) wic is obtaied from equatio (27). Terefore, te sceme is cosistet. Stabilit aalsis of te ew stadard-fiite differece sceme Teorem 4 Let = ( ) ad l = l( ) deote two differet umerical solutios of differetial equatio () wit te iitial coditios specified as ( ) = η adl ( ) = η respectivel, suc tat η η <ε, ε >. If te two umerical estimates are geerated b te iterpolatio sceme (27), we ave φ ( (44) l l φ (, l; (45) Te coditio tat l k η η (46) is te ecessar ad sufficiet coditios tat our umerical sceme (8) be stable ad coverget. 77

5 Am. J. Sci. Id. Res., 2, 2(2): Proof Let ad = Af(, ) (47) l = l Af (, l) (48) especiall as te step-legt teds to zero. For applicatio of te sceme (8), refer to Ibijola, E. A. ad Suda, J. (2), Australia Joural of Basic ad Applied Sciece, 4(4): REFERENCES te, [ ] l = l A f(, ) f(, l ) (49) Let be a poit i te iterior of te iterval wose ed poits are ad l, if we appl te Mea Value Teorem, as before, we ave f(, f (, ) f(, l) = ( l) (5) Let us defie f (, L = (5) Te, f(, l l A ( l ) (52) l l AL( l) (53) Takig te absolute value of bot sides of equatio (53) gives l = ( l ) AL( l ) AL l (54) Give ε >, ad ( ) = η, l ( ) = η ad AL = K, te l K η η (55) Te we coclude tat our metod (8) is stable ad ece coverget. CONCLUSION Sice, it as bee establised tat te sceme is coverget, cosistet ad stable, it is obvious its umerical solutio will sow a measure of covergece towards te eact (teoretical) solutio, Fatula, S. O. (976), A New Algoritm for Numerical Solutio of Ordiar Differetial Equatios. Computer ad Matematics wit Applicatios, 2: Fatula, S. O. (988), Numerical Metods for Iitial Value Problems i Ordiar Differetial Equatios. Academic Press Ic, New York. Gakov, F. D. (966), Boudar Value Problems. Dover Publicatios Ic, New York. Herici, P. (962), Discrete Variable Metods i Ordiar Differetial Equatios. Jo Wile ad Sos, New York. Humi, M. ad Miller, W. (988), Secod Course i Ordiar Differetial Equatios for Scietists ad Egieers. Spriger-Verlag; New York. Ibijola, E. A. ad Suda, J. (2), A Comparative Stud of Stadard ad Eact Fiite-Differece Scemes for Numerical Solutio of Ordiar Differetial Equatios Emaatig from te Radioactive Deca of Substaces. Australia Joural of Basic ad Applied Scieces, 4(4): Lambert, J. D.(973), Computatioal Metods i Ordiar Differetial Equatios. Jo Wille ad Sos, New York. Lambert, J. D. (99), Numerical Metods for Ordiar Differetial Sstems: Te Iitial Value Problem. Jo Wille ad Sos, New York. Mickes, R. E. (98), No-Liear Oscillatios. Cambridge Uiversit Press, New York. Mickes, R. E. (99), Differece Equatios; Teor ad Applicatios. Va Nostrad Reiold, New York. Mickes, R. E. (994), No-Stadard Fiite Differece Models of Differetial Equatios. World Scietific, Sigapore. Mickes, R. E. (999), Applicatios of No-Stadard Metod for Iitial Value Problems. World Scietific, Sigapore. Ross, S. L. (964), Differece Equatios. Blaisdeu; Waltam, MA. 78