On Exact Finite-Difference Scheme for Numerical Solution of Initial Value Problems in Ordinary Differential Equations.

Transcription

1 O Exact Fiite-Differece Sceme for Numerical Solutio of Iitial Value Problems i Ordiar Differetial Equatios. Josua Suda, M.Sc. Departmet of Matematical Scieces, Adamawa State Uiversit, Mubi, Nigeria. josuasuda000@aoo.com ABSTRACT Tis paper presets a umerical metod called te Exact Fiite-Differece Sceme for te solutio of Ordiar Differetial Equatios of firstorder. Te eed for exact fiite differece sceme came up due to some sortcomigs of te stadard metods; i wic te qualitative properties of te exact solutio are ot usuall trasferred to te umerical solutio. Tese sortcomigs create problems wic ma affect te stabilit propert of te stadard approac. Te exact fiite-differece sceme as te propert tat teir solutios do ot ave umerical istabilities. (Kewords: exact fiite-differece sceme, ostadard fiite differece sceme, iitial value problems, ODEs, models) 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 expressed 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 approximate solutio b meas of various perturbatio metods (Rose, 964 ad 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. For a umerical metod to be coverget, it as to be a sufficietl accurate represetatio of te differetial sstem (Lambert, 99). It as bee observed tat te exact fiite differece sceme (wic is a special case of o-stadard fiitedifferece sceme) is oe tat does ot exibit umerical istabilit. Trougout tis work, we sall cosider te geeral first-order differetial equatio: d f ( x,, λ), t ( 0) 0 () dt were f (, x, λ ) is suc tat Equatio () as a uique solutio over te iterval, 0 t T ad for λ i te iterval λ λ λ. Equatio () occurs i psical ad biological scieces, maagemet scieces ad egieerig. I fact, te importace of solvig equatios of te form () caot be over empasized. For damical sstems of iterest, i geeral, T, i.e. te solutio exist for all time. Tis solutio ca be writte as: () t φ( λ,, t,) t () wit 0 0 φ( λ,, t, t) (3) A discrete model of Equatio () ca be writte as: g( λ,,, t), t + (4) Te solutio to equatio (4) ca be expressed as, η( λ,,, t, t ) (5) wit 0 0 η( λ,,, t, t ) (6) Te Pacific Joural of Sciece ad Tecolog 60 ttp:// Volume. Number. November 00 (Fall)

2 Defiitio (Mickes 994) Equatios () ad (4) are said to ave te same geeral solutio if ad ol if, ( t ) (7) for arbitrar values of. Defiitio (Mickes 994) A exact fiite differece sceme is oe for wic te solutio to te differece equatio ave te same geeral solutio as te associated differetial equatio. Tese defiitios lead to te followig result. Teorem (Mickes 994) Te differetial equatio () as a exact fiitedifferece sceme give b te expressio: [ λ, t t ] +,, + φ (8) were φ is tat of Equatio (). old weever te rigt-side of (9) is well defied. ii) Te teorem is ol a existece teorem. It basicall sas tat if a ordiar differetial equatio as a solutio, te a exact fiite-differece sceme exists. iii) iv) A major implicatio of te teorem is tat te solutio of te differece equatio is exactl equal to te solutio of te ordiar differetial equatio o te computatio grid for fixed, but arbitrar step-size. Te teorem ca be easil geeralized to sstems of coupled, first-order ordiar differetial equatios. Te discover of exact discrete models for ordiar differetial equatios is of great importace, primaril because it allows us to gai isigts ito te better costructio of fiitedifferece scemes. Te also provide computatioal ivestigator wit useful becmarks for compariso wit te stadard procedures. Proof Te group propert of te solutio to () gives, [, ( t), t t ] ( t + ) φ λ, + (9) If we ow make te idetificatios, t t, ( t) (0) te (9) becomes, + ( + φ λ,, t, t ) () Tis is te required ordiar differece equatio tat as te same geeral solutio as (). Remarks i) If all solutios of () exist for all time, i.e. T, Te () olds for all t ad. Oterwise, te relatio is assumed to THE GENERAL THEORY OF NON-STANDARD METHODS Cosider te differetial equatio: d f (, λ) () Were λ is a -parameter vector. Equatio () ca be writte i te form: + φ(, λ) If d F(,, λ, ) (3) + + (4) Te Equatio (3) is a geeralizatio of Equatio (4). I tis case, Te Pacific Joural of Sciece ad Tecolog 6 ttp:// Volume. Number. November 00 (Fall)

3 d + φλ (, ) (5) Were φ( λ, ), te deomiator fuctio as te propert tat: φλ (, ) + o( ) (6) λ fixed, 0 Te above formulatio is based o te traditioal defiitio of te derivative wic is of te form d Lim ( x+ ψ ( )) ( x) (7) 0 ψ ( ) Were ψ i ( ) + o( ), 0, i,. Example of fuctios ψ ( ) tat satisf coditio (7) above are: si ( ) e ψ ( ) e λ e λ etc (8) Te values of ψ i, i,,..., depeds o te differetial equatio uder cosideratio. It must be stated ere tat if 0, te d x ( + ψ( )) x ( ) ( x + ) ( x) Lim 0 ψ ( ) Lim 0 (9) NON-STANDARD FINITE DIFFERENCE MODELLING RULES Te geeral form of o-stadard metod ca be writte as: + F (, ) (0) No-stadard fiite differece scemes were developed usig a collectio of rules set b Mickes as follows: Rules (Mickes 994) Te order of te discrete derivative must be exactl equal to te order of te correspodig derivatives of te differetial equatio. If tis rule is violated, tis ca lead to umerical istabilit i te form of oscillatios wic ma be bouded or ubouded. Te matematical reaso for te above occurrece is tat discrete equatios ave large class of solutios ta differetial equatios. As a illustratio, let us cosider te followig first order differetial equatio: d () If we model () b a cetral differece sceme of te form: + () It will be discovered tat tis modelig as extra solutio tat is strage because Equatio () is of secod order wile () is of first order, tus te priciple of uiqueess is violated ad tis leads to te existece of umerical istabilit. Rule (Mickes 994) Deomiator fuctio for te discrete derivatives must be expressed i terms of more complicated fuctio of te step-sizes ta tose covetioall used. Tis rule allows te itroductio of complex aaltic fuctio of i te deomiator. For istace, cosider, d ( ) (3) Tis is i form of a logistic equatio. If te deomiator fuctio D is give b, D (4) e Te Pacific Joural of Sciece ad Tecolog 6 ttp:// Volume. Number. November 00 (Fall)

4 te substitutig Equatio (4) i Equatio (5) gives, e + ( ) + (5) It must be stated ere tat te selectio of a appropriate deomiator is a art (Mickes, 999). We must examie te differetial equatio for wic te exact scemes are kow. Close examiatio of differetial equatio for wic exact scemes are kow, sows tat te deomiator fuctio geerall are fuctios tat are related to particular solutio or properties of te geeral solutio to te differece equatio. Rule 3 (Mickes 994) Te o-liear terms must i geeral be modeled (approximated) o-locall o te computatioal grid or lattice i ma differet was, for istace, i Equatio (5), it is assumed tat +. Te o-liear terms, 3 ca be modeled as follows: + (6) ( + + ) (7) + (8) 3 3 ( + + ) (9) Rule 5 (Mickes 994) Te fiite-differece equatio sould ot ave solutios tat do ot correspod exactl to te solutio of te differetial equatios. Te ostadard metods sall be applied to some problems as sow below: Example Te o-stadard fiite-differece sceme for te solutio of ', (0) (30) is preseted usig te followig approximatios: + (3) + + (3) Usig Equatio (3), Equatio (30) ca be writte as: + φ( ) (33) + wic is i te form of Equatio (5). φ( ) (34) + + ( ( ) ) + φ (35) Tis ca be writte i a compact form as: Te particular form selected from Equatios (6) to (9) depeds o te full discrete model. + φ( ) (36) Rule 4 (Mickes 994) Special solutios of te differetial equatios sould also be accompaied b special discrete solutios of te fiite-differece models. Equatio (36) is of te form (0) wic is i ostadard form. Now, usig Equatio (3), Equatio (30) ca be writte as: ( ) φ( ) (37) Te Pacific Joural of Sciece ad Tecolog 63 ttp:// Volume. Number. November 00 (Fall)

5 + φ( ) + + (38) + + φ( ) + φ( ) + (48) Tat is, φ( ) + φ( ) (39) + + Equatio (39) simplifies to: + ( + φ( ) ) φ( ) Equatio (40) ca be writte i te form: (40) Teorem (Aguelov ad Lubuma 003) Te fiite differece sceme (0) is stable wit respect to mootoe depedece o iitial value, if: F (, ) 0, R, >0 (49) ( + φ( ) ) + + φ( ) + Equatio (4) is also of te form (0). Example Cosider te differetial equatio: (4) Teorem 3 (Aguelov ad Lubuma 003) Te o-stadard sceme (36) is stable wit respect to mootoe depedece o iitial value ad mootoe of solutio ad terefore is elemetar stable. Proof d ( ) (4) No-stadard differece sceme is costructed for te solutio of te Equatio (4) b approximatig +. Equatio (4) becomes: + φ( ) ( ) (43) + + φ( ) ( ) (44) φ( ) φ( ) (45) Cosider te sceme (36): F (, ) + (50) φ( ) Here, F (, ) φ( ) (5) ( ) F [ φ( ) ] [ φ( ) ] [ φ( ) ] (5) + φ( ) + φ( ) (46) Tat is, ( + φ( ) ) + φ( ) (47) + + wic leads to, φ( ) + φ( ) [ φ( ) ] 0 [ φ( ) ] F (, ) 0 (53) (54) (55) Te Pacific Joural of Sciece ad Tecolog 64 ttp:// Volume. Number. November 00 (Fall)

6 Here, sice φ( ) is positive ad >0, tis implies tat te sceme (36) is stable wit respect to mootoe depedece o iitial value. DERIVATION OF THE EXACT FINITE- DIFFERENCE SCHEME I tis sectio, te exact-fiite differece sceme capable of producig a exact solutio to problems i form of Equatio () sall be derived. Te discover of exact discrete models for particular ordiar differetial equatios is of great importace, primaril because it allows us to gai isigts ito te better costructio of fiitedifferece scemes. Te also provide te computatioal ivestigator wit useful becmarks for compariso wit te stadard procedures. Above all, a major advatage of avig a exact differece equatios model for ordiar differetial equatio is tat questios related to te usual cosideratio of cosistec, stabilit, ad covergece eed ot arise (Mickes 994). Cosider te equatio of te form (). If we assume tat te exact (teoretical) solutio of (), at poit x x deoted b ( x ) as te same geeral solutio wit umerical solutio (i.e. te ew exact fiite sceme), at poit x x deoted b, te; ( x ) (56) Tis implies tat at poit x x +, ( x ) (57) + + Te followig determiat gives te required differece equatio: ( x) ( x ) (58) Tis is te exact fiite sceme capable of solvig a equatio of te form (). It is importat to ote tat te sceme (59) is of te form (0). Oe of te sortcomigs of a exact fiite differece sceme is tat it is ecessar tat we must kow te teoretical solutio before we ca costruct te metod. Te advatage of exact fiite differece sceme is tat it produces exact solutio to te differetial equatios uder cosideratio. Aoter advatage is tat, questios related to te usual cosideratio of cosistec, stabilit ad covergece eed ot arise (Mickes 994). Costructio of Exact-Fiite Differece Scemes We sall ow cosider ow te exact fiitedifferece sceme is beig costructed usig te followig test problems. Problem Cosider te Iitial Value Problem, ' x x 4, (0) 3 (60) wit te teoretical (exact) solutio, x ( ) + e x (6) At te poit x x +, we ave, x e + (6) ad at te poit x x +, we ave, x e (63) Te followig determiat gives te required differece equatio, It is obvious from equatio (58) tat, x ( ) + + x ( ) (59) + + e x x + e + 0 (64) Te Pacific Joural of Sciece ad Tecolog 65 ttp:// Volume. Number. November 00 (Fall)

7 x+ ( + e ) + x ( + e ) (65) + 00e 0.006x + 00e 0.006x 0 (7) were x ad x+ ( + ). Te exact fiite-differece sceme (65) is capable of solvig Equatio (60). Problem Cosider te Iitial Value Problem ', (0) (66) wit te teoretical (exact) solutio, x ( ) x (67) Te followig determiat gives te required differece equatio, + x x ( ) [ ( + ) ] (68) (69) Equatio (69) is te exact fiite-differece sceme for solvig Equatio (66). Problem 3 Cosider te Iitial Value Problem, ' 0.006, (0) 00 (70) wit te teoretical (exact) solutio, x ( ) x e (7) Te determiat below gives te required differece equatio, e + (73) Tis is te exact fiite-differece sceme for Equatio (70). For te applicatios of te costructed scemes above, see Ibijola, E. A. ad Suda, J. (Aust. J. of Basic & Appl. Sci., 4(4):64-63, 00). CONCLUSION From te presetatio above, we coclude tat te Exact Fiite-Differece Sceme is computatioall reliable ad efficiet. Tis is because it performs well o iitial value problems of ordiar differetial equatios, i fact te umerical solutio does ot exibit umerical istabilit. REFERENCES. Humi, M. ad Miller, W Secod Course i Ordiar Differetial Equatios for Scietists ad Egieers. Spriger-Verlag: New York, NY.. Ibijola, E.A. ad Suda, J. 00. A Comparative Stud of Stadard ad Exact 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, NY. 4. Lambert, J.D. 99. Numerical Metods for Ordiar Differetial Sstems: Te Iitial Value Problem. Jo Wille ad Sos: New York, NY. 5. Mickes, R.E. 98. No-Liear Oscillatios. Cambridge Uiversit Press: New York, NY. 6. Mickes, R.E Differece Equatios; Teor ad Applicatios. Va Nostrad Reiold: New York, NY. 7. Mickes, R.E No-Stadard Fiite Differece Models of Differetial Equatios. World Scietific: Sigapore. Te Pacific Joural of Sciece ad Tecolog 66 ttp:// Volume. Number. November 00 (Fall)

8 8. Mickes, R.E Applicatios of No-Stadard Metod for Iitial Value Problems. World Scietific: Sigapore. 9. Rose, S.L Differece Equatios. Blaisdeu; Waltam, MA. ABOUT THE AUTHOR Josua Suda is a Lecturer at te Adamawa State Uiversit, Mubi-Nigeria. He olds a Masters of Sciece (M.Sc.) degree i Numerical Aalsis. His researc iterests are i Numerical Aalsis. SUGGESTED CITATION Suda, J. 00. O Exact Fiite-Differece Sceme for Numerical Solutio of Iitial Value Problems i Ordiar Differetial Equatios. Pacific Joural of Sciece ad Tecolog. (): Pacific Joural of Sciece ad Tecolog Te Pacific Joural of Sciece ad Tecolog 67 ttp:// Volume. Number. November 00 (Fall)