Africa Joural of Matematics ad omputer Sciece Researc Vol. (), pp. -, Marc Available olie at ttp://www.academicourals.org/ajmsr ISSN 6-97 Academic Jourals Full Legt Researc Paper A comparative stud of a class of implicit multi metods for umerical solutio of o-stiff ad stiff first order ordiar differetial equatios Famurewa O. K. E*, Ademilui R. A. ad Awoemi D. O. Departmet of Matematical Scieces, Federal Uiversit of ecolog, Aure, Odo State, Nigeria. Accepted Jauar, is wor describes te developmet, aalsis, implemetatio ad a comparative stud of a class of Implicit Multi- Liear Multistep metods for umerical solutio of o-stiff ad stiff Iitial Value Problems of first order Ordiar Differetial Equatios. ese multi- metods icorporate more aaltical properties of te differetial equatio ito te covetioal implicit liear multistep formulae ad var te step-size () as well as te order of te (l) to obtai more accurate ad efficiet metods for solutio of o-stiff ad stiff first order ordiar differetial equatios. e basic properties of tese metods were aalzed ad te results sowed tat te metods are accurate, coverget ad A-stable. Hece, suitable for te solutio of o-stiff ad stiff iitial value problems of ordiar differetial equatios. A comparative stud of te ewl developed metods are carried out to determie te effect of icreasig te step-size () ad te order of te (l). e result sowed a remarable improvemet i accurac ad efficiec as te step-size () ad te order of te (l) are icreased. Ke words: Implicit, Multi-, Multi-step, No-stiff, Stiff, Ordiar ad Differetial equatio. INRODUION Differetial equatios occur i coectio wit te matematical descriptio of problems tat are ecoutered i various braces of sciece lie Mecaics, emistr, Biolog ad Ecoomics. (Awoemi, 99).osequetl, it costitutes a large ad ver importat aspect of toda s matematics. oug, tese problems exist b teor or priciple, teir matematical aalses give rise to differetial equatios, because te obects ivolved obe certai psical ad cemical laws ivolvig rates of cage (Ross, 99; Auziger et al., 99; ourat, 7). Ol a few of tese differetial equatios ca be solved aalticall, tis reaso gave te searc for umerical approximatio. Ordiar differetial equatios (ODEs) ca be classified ito two: Iitial value problem (IVP) or boudar value problem (BPV) depedig upo te give coditio(s) (Ademilui ad Kaode, ). A differetial equatio togeter wit iitial coditio prescribed at oe poit is called IVP. For *orrespodig autor. E-mail: famurewaav@aoo.com. example te differetial equatio: x, () A differetial equatio togeter wit coditios specified at two eds is called BVP. For example, te differetial equatio: x, (),() Wit coditio prescribed at two poits x ad x is called BVP us, a differetial equatio of te form: f ( x ), ( x ), a x b, is a first order IVP were f is assumed to be Lipscitz cotiuous (Gozalez et al., ). A Liear Multi-step Metod (LMM) for umerical solutio of first order ordiar differetial equatios of te id () is a computatioal metod of te form: ()
Famurewa. () for approximatig at te successive poits (x, ), were ad are costats to be determied (Auziger, et al., 99; Jai, 9). I tis stud, we cosider te developmet of metods for wic ad respectivel wit te iclusio of more aaltical properties of te differetial equatio b wa of more properties of te differetial equatio. e stud also attempts to determie te effect of icreasig te order of te s as well as varig te step-size of te Liear Multistep Metods of te form: l i i i ; () i wic ivolves more properties of te differetial equatio. e aim of tis stud is to compare te accurac ad stabilit of some implicit multi- liear multi-step metods. DERIVAION OF HE MEHODS Liear multistep metods of te Form ca be classified ito explicit ad implicit metods (Lambert, 97). e metod is explicit we ad implicit we. I tis stud, we are cocered wit te developmet, aalsis, implemetatio ad a comparative stud of a famil of implicit multi liear multistep metods. at is, metods for wic. o acieve tis, te local trucatio error Formula to determie parameters s ad i s of te Formula for step umbers ad was cosidered. osequetl, it is assumed tat te local trucatio error for step applicatio of te formula to problem (Equatio ) ca be defied as: i Equatio ad combie terms i equal powers of, we ave: p p o... p... P ( ) () were: P P! P ( P ) ( ) ( ) P! P! Oe step first metod Settig, l i Equatio gives: ( P ) (6) wit local trucatio error: (7) e alor s expasio of: iii... ( ) ()!! ad;!! iii ivi ii...( ) (9) Substitutig tese ito Equatio 7 ad combie terms i equal powers of, gives:... ( ) o were: l i i i i were l is te order of te of i Adoptig alors series expasio of variables, ( ), ad i ( )L ( ) r i r r! give as: r i, ()m () 6 Imposig accurac of order o ad ( ). at is,, to ave
Afr. J. Mat. omput. Sci. Res. 6 Solvig tis set of equatios wit, gives:, ad Substitutig tese values ito Equatio 6 ad simplifig to obtai a oe-step first metod of te form: ( ) () wic coicides wit te rapezoidal metod (Lambert, 97). Oe step secod metod Settig, l i () gives: [ ] [ ] () wit local trucatio error: [ ] [ ] () Adoptig te alor s series expasio of ad as i Equatios ad 9 respectivel ad; iv vi ii ii iii... ( ) ()!! i Equatio, combiig terms i equal powers of gives: iii o were, ( ) iv 6 6 6 Imposig accurac of order o, to ave ad ( ). osequetl, te followig sstem of liear equatios were obtaied: 6 6 6 Solvig tis set of equatios wit gives:,,, ad. Substitutig tese values ito Equatio ad simplifig to obtai a oe step liear multi- multistep formula: [ ] [ ] () wo step first liear multi-step metod Settig, l i (), gives: [ ] () wit local trucatio error:
Famurewa. [ ] (6) Adoptig te alor s series expasio of,, ad i as give i Equatios, 9, 7 ad i Equatio 6: ii iii i... ( ) (7)!! iii iv i i ii...( ) ()!! ad combie terms i equal power of gives: ( ) were; 6 6 Imposig accurac of order o, we ave ad ( ) at is, 6 6 Solvig tis set of equatios wit gives:,,, ad Substitutig tese values ito Equatio ad simplifig we obtai a two-step first formula of te form: ( ) (9) wic coicides wit Simpso s oe tird rule (Lambert, 97). wo step secod liear multi-step metod Settig, l i () gives: [ ] [ ] () wit local trucatio error: [ ] [ ]. () Adoptig te alor s series expasio of,, ad as i Equatios, 9, 7 ad i Equatio ad combie terms i equal powers of gives:... 7 vii 7 ( ) Were; 6 6
Afr. J. Mat. omput. Sci. Res. 6 7 6 ( 6 ) ( ) ( 6 ) 7 Imposig accurac of order 7 o, to ave:, 6 7 ad ( ) at is, 6 6 6 Solvig tis set of equatios gives: Substitutig tese values ito Equatio ad simplifig we obtai a two-step secod formula of te form: ( ) ( ) () Basics properties of te metods Accordig to Gear (97), a good umerical metod for solutio of ordiar differetial equatios is required to be accurate, cosistet, zero-stable, coverget ad absolutel-stable, tese were ivestigated. Order of accurac ad error costat of te metods Errors are ofte geerated we umerical formula is used to solve a differetial equatio. ese errors occur as a result of usig approximate values of fuctio, coupled wit umerical trucatio. e magitude of te error determies te degree of accurac of te scemes. If te magitude is adequatel small, te metod is said to be accurate, oterwise it is iaccurate (Babatola ad Ademilui, 7; Dalquist, 97). Its effect o umerical solutio is to mae it deviate sigificatl from te exact solutio, wic ca mae te solutio ustable. Accordig to Lambert (97) ad Fatula (9), a liear multi step metod is said to be of order P if te order of te local trucatio error is P. Oe-step first metod For te Oe-Step First Derivative Metod () te local trucatio error: iii o were; ( ) iv 6 wit;, ad Substitutig tese values ito Equatio to ave: ()
Famurewa. 6 implig tat,, ece metod () is of order wit error costat Oe-step secod metod For te oe-step secod metod () te local trucatio error: iii o were; ( ) iv wit; 6 6 6,,., ad Substitutig tese values ito Equatio gives: 6 7 7 Implig tat;, 7 Hece, metod () is of order wit error costat 7 wo-step first metod For te two-step first metod (9) te local trucatio error: iii o were; ( ) iv 6 6 wit;,,, ad Substitutig tese values ito Equatio gives:
6 Afr. J. Mat. omput. Sci. Res. 6 9 7 9 9 Implig tat;, ad 9 Hece, metod (9) is of order wit error costat 9 wo-step secod metod For te two-step secod metod () te local trucatio error:... 7 vii 7 ( ) were; 6 6 6 7 ( 6 ) ( ) ( ) 6 wit; Substitutig tese values ito Equatio 7 gives: 6 6 6 6 6 6 6 7.7 67 7 9 Implig tat 6 7.7 ad Hece, metod () is of order 7 wit error costat.7 osistec Accordig to Lambert (97) ad (Awoemi, 999 ad ), a liear multi-step metod of tpe () is cosistet if te parameters s ad i s satisf te followig coditios:
Famurewa. 7 i) Order P ii) ad l - iii) i Also te secod coditio is satisfied; l (iii) ad Oe-step first metod (i) Sice te Oe-step first metod () is of order, te te first coditio above is satisfied. (ii) Wit,, l Also, te secod coditio is satisfied. (iii) l l i ad ad ad ad Hece, te tird coditio is satisfied. Now tat all te coditios are satisfied, te te oe-step first metod is cosistet. Oe step secod metod (i) e oe step secod metod () is of order, te te first coditio is satisfied. (ii) Wit,,,, l i ad ad meaig tat te tird coditio is satisfied. Now tat all te coditios are satisfied, te te oe-step secod metod is cosistet. wo step first metod (i) e wo step secod metod (9) is of order, te te first coditio is satisfied.,,, (ii) Wit, ad ` - Also, te secod coditio is satisfied. (iii) ad i ad, ad
Afr. J. Mat. omput. Sci. Res. meaig tat te tird coditio is satisfied. Now tat all te coditios are satisfied, te te two-step first metod is cosistet. wo step secod metod (i) e two-step secod metod () is of order 7, te te first coditio is satisfied. (ii) Wit,,,,,, 7 - Also, te secod coditio is satisfied; (iii) ad i - ad 7 ad meaig tat te tird coditio is satisfied. Now tat all te coditios are satisfied, te te two-step secod metod is cosistet. Zero stabilit Accordig to Auziger et al. (99), Baaev ad Osterma (), Babatola ad Ademilui (7), a liear multistep metod of te form: o ( ) wit first caracteristic polomial; ρ(r) r - o r is said to be zero-stable if te root of te first caracteristic polomial ρ(r) as modulus less ta or equal to. Oe- step first metod e oe step first metod. ( ) wose first caracteristic polomial is: ρ(r) ( r ) r r r ad its roots are r or r Sowig tat te roots are witi a uit circle, ece it is zero stable. Oe step secod metod e oe step secod metod: ( ) ( ) wit first caracteristic polomial is: ρ(r) ( r ) r r r Solvig we ave r or r Sice te roots are witi a uit circle, te metod is zero-stable. wo- step first metod e wo Step first Derivative Metod: [ ] wose first caracteristic polomial ρ(r) r r r ( r ) Solvig gives r, r or r Sice te roots are witi a uit circle, te metod is zero-stable. wo step secod metod e two step secod metod:
Famurewa. 9 ( ) ( ) wose first caracteristic polomial: ρ(r) r r r ( r r ) r r ( r ) Solvig gives r or r (twice) Sice te roots are witi a uit circle, te metod is zero-stable. overgece Accordig to Palecia (99) Auziger et al. (996) ad Awoemi (), a ecessar ad sufficiet coditio for a liear multistep metod to be coverget is tat, it must be cosistet ad zero-stable. From te aalsis above, te metods are cosistet ad zero stable, ece te metods are coverget. Absolute stabilit of te metods A liear multi-step metod is said to be absolutel stable if te regio of its stabilit covers te wole left alf of te complex plai. (Palecia,99).o ascertai te regio of A stabilit of te metods, boudar locus metod ad Dalquist Stabilit model test equatio ( ) are adopted. λ Oe step first metod Applig te oe step secod metod: ( ) to solve te test equatio gives: λ ( λ λ ) λ λ λ λ Settig Z we obtai: z z u is called te stabilit fuctio. is metod will produce a coverget approximate if: were ( z) u ( z) < z tat is., < z z < < z Simplifig, we ave sets A ad B wit; A { z / z < } ad B { z/ z < } e regio of A stabilit is te itersectio of sets A ad B as sow i te doubl saded portio of te regio i Figure. Hece, te metod is A stable. Oe step secod metod Applig te oe step secod metod: ( ) ( ) to solve te test equatio gives: ( λ λ ) ( λ λ ) Simplifig to obtai z < or z <. e regio of A stabilit is sow b te doubl saded portio of te regio i Figure. Hece te metod is A stable.
Afr. J. Mat. omput. Sci. Res. tat is (-, ) Figure. Regio of absolute stabilit of oe-step first metod. tat is(-, ) Figure. Regio of absolute stabilit of oe-step secod metod. wo step secod metod e two step first metod: [ ] wit first caracteristic polomial: ρ(r) r ad secod caracteristic polomial: δ ( ) r ( r r ) Applig te boudar locus metod; implig ( ) r ρ δ ( r) ( r) iθ were r e osθ isi θ ( θ ) os θ i si θ ( os θ i si θ ( os θ i si θ ) ) Ratioalizig, simplifig ad cosiderig ol te real part of: ( θ) x( θ) i( θ), θ gives x ( θ ) (, ) Hece te metod as zero stabilit ol, terefore it is ot A stable. wo step secod metod e two step secod metod: ( ) ( ) ( r) r ( r r ) wit first caracteristic polomial: ρ(r) r - r ad secod caracteristic polomial:
Famurewa. tat is (-, ) Figure. Regio of absolute stabilit of two-step secod metod. Hece te metod is A stable. able. Results obtaied for problem i respect of metods to. X Exact solutio Oe step first Oe step secod wo step first wo step secod.....667.667.6...7976...7..997.96.997699.997699.997766..69..6.6.6996..797.799.79777.79777.797.6..7.9.9.76.7.7.77.7.7.76..6.6779.6796.6796.6.9.96.6.97.97.96..66.669.69.69.6667 δ ( r) ( r ) ( r) ( θ ) ( r) ( r) ρ iθ were r e osθ isi θ δ ( osθ i si) ( osθ isiθ ) Ratioalizig, simplifig ad cosiderig ol te real part of: ( θ ) x( θ ) i( θ ), θ gives x ( θ) (, ) te regio of A stabilit is sow b te doubl saded portio of te regio i Figure. est problems o test te suitabilit ad performace of te scemes, te formulae are traslated ito computer algoritms usig FORRAN programmig laguage. ese FORRAN programmes are used to solve some sample first order iitial value problems of (o-stiff ad stiff) ODEs. e results are preseted i ables to ad Figures to 6. e mai aim is to determie te accurac of te metods as te order of te ad step umber were icreasig. Problem A o stiff I. V. P. ( ), x [,wit ]. x, x Exact solutio: x ( ) e x
Afr. J. Mat. omput. Sci. Res. able. Results obtaied for problem i respect of metods to. X Exact solutio Oe step first Oe step secod wo step first wo step secod...67.6.67.6.67..667.66.6679.669.666..77.7796.779.77999.77..969.66.69.66.99..996.969.996.996.99.6.969.976.969.9696.9697.7.996.96.9999.996.999..9979.99.9977.99.996.9.9967.99.996967.99667.9966..997.99967.99.99.99977 able. Results obtaied for problem i respect of metods to. X Exact solutio Oe step first Oe step secod wo step first wo step secod...9d-.967d-.96d-.9666d-.9d-. 6.6D- 6.9966D- 6.6D- 6.7D- 6.69D-. 9.77696D- 9.7969D- 9.77969D- 9.7796D- 9.776966D-..69D-7.76D-7.6D-7.6D-7.69D-7..96D-7.66D-7.99D-7.9D-7.96D-7.6.9D-7.67D-7.6D-7.6D-7.9D-7.7.6D-7.69D-7.6D-7.D-7.67D-7..76D-7.976D-7.76D-7.77D-7.77D-7.9.79D-7.7D-7.7D-7.79D-7.7996D-7..9D-7.666D-7.996D-7.96D-7.96D-7 able. Results obtaied for problem i respect of metods to. X Exact solutio Oe step first Oe step secod wo step first wo step secod..699.76.699.6996.696..979.6...9..969.96.96.966.969..9.796.97.7...797.767.76.79.76.6.66.67.669.66.66.7.699.9.669.666.69...69.9.996.9.9.76.769.96.9.76..6.69.6.6.69 Problem A stiff I. V. P. ( x ), ( ) wit. x Exact solutio: (x) Problem A stiff I. V. P. x e x
Famurewa...7.6. Oe Step Adams wo Step Addisos. E r ro s... -....6.. Figure. Errors of secod metods ad some existig l mm wit respect to problem oe. X..7.6 Oe Step Adams wo Step Addisos. Errors.... -.......6.7..9 X Figure. Errors of secod metods ad some existig l mm wit respect to problem two.
Afr. J. Mat. omput. Sci. Res.... Oe Step Adams wo Step Addisos. Errors.6.....6.. X Figure 6. Errors of secod metods ad some existig l mm wit respect to problem tree. ( ), ( ) wit. x Exact solutio: (x) Problem e x A o-liear I.V.P. (Beroulli differetial equatio) x, Exact solutio (x) oclusio ( ), x [,] wit. x e I tis stud, a class of implicit multi- liear multi-step metods as bee developed for umerical solutio of first order ordiar differetial equatios. Aalsis of te basic properties sowed tat te metods x are cosistet, zero stable, coverget ad absolutel stable. Suggestig tat te metods are suitable for te solutio of o stiff ad stiff Iitial Value Problems of Ordiar Differetial Equatios ad tat secod metods gave better accurac ta first metods. REFERENES Ademilui RA, Kaode SJ (). Maximum order secod brid multistep metods for itegratio of iitial value problem i ordiar differetial equatios. J. Niger. Assoc. Mat. Ps. : 6. Auziger W, Fra R, Kirliger G (99). A ote o covergece cocepts for stiff problems. omput., : 97-. Auziger W, Fra R, Kirliger G (99). A existece of B covergece for R K Metods. Appl. Numer. Mat., 9: 9 9. Auziger W, Fra R, Kirliger G (99). Moder covergece teor for stiff iitial value problems. J. omput. Appl. Mat., : -6. Awoemi DO (99). O some cotiuous liear multistep metods for iitial value problems. P.D esis (Upublised), Uiversit of Ilori, Nigeria. Awoemi DO (999) A class of cotiuous metods for geeral secod order iitial value problem i ordiar differetial equatio. It. J. omput. Mat., 7: 9 7. Auziger W, Fra R, Kirliger G (996). Extedig covergece teor for oliear stiff problems. Part I, BI, : 6-6. Babatola PO, Ademilui RA (7). Oe stage implicit Ratioal R K
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