Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method

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1 Iteratioal Joural of eoretical ad Applied Matematics 7; (: -5 ttp://wwwsciecepublisiroupcom//itam doi: 648/itam75 Applicatio of a wo-step ird-derivative Block Metod for Starti Numerov Metod Oluwaseu Adeyeye *, Zuri Omar Departmet of Matematics, Uiversiti Utara Malaysia, Keda, Malaysia address: adeyeye_oluwaseu@assuumedumy (O Adeyeye, zuri@uumedumy (Z Omar * Correspodi autor o cite tis article: Oluwaseu Adeyeye, Zuri Omar Applicatio of a wo-step ird-derivative Block Metod for Starti Numerov Metod Iteratioal Joural of eoretical ad Applied Matematics Vol, No, 7, pp -5 doi: 648/itam75 Received: September 6, 6; Accepted: December, 6; Publised: Jauary, 7 Abstract: Numerov metod is oe of te most widely used aloritms i pysics ad eieeri for solvi secod order ordiary differetial equatios e umerical solutio of tis metod as bee improved by differet autors by usi differet starti formulas but i recet years, tere as bee a deart i tat tred wic iformed te itroductio of a twostep tird-derivative block metod i tis paper to start Numerov metod wit te aim of etti better results ta previous approaces e selectio of te steplet as two is to ave a uiform basis for compariso wit oter existi two-step starti formula i literature Altou, te accuracy of te two-step metod adopted i tis article was eaced by te itroductio of ier derivative Hece, tis paper presets a two-step tird-derivative block metod wic displayed better accuracy we adopted for starti Numerov metod as sow i te umerical results us, te tird-derivative block metod, as a starti formula, is see to be quite suitable for starti Numerov metod we applied to pysical models Keywords: Numerov, wo-step, ird-derivative, Block Metod, Secod Order, Iitial Value Problems Itroductio is paper cosiders te solutio of te secod order iitial value problem of te form ( ( α ( y f x, y, y a, y a β ( were α ad β are costats Previously, te differetial equatios of te form ( above ave bee solved by reductio to a system of two first order iitial value problems However, tis approac ivolves bot uma ad computatioal burde ad ece te eed to itroduce direct umerical metods for te solutio of ( ad te Numerov metod is oe of suc direct metods e Numerov metod as ive below, y y y ( f f f ( is a very efficiet two-step metod for directly solvi secod-order differetial equatios of te form ( ad a suitable example is te Scrodier equatio as discussed by [7, 5, 7], ad te Poisso equatio as discussed i [] Also, some studies ave sow tat Numerov-type metods also bei applied D Quasi-liear Elliptic Boudary Value Problems [], D biarmoic ad triarmoic equatios [9] amost may oters Hece, te importace of te Numerov metod caot be overempasized ad te presetatio of a formula tat will adequately start it to ive better accuracy is a area of iterest wic is te aim of tis paper e Numerov metod as a order of accuracy of four wit a correspodi error costat of C 6 4 I te paper by [], it was stated tat tis level of accuracy tat te Numerov metod possesses toeter wit its applicatio to ( resulti i a tridiaoal set of alebraic equatios as made te metod popular However, te questio o ow to adequately start te metod ad te quest to discover a better starti value dates back to te work of [6] [6] observed tat, for te applicatio of te Numerov metod i solvi (, tere is a eed for two previous values of te solutio i order to calculate a ew oe Hece, [6] adopted a starti formula of order tree obtaied usi aylor series approac [6] also developed a parallel aloritm wit te aim of icreasi te accuracy of te Numerov metod is

2 Iteratioal Joural of eoretical ad Applied Matematics 7; (: -5 aloritm was furter exteded by [] wo preseted a formula for starti te Numerov metod more accurately usi a order four ybrid metod developed based o multistep collocatio approac to start te metod However, sice te work of [], tere as bee a deart to te presetatio of aloritms to adequately start te Numerov metod wic presets te area of iterest i tis paper e Numerov metod, as widely kow, is a two-step metod ad it is suitable to develop a two-step metod to adequately start it wit better accuracy However, tere is eed to boost te two-step starti metod wit te itroductio of ier derivative wic ives rise to te twostep tird-derivative metod It is observed tat very few cotributios i literature exist o te applicatio of ier derivative metod; owever, tis metod dates back to te early 4s as discussed by [] I is work, te autor stated some particular cases of te itroductio of secod derivative i a first order liear multistep metod wic icludes te metod of order four cosidered by [] ad i coectio wit stiff equatios by [] amost oters [] developed a secod derivative metod to solve stiff first order ordiary differetial equatios ad stated tat te motivatio for itroduci te secod derivative is to eerate results wit better accuracy we te metod is applied to solve stiff first order iitial value problems Similarly, i recet works, [5] developed a tird-derivative metod (DM to solve secod order iitial ad boudary value problems ad also, [4] developed a fourt derivative metod (FDM to solve tird order boudary value problems I teir separate works, te itroductio of ier derivative ave results wit very ood accuracy we compared wit exact solutio of te differetial equatio cosidered Hece, te motivatio to adopt a two-step ier derivative metod for starti te Numerov metod wit expected accuracy is paper presets a tird-derivative block metod wic is also a two-step metod as te Numerov metod for starti te Numerov metod ad obtaii better level of accuracy ta previously proposed starti metods e outlie for tis article is as follows: te derivatio of te tird-derivative block metod is sow i Sectio ad its umerical results are sow i Sectio, te discussio of results is ive i Sectio 4 ad te article is cocluded i Sectio 5 Derivatio of te ird-derivative Block Metod e discrete sceme for te tird-derivative block metod is derived from te liear multistep metod form ive below were ( y α y β f λ (,, f f x y y, wic by otatio is (, (, ( df x y x y x, dx (,, x y y Expadi idividual terms i ( usi aylor series expasio ad substituti te expasios back i ( ives te followi writte i matrix form as below (coefficiets of ( m y x are equated α α (! β (! 4 β ( ( ( 4!!! β 5 ( ( ( ( ( λ 5!!!!! ( ( ( ( ( 6! 4! 4!!! λ ( ( ( ( ( 7! 5! 5! 4! 4! (! (! 4 ( 4! 5 ( 5! 6 λ ( 6! 7 ( 7! e values of α, α, β, β, β, λ, λ ad λ are obtaied usi matrix iverse metod as ive below ( α, α, β, β, β, λ, λ, λ ( ,,,,,,, Substituti (4 i ( ives te discrete sceme ( ( 5 4 e additioal metods eeded to obtai te desired block are to be obtaied from y α y β f λ y α y β f λ y α y β f λ (4 (5, (6 were eac of te terms i (6 are similarly expaded usi aylor series expasio ad substituti tese expasios back i (6 ives a series of liear equatios system tat ca be solved usi matrix iverse approac to obtai te ukow derivatives

3 Oluwaseu Adeyeye ad Zuri Omar: Applicatio of a wo-step ird-derivative Block Metod for Starti Numerov Metod e substitutio of te obtaied coefficiets ito (6 provides te derivatives of te discrete sceme as preseted below ( 4 ( 7 ( 59 8, ( 68 ( ( , ( 7 ( 6 ( 5 8 y y y f f f Combii te equatios formulated i te discrete sceme derived i equatio (5 toeter wit te derivatives obtaied i equatio (7, ives a expressio tat ca be preseted i te followi matrix form y y y y y ( f f f 5 ( 4 y y ( f 7 f f 4 ( y ( 87 f 66f 7 f 68 ( y ( f 6f f 7 ( Adopti matrix iverse metod, A, x A B were y ( f f f 5 ( 4 ( 7 4 ( 58 8 y ( 87 f 66 f 7 f 68 ( y ( f 6f f 7 ( y y f f f 68 B 84, (7 x y, y, y, y is determied ad expressed as ive below ( 7 ( 59 8, 5 ( 79 9 ( 6 4, 4 ( 8 ( 4, ( 7 6 f ( y y f f f y y f f e block metod i equatio (8 is te ew two-step tird-derivative starti formula to implemet te Numerov metod Note tat te starti formula proposed by [6] takes te form ive below ( (8 y y y f f f, (9 wile te starti formula from [] takes te form ( y y y f f f f ( Basic Properties of te ird-derivative Block Metod e properties tested for te block metod will be limited to te required to esure coverece is follows from te coditios stated by [4] tat a liear multistep metod is coveret if it is cosistet ad zero-stable Defiitio A liear multistep metod is cosistet if it as order p e obtai te order of te tird-derivative block metod, te correctors i equatio (8 above are cosidered ad te liear operators defied as 4 ( 7 ( 59 8 y y f f f L y, 68 ( y y 5 ( 79f f 9f L y 5 ( 6 4 followi te cocept of [8] Expadi idividual terms i ( usi aylor series expasios defied as

4 Iteratioal Joural of eoretical ad Applied Matematics 7; (: -5 ( ( ( ( y y x y x y x ( ( y x! f y x y x y x ( ( ( ( iv ( ( ( ( y x y x y x ( e substituti ( i ( ad collecti like terms ives ( i te form ( ( ( ( ( p ( ( p ( p L C y x C y x C y x p p L C y x C y x C y x p o satisfy Defiitio, te block metod as order p if C sice te error costat for te block metod is expected at C p us for te expressio of L, C C 7 C! C! ad for expressio L C C 79 9 C! C! Hece, te block metod is cosistet Movi o to te property of zero-stability, first equatio ( is coied to take te form were I Y AZ BF B F CG C G k k k y I, Yk, A, y y 4 f Z, B,, 79 F y 5 f f 68 B, F, 9 k C, 5 5 f 5 G, C , G 6 4 k 5 5 Now, te tird-derivative block metod is zero-stable if te p r ri A are roots of te caracteristic polyomial ( simple or less ta oe Upo substitutio, p r r r r ( ( wit roots r, ad tis implies te block metod is zerostable O satisfyi all criteria for coverece, te tird-derivative block metod is said to be coveret ad ece suitable for applicatio to solve ordiary differetial equatios 4 Numerical Results I tis sectio, two secod order ordiary differetial equatios will be cosidered, wic are already solved i literature by past autors Compariso will be made by computi te absolute value differece betwee te exact solutio ad te computed solutio Problem ( ( y y, y y, Exact Solutio: y ( x cosx x si x Source:[] Problem ( ( y y, y, y, Exact Solutio: y ( x Source: [] x e able Absolute Errors for Problem were starti values are provided for te Numerov metod usi (8, (9 ad ( x Starti Value (9 Starti Value ( Starti Value (8 7E-7 69E-9 68E- 47E-7 76E-8 456E-9 6E-7 6E-8 454E E-7 47E-8 969E-9 5 E-6 E-7 955E-9 6 E-6 87E-7 5E-8 7 4E-6 7E-7 46E-8 8 6E-6 49E-7 E E-6 579E-7 9E-8 9E-6 77E-7 4E-8 able Absolute Errors for Problem were starti values are provided for te Numerov metod usi (8, (9 ad ( x Starti Value (9 Starti Value ( Starti Value (8 4E-7 496E-9 59E- 45E-7 49E-8 77E-9 6E-7 668E-9 78E E-7 4E-8 69E-9 5 E-6 677E-7 74E-9 6 5E-6 8E-7 974E-9 7 6E-6 77E-7 E-8 8 9E-6 E-7 4E-8 9 E-6 E-7 9E-8 56E-6 95E-7 5E-8

5 4 Oluwaseu Adeyeye ad Zuri Omar: Applicatio of a wo-step ird-derivative Block Metod for Starti Numerov Metod 5 Discussio of Results ese umerical problems cosidered are te same cosidered i te work of [] Wit referece to able ad able above, altou te starti value adopted by [] as see i equatio ( is also a two-step metod but wit oe ybrid poit, tis two-step tird-derivative metod performs better avi closer accuracy to te exact solutio I te same vei, te two-step tird-derivative metod also ives more accurate results ta te two step starti formula by [6] as stated i equatio (9 6 Coclusio is paper presets a ew approac for starti Numerov metod wit better accuracy e starti metod preseted by [, 6] ad te tird-derivative metod preseted i tis paper are all two-step metods wic ives a ood basis for te compariso of tese starti values However, from te results i ables ad, te accuracy of te ew starti formula o te Numerov block metod sows better accuracy Hece, tis ew starti metod (tird-derivative block metod is a better formula for adequately starti te Numerov metod Refereces [] Adee, S O, P Oumayi, U W Sirisea ad Y A Yaaya 5 Note o starti te Numerov metod more accurately by a ybrid formula of order four for a iitial value problem Joural of Computatioal ad Applied Matematics, 75 (: 69-7 doi: 6/cam466 [] Ele, B L 968 Hi order A-stable metods for te umerical solutio of systems of DEs BI Numerical Matematics, 8 (4, doi: 7/BF947 [] Erit, W H 974 Secod derivative multistep metods for stiff ordiary differetial equatios SIAM Joural o Numerical Aalysis, (, - doi: 7/79 [4] Fatula, S O 988 Numerical metods for iitial value problems i ordiary differetial equatios, Academic Press, New York [5] Jator, S N, ad Li, J A aloritm for secod order iitial ad boudary value problems wit a automatic error estimate based o a tird derivative metod Numerical Aloritms, 59 (, -46, doi: 7/s [6] Gozalez, J Q ad D ompso 997 Getti started wit Numerovs metod Computers i Pysics, (5: ttp://dxdoior/6/6859 [7] Kouetsof, A A ew two-step ybrid metod for te umerical solutio of te Scrodier equatio Joural of Matematical Cemistry, 47 (: doi: 7/s [8] Lambert, J D Computatioal metods i ordiary differetial equatios, Wiley: Lodo, 97 [9] Misra, B N ad R K Moaty Sile cell Numerov type discretizatio for D biarmoic ad triarmoic equatios o uequal mes Joural of Matematical ad Computatioal Sciece, (: 4-5 ttp://wwwscikor/idexpp/mcs/article/viewfile/774/8 [] Moaty, R K ad R Kumar 4 A ovel umerical aloritm of Numerov ype for D quasi-liear elliptic boudary value problems Iteratioal Joural for Computatioal Metods i Eieeri Sciece ad Mecaics, 5 (6: doi: 8/ [] Norto, M S 9 Numerovs Metod for approximati solutios to Poissos equatio ttps://wwwsiueedu/_morto/numerovpdf (Accessed o October 9, 5 [] Obreckoff, N 94 O mecaical quadrature (Bularia Frec summary Spisaie Bular Akad Nauk, 65, 9-89 [] Oumayi, P, U W Sirisea ad S Adee Some teoretical cosideratios of cotiuous liear multistep metods for u ( v ( ad Applied Scieces, (: -5 f x, u, v, Baale Joural of Pure [4] Sai, R K, Jator, S N, & Ka, N A Cotiuous fourt derivative metod for tird order boudary value problems Iteratioal oural of pure ad applied matematics, 85 (5, 97-9 doi: 7/ipamv85i59 [5] Simos, E 9 A ew Numerov-type metod for te umerical solutio of te Scrodier equatio Joural of Matematical Cemistry, 46 (: 98-7 doi: 7/s [6] Yusup, Y ad P Oumayi 5 New multiple FDMs trou multistep collocatio for y f (x, y Proceedis of te Natioal Matematical Ceter, Abua Nieria [7] Vio-Auiar, J ad H Ramos 5 A variable-step Numerov metod for te umerical solutio of te Scrodier equatio Joural of Matematical Cemistry, 7 (: 55-6 doi: 7/s