Iteratioal Joural of Iovative Mathematics, Statistics & Eerg Policies ():-6, Ja.-Mar. 7 SEAHI PUBLICATIONS, 7 www.seahipa.org ISSN: 67-8X A Class of Bleded Bloc Secod Derivative Multistep Methods for Stiff Sstems Omagwu S., Chollom J. P, Kutchi.S.Y. Departmet of Mathematics & Statistic, Kadua Poltechics, Kadua Nigeria & Departmet of Mathematics, Uiversit of Jos, Jos Nigeria E-mail of the correspodig author (samsoomagwu@ahoo.com) ABSTRACT I this paper, the formulatio of a class of secod derivative bleded bloc multistep methods for step umbers =,, is cosidered through the Eright ad the multistep collocatio approaches. With these approaches, we hope to improve the stabilit regios of the Adams Moultos Methods ad thereb maig them suitable for the solutio of stiff ordiar differetial equatios. These ew methods proposed i this paper tur out to be A-stable. Numerical examples obtaied demostrate the accurac ad efficiec of the ew bleded bloc multistep methods. Kewords: A-stable, Adams Moultos methods, Bleded Bloc methods ad Stiff ODEs,. INTRODUCTION Most real life problems whe modelled mathematicall result i ordiar differetial equatios. Some of the equatios do ot have aaltic solutios as such the eed for good umerical methods to approximate their solutios. I this paper we are cocer with the umerical solutio of the stiff iitial value problem () usig the secod derivative liear multistep. '( x) = f ( x, ( x)), ( x ) = () :[ x, x ] Â ad f :[ x, x ] Â Â m m m o the fiite iterval I = [ x, x N ] where N N is cotiuous ad differetiable. The secod derivative -step method taes the followig form a = h b f h g g = = = å å å () where a, b ad g are parameters to be determied ad g = f '. Several methods have bee developed to overcome this barrier theorem. Researchers lie Gear (96),Lambert(98), Butchers(966), the secod derivative methods of Eright (97),Gei (97),Gamal ad Ima (998), Sahi et-al(),mehdizadeh et-al (), Ehigie ad Ouuga() ad the third derivative method of Ezzeddie ad Hoati (), either relax the coditio to obtai A stable methods or icorporate off-step poits to improve the stabilit of the methods. I this wor, we cosider the secod derivative bleded bloc multistep methods for step umbers =,,, 6. With this approach, we hope to improve the stabilit regios of the covetioal Adams Moulto method ad thereb maig them suitable for the solutio of ()
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7. Bleded Liear Multistep Methods A good umerical method for the solutio of stiff sstems of ODEs must have reasoabl wide regio of absolute stabilit. A-stabilit requiremet puts a severe limitatio o the choice of suitable umerical methods for solvig stiff ODEs. Most covetioal umerical methods are ot A-stable for but bledig two existig methods has the chaces of producig oe. Bleded liear multistep methods is a process where two existig liear multistep methods are combied b multiplig the Jacobia of the bleder with a costat multiplier to produce a sigle method without alterig the order of the Bleded method for improved performace. Eright (97) bleded the Adams Moultos method of order ad the Bacward Differetiatio Formula of order to obtai the Eright Formula of order. Eright AMF hjbdf Where J is the Jacobia ad is a costat which determies the agle of the regio of absolute stabilit of the method where v i! ad v i s ss... s i ds Seel ad Kog (977) exteded the Eright Bleded Formulas ad propose a sub class of the Bleded formulas as suitable for a geeral purpose ODE solver. The method possesses ehaced stabilit properties with good regio of absolute stabilit. Thus, the method proposed i this paper is oe that combies these desirable qualities for the direct solutio of stiff sstem of ODEs.. FORMULATION OF THE METHOD. Three- Step Bleded Multistep Methods The Bleded Liear Multistep Methods for =,, are derived i this sectio usig Eright approach ad the same processes used b Eright was adopted to obtai the values of the parameters below
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 The Costat To obtai the values of the costat g * we use the equatio v i v s ss s i ds where i...! (.) v v v!!! v s( s )( s ) s( s )( s )( s ) s( s )( s )( s )( s ) v v v v v v v v v v v ds v 9 8 v 7 6 v ds 7 ds 86 8 Adams Moulto Method = is bleded with the Bacward Differetiatio Formula (BDF) =. Usig the Eright s approach Adams Moulto = 9 9 - - - = BDF, = - - - = 6 Bledig the two gives 6
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 9 9 Jh 6 * 9 substitutig γ = from (.) ito the above method 8 ad simplifig ields 7 9 = - 8 7 '' - - h 96. Four- Step Bleded Liear Multistep Methods Adams Moulto Method = is bleded with the Bacward Differetiatio Formula =.Usig the Eright s approach. Adams Moulto = 9 7 BDF Bledig the two Jh 9 7 gives 6 6 6 6 7 7 substituig the value γ from (.) ito the above methodad simplifig ields 7 7 '' h 76 8 9 76. Five- Step Bleded Liear Multistep Methods Adams Moulto Method = is bleded with the Bacward Differetiatio Formula =. Usig the Eright s approach. Adams Moulto Method K= 7
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 7 6 Bledig the two gives BDF 7 Jh 6 7 7 7 7 7 7 9 88 9 88 87 86 from (.) ito 8 9 7 76 77 86 '' h 68 8 substituig the value γ theabove methodad simplifig ields 7 8 (.). BLENDED BLOCK LINEAR MULTISTEP METHODS (BBLMM) The bleded bloc liear multistep method for implemetatio purposes is costructed usig the Oumai et al (99, 999) approach where the geeral cotiuous step LMM is expressed as: ( x) t m ( x) h ( x) f (.) ad extedig it to the secod derivative gives the geeral form of the cotiuous step d derivative liear multistep method as: ( x) t ( x) h m ( x) f h m ( x) '' (.) where h is the step legth, m the umber of distict collocatio poits ad t, the umber of iterpolatio poits. But for the purpose of this wor the geeral form of our cotiuous step d derivative BBLMM is expressed as: 8
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 ( x) m '' ( x) h ( x) f h ( x) (.) wher e h is the step legth, m the umber of distict collocatio poits. With t m i ( x), ix,,,..., t i ( x) tm i, i i x,,,..., m (.) ( x) t m i i, ix,,,..., m beig the cotiuous coefficiets of the method... Three- Step Bleded Bloc Liear Multistep Methods (BBLMM) To costruct the three-step BBLMM, the geeral form of the Three Step Bleded Liear Multistep methods expressed i the form of (.) is % '' ( x ) = a ( x) h éb ( x) f b( x) f b( x) f b( x) f ù ë û h l ( x) (.) Where a ( x), b ( x), b ( x), b ( x), b ( x), l ( x) are the cotiuous coefficiets of the method. I order to determie the cotiuous coefficiets i (), we use the approach i Oumai et.al. (99) to obtai the cotiuous coefficiets as give below. a ( t x ) =, æ t 9t t t hö hb ( t x) = t - - - ç h h 8h 9h hb hb hb æ hö ç h h h h 9t 7t t t 7 ( t x) = - - - æ ö ç h h 8h h - 9t t 7t t h ( t x) = ç - - æ t h ö - ç h 8h h 8h t 9t t ( t x) = - æ - t t t t hö hl ( t x) = - ç 8h h h Substitutig (.6) ito (.) ields the cotiuous form of the-step BLMM (.7). (.6) 9
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 æ ö % ( t x ) = - - - f ç h h 8h 9h ç t 9t t t h t æ 9t 7t t t 7 ö ç h - - - f ç h h h h æ - 9t t 7t t hö - - f ç h h 8h h æ t 9t t t h ö ç - - f ç h 8h h 8h æ - t t t t hö - g ç 8h h h (.7) Evaluatig the cotiuous scheme (.7) at t =, h,h ields the Three-step BBLMM 7 = - - - - h '' 9 9 8 = - - - h 8 8 '' (.8) 7 9 7 9 = - - h 8 8 ''.6. Four- Step Bleded Bloc Liear Multistep Methods (BBLMM) To costruct the four- step BLMM, the geeral form of the Four- Step Bleded Liear Multistep methods expressed i the form of (.) is % ( x ) = a ( x) h é ëb ( x) f b ( x) f b ( x) f b ( x) f b ( x) f h l '' ( x) (.9) ù û The cotiuous coefficiet are obtaied usig the same procedure i (.7) ad substitutig same ito (.9) gives the cotiuous form of the four step bleded liear multistep method.
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 æ 7t 9t t ö ç t - - 6h h 8h % ( t x ) = f 7t h t ç - - h 6 76h 6 æ 6 8t 6t t t t 7 ö ç h - - - f ç h 7h 6h 9h 8h 6 æ- t t t t t 99hö - - - f ç h h h h 8h 6 æ 6 8t t 7t t t 9 ö ç h - - - f ç h 9h h h 6h æ 6 7t 69t 97t t t ö ç - - - 6h h h 7h 78h f ç 9h ç- ç æ 6 t t t t t h ö ç g - - - (.) ç 6h 96h h h Evaluatig the cotiuous scheme (.) at t =, h, h, h ields the Four-Step BBLMM 7 9 7 = - - - - 9 9 7 99 9 9 = - - - - h 6 6 6 '' 7 8 = - - 9 6 9 8 '' - h 78 6 7 = - - 76 8 7 '' h (.) 9 76.. Five- Step Bleded Bloc Liear Multistep Methods (BBLMM) To costruct the five- step BLMM, the geeral form of the Five- Step Bleded Liear Multistep methods expressed i the form of (.) is % ( x ) = a ( x) h éb ( x) f b( x) f b( x) f b( x) f b( x) f b( x) f ù ë û '' h l ( x).... (.) The cotiuous coefficiet are obtaied usig the same procedure i (.7) ad substitutig same ito (.) gives the cotiuous form of the five step bleded liear multistep method.
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 æ 6 7 9t 99t t t 8h t t ö ç t 6 % ( t x ) = - - - - - f ç h 8h 8h 7h 8h h æ 6 7 t t 9t 7t 9t t ö ç 6 h - - - - f ç 8h h 8h 6h 76h 67h æ 6 7 - t 9t 67t t t t ö ç - - 6 h - f ç 6h 8h h h h h 9 æ 6 7 t t t 9t 7t t 76 ö ç 6 h - - - - f ç 6h 6h 7h h h 68h æ 6 7 t t t t t t ö ç 6 h - - - - f ç 8h 7h 8h h 9h 68h æ 6 7 9t 69t 9t 9t 7t 7t 8hö - - - 6 f ç h 8h 8h 7h 96h h 7 æ 6 7 t 7t t 7t t t 6h ö ç - - - g ç 8h h h 8h 8h (.) Evaluatig the cotiuous scheme (.9) at BBLMM t =, h, h, h,h ields the Five-Step 8 = - - - 9 76 8 6 '' - - - h 7 77 7 = - - 8 8 8 9 86 '' - - - h 6 8 = - - - 6 9 77 6 '' - - h 6 6 76 6 9 89 677 = - - 8 8 76 9 867 7 '' - - - h 6 6 68 8 9 7 = - 76 7 87 77 86 - - h 8 68 8 '' (.6) Stabilit Aalsis of the methods(see Ehigie ad Ouugha () The three step method has order Zero-stabilit of the Bloc Methods P = (,,) T ad error costat of æ- 889 8-6 ö C6 = ç,, ø T Followig the wor of Ehigie ad Ouuga (), we observed that the three step bloc method is zero stable as the roots of the equatio are less tha or equal to. Sice the bloc method is cosistet ad zero-stable, the method is coverget (Herici 96).
im(z) Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 Regio of Absolute Stabilit: Solvig the characteristic equatio for r, we obtai the stabilit fuctio æ 6 (8(8 z 766z - 8677z - 778z ö ç 8 7 87z 879)) / (8968z - 986879z 6 R( z) = - 679898z 78878z 878898z - 8778988z - 8698676z 87968z ç 8878) è ø RAS FOR A CLASS OF BLENDED LINEAR MULTISTEP FOR K= TO K= K= K= K= K= - - -.... Re(z) Numerical Experimet Problem Problem Problem
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 Fig: Solutio to Problem usig the Three Step Bleded Bloc Liear Multistep Secod derivative Methods.9.8.7 O ODE solver ODE solver ODE solver Bleded Bloc Bleded Bloc Bleded Bloc.6..... Figure : Solutio of problem Computed with Three - Step Bleded Bloc Adams Moutos Methods Fig : Solutio to Problem usig the Four Step Bleded Bloc Liear Multistep Secod derivative Methods.9.8.7 ODE solver ODE solver ODE solver Bleded Bloc Bleded Bloc Bleded Bloc.6..... Figure : Solutio of problem Computed with Four - Step Bleded Bloc Adams Moutos Methods Fig: Solutio to Problem usig the Five Step Bleded Bloc Liear Multistep Secod derivative Methods.9 ODE solver ODE solver ODE solver.8 Bleded Bloc Bleded Bloc Bleded Bloc.7.6..... Figure : Solutio of problem Computed with Five - Step Bleded Bloc Adams Moutos Methods.9.8 Exact Exact Bleded Bleded.7.6..... Figure : Solutio of problem Computed with Three - Step Bleded Bloc Adams Moultos Methods
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7.9.8 O Exact Exact Bleded Bleded.7.6..... Figure 6: Solutio of problem Computed with Four - Step Bleded Bloc Adams Moultos Methods.9.8 Exact Exact Bleded Bleded.7.6..... Figure 7: Solutio of problem Computed with Five - Step Bleded Bloc Adams Moutos Methods. Exact Exact Bleded Bleded.. -. - -. Figure 8: Solutio of problem Computed with Three - Step Bleded Bloc Adams Moutos Methods
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7. Exact Exact Bleded Bleded.. -. - -. Figure 9: Solutio of problem Computed with Four - Step Bleded Bloc Adams Moutos Methods. Exact Exact Bleded Bleded.. -. - -. Figure : Solutio of problem Computed with Five - Step Bleded Bloc Adams Moutos Methods CONCLUSION A class of bleded bloc liear Multistep secod derivative methods have bee costructed through the Eright ad multistep collocatio approaches. The Regio of absolute stabilit of the covetioal Adams Moulto methods has bee greatl ehaced. (A-stable). These methods are all coverget. Numerical results reveal the efficiec of the methods i solvig stiff sstems ODEs REFERENCES Dahlquist, G. (96). A Special Stabilit problem for Liear Multistep Methods. Bit, 7-. Gear, C.W. (96). Hbrid Methods for Iitial Value Problems i Ordiar Differetial Equatios. SIAM J. Numer. Aal. Ser B,, 69 86. Lambert, J.D. (97). Computatioal Methods i Ordiar Differetial Equatios. Joh Wille. New Yor. Butcher, J.C. (966). O the Covergece of Numerical Solutios to Ordiar Differetial Equatios. Math. Comp., -. Chollom J.P. (): A stud of Bloc Hbrid Methods with li o two-step Ruge Kutta Methods for first order Ordiar Differetial Equatios. PhD Thesis (Upublished) Uiversit of Jos, Jos Nigeria. Chollom J.P., Olatubusi I.O. ad Omagwu S (): A Class of A-Stable Bloc Explicit Methods for the Solutio of Ordiar Differetial Equatios. Research Joural of Mathematics ad Statistics (): - 6. Dalquist, G. (96). A Special Stabilit problem for liear Multistep Methods BIT. 7- Ehigie J.O. & Ouuga S.A. (). L( )-Stable Secod Derivative Bloc Multistep Formula for stiff iitial Value Problems. Iteratioal Joural of applied Mathematics (). 7 - Eright, W.H (97). Numerical solutio of stiff differetial equatios 6(97), -, Dept of Computer Sciece, Uiversit of Toroto, Toroto, Caada. 6
Omagwu et al... It. J. Io. Maths, Statistics & Eerg Policies ():-7, 7 Fatula, S.O. (99). Bloc Method for Secod Differetial Equatios. Iteratioal Jourals of Computer Mathematics., -66 Sahi, R.K.,Jator,S.N ad Kha, N.A. (). A Simpso s tpe secod derivative method for stiff sstems, Iteratioal oural of pure ad applied mathematics. 8(), 69-6. Mehdizadeh, K. M., Nasehi,O,N. ad Hoati, G. (). A class of secod derivative multistep methods for stiff sstems, Acta uiversitatis Apulesis, 7-88. Oumai, P., Awoemi, D.O., Jator, S.N. ad Sirisea, U.W. (99).New Liear Multistep Methods with cotiuous Coefficiets for first order Iitial Value Problems. Joural of the Nigeria Mathematical Societ., 7. Herici, P. (96). Discrete Variable Method i Ordiar Differetial Equatios. Joh Wille. New Yor 7