Four steps implicit scheme for the solution of second order ordinary differential equation was derived

through interpolation and collocation method. Newton polynomial approximation method was used to

generate the unknown parameters in the corrector. The method was tested with numerical examples

and it was found to be efficient in solving second order ordinary differential equations

Adesanya, A.O.

Anake, T. A.

Oghonyon, J. G.

Osilagun, J.A.

English

Covenant University Repository

FOUR STEPS IMPLICIT METHOD FOR THE  SOLUTION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS  *1OSILAGUN, J. A.,  2ADESANYA, A.O.  2ANAKE, T. A.  AND 2OGHOYON, G. J.           1Department of Mathematical Sciences, Olabisi Onabanjo University,          Ago Iwoye, Ogun State, Nigeria.            2Department of Mathematics, Covenant University, Sango Ota, Ogun State, Nigeria.          *Corresponding author:  torlar10@yahoo.com   Tel: +2348086049694           plicit block method based on Newton’s backward divided difference formula.     Adesanya et al. (2009) proposed a two step method for the general solution of second order which is self stating and adopted New-ton’s polynomial to generate the starting value. Awoyemi et al. (2009) recently pro-posed a self starting Numerov’s method. This method solves both initial and bound-ary value problems of ordinary differential equation. Yahaya (2007) constructed a Nu-merov method from a quadratic continuous polynomial solution. This process led to method which is applied to both initial and boundary value problems.   In this research work, we propose a block method for step length of four. This method adopts Newton’s polynomial approximation to generate the starting value and solves gen-eral second order ordinary differential equa-ABSTRACT Four steps implicit scheme for the solution of second order ordinary differential equation was derived through interpolation and collocation method. Newton polynomial approximation method was used to generate the unknown parameters in the corrector. The method was tested with numerical examples and it was found to be efficient in solving second order ordinary differential equations.  Keyword: collocation, interpolation, continuous, implicit, Newton’s polynomial, corrector.   INTRODUCTION The second order initial value ordinary dif-ferential equation of the form  ,                                       (1) is considered in this paper. Differential equation is often used to model scientific and technological problems which most times, these equations do not have analytic solution, hence an approximate numerical method is required to solve these problems. Block methods for numerical solution of ordinary differential equations have been proposed by several researchers such as Milne (1953), Shampine and Watt (1969), Worland (1976) and Omar (1999). Rosser (1967) introduced the 3 point implicit block method based on integration formulae which is basically Newton’s cote type. Zanariah et al. (2006) proposed 3 points im-),,( ''' yyxfy  1'0 )(,)(   ayayJ. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 Journal of Natural  Sciences, Engineer-ing and Technology  52 © UNAAB 2009 tion directly.                                                                  METHODOLOGY We consider an approximate solution to (1) in power series                                (2.2)   are constants to be determined. We Consider a linear multistep method of the form  kjjj xaxy0)()( 12)1(0,,  kja jjj                                                            (2)  where , m=the distinct collo-cation point, t is the interpolation point, for our method, the step length k=4   10102 )()()(trmrrnrrnr fxhyxxy ],[ rnn xxx ,                                                                     (3) , r=0, 1, 2…, m-1                                                                    (4) ,  r=0, 1, m-1                                        (5) To get  and , According to Yahaya (2007), he derived a matrix of the form  where: is an identity matrix of dimension .                                 (6) 10,1 )()(miiirir xpx  1,....,2,1,0  mr10,1 )()()(mtiiri xpxx rnrn yxy  )( ]1...,2,1,0[  tr ,)(''rnfxy )(xj )(xjIDC I )()( mtmt .)2)(1(.200............200.1......1.122112111121112mtmmttntntnitnnnitnnnxmtmtxxxxxxxxxD1OSILAGUN, J. A., 2ADESANYA, A.O.2ANAKE, T. A. AND 2OGHOYON, G. J. J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 53                                            Development value for the unknown Theorem (3.0): Assuming that  and for k=0, 1, n are distinct values, then  , where is a polynomial that can be used to approximate   For Newton’s polynomial                                                                                                                                               , is the remainder and has the form                                                          (Awoyemi et al.,2009)  Development of four steps method  In developing the method with step length k=4, we consider D=                                               1,20,21,1,0,1,220,221,21,20,21,120,121,11,10,1....................mmtmttmtmtmtmtmthhhhhhc],[1 bacf n ],[ baxk )()()( xRxyxf n )(xy)(xf))...()((......))(()()( 110102010  nn xxxxxxaxxxxaxxaaxy)()( xyxf  )(xRn))()....()(()!()( 11011nnnnn xxxxxxxxxfxR  443424443332334232222413121143263534333233625242322223020126200302012620030201262003020126200302012620011nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxFOUR STEPS IMPLICIT METHOD FOR THE SOLUTION OF GENERAL … J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 (7) (8) (9) (10) 54 This gives a continuous scheme                                                                                                                                                                                                            (11) where:                      Evaluating (11) at i.e. when , gives                            (12)                                                                                                                            (13)                                                                                                                  (14)                                                                                  Evaluating the first derivative of (4,1) at t  = -3, gives                           (15)  t2)1(3  t tttttht 11205621440)( 345620  tttttht 1830592360)( 345621  tttttht 55605122240)( 345622  ttttttht 1181805025152360)( 2345623  tttttht 2160551821440)( 345624 hxxt n 34nx x  3,2,1 t)41420419(240480240 12342234 nnnnnnnn fffffhyyy  )2419424(240480240 12342123 nnnnnnnn fffffhyyy  )17252402523(240720480 12342123 nnnnnnnn fffffhyyy  )475190896628433(144014401440 1234223'nnnnnnnn fffffhyyhy  1OSILAGUN, J. A., 2ADESANYA, A.O.2ANAKE, T. A. AND 2OGHOYON, G. J. J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 55 Solving (12)-(16) using block method gives                                (16) Hence, from (16)                                                                                                                                                                                                                                                                                                                                               ''3'2'1321243212321432140003000200010007208690001601470009053000144036700080574557712019112491609838027401173014583158480736029240478310001000100010001000010000100001nnnnnnnnnnnnnnnnnnnnyyyyhffffhffffhyyyyyyyy  '432121 367211162825401440 nnnnnnnhyfffffhyy    '432122 2533163014490 nnnnnnnhyfffffhyy    '432123 314796054468160 nnnnnnnnhyfffffhyy    '32124 456644819245 nnnnnnnhyffffhyy  FOUR STEPS IMPLICIT METHOD FOR THE SOLUTION OF GENERAL … J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 (17) (18) (19) (20) 56 Developing the unknown for k=4 We evaluate the first derivative of (8), and neglecting and higher values.                                                                                                                                                                                                                                                                                                                                               Solving (24)-(28)  for , i = 1(1)4 gives,                                                                                                                                                                                          Substituting for in (15) and substituting in (16)-(19) and solving for ,  gives,                                                                                                                                                            6annnnnnn yyyyyyhy 137300220120751260 12345'  nnnnnnn yyyyyyhy 12651206020360 12345'1  nnnnnnn yyyyyyhy 330206015260 12345'2  nnnnnnn yyyyyyhy 215602030360 12345'3  nnnnnnn yyyyyyhy 32060120651560 12345'4  )225924723824205386(6348''4'3'2'11 nnnnnnn yyyyyhyy  )17412135288705(529''4'3'2'12 nnnnnnn yyyyyhyy  )729105157825202598(2116''4'3'2'13 nnnnnnn yyyyyhyy  )504476258012722188(1587''4'3'2'14 nnnnnnn yyyyyhyy  2ny inf  4)1(1i)4837481614361(141 ''4'3'2'11 nnnnnnn yyyyyhff  )7231241942134(161 ''4'3'2'12 nnnnnnn yyyyyhff  1OSILAGUN, J. A., 2ADESANYA, A.O.2ANAKE, T. A. AND 2OGHOYON, G. J. J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) 57                                                                                                     Substituting (25)-(28) into the second derivative of (8) and comparing with (29)-(32) give                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Analysis of the scheme Convergence  The convergence analysis of this scheme is determined using the approach of Fatunla (1992), where each block integrator is represented as a single step block r- point multistep method of the form              where:  are r by r matrix respectively with element  specifically, is an  identity matrix, are vectors of numerical estimate describe below. With r-vector  )4829163674923(1411 ''4'3'2'13 nnnnnnn yyyyyhff  )37373166316(1 '' 4'3'2'14 nnnnnnn yyyyyhff  nnn hfyy 222193.0''1 nnn hfyy 111768.0''2 nnn hfyy 188485.1''3 nnn hfyy 311393.1''4 FOUR STEPS IMPLICIT METHOD FOR THE SOLUTION OF GENERAL … J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 (31) (32) (33) (34) (35) (36)  58  Awoyemi (2009) for details       Analysis of the block   det  det    Hence the block is zero stable.  Numerical example We test the efficiency of our scheme on linear and non linear second order differential equation. Problem 1:      010001000100010001000010000100001010001001001000)1(3 1,0,0,00)( 2'''  yxy' 1(0) 1, (0) , 0.1/ 402y y h  xxxy22ln211)(Exact solution  1OSILAGUN, J. A., 2ADESANYA, A.O.2ANAKE, T. A. AND 2OGHOYON, G. J. J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61  59 S/N Expected solution New method error 1 1.00125000065104 1.001250000186 4.650D-10 2 1.00250000520835 1.0025000039971 1.211D-09 3 1.00375001757828 1.003750013474 4.030D-09 4 1.00500004166729 1.005000047982 6.314D-09 5 1.00625008138212 1.0062500803586 8.462D-09 6 1.00750014062974 1.00750002579038 1.148D-09 7 1.00875022331755 1.008750239662 1993D-09 8 1.01000033335333 1.0100000783829 2.550D-09 9 1.01125047464542 1.0112504890376 4.256D-09 10 1.01250065110271 1.0125006101014 4.100D-08 Problem 11      xxeyy 3'' 40/1.0,325)0(,323)0( '  hyy)3exp(3234)(xxxyExact solution  S/N Expected result y(x) New method yn(x) Error y(x)-yn(x) 1 -0.094140915761848 -0.0941409131568 2.61D-09 2 -0.094532404142338 -0.0945324074753 3.33D-09 3 -0.094924451608388 -0.0949244224215 2.92D-08 4 -0.095317044390700 -0.0953170247449 2.02D-08 5 -0.095710168480980 -0.0957101793552 1.08D-08 6 -0.096103809629113 -0.0961039982252 1.88D-07 7 -0.09649533403163 0.0964952920353 4.23D-08 8 -0.096892584872264 -0.0968923659413 2.18D-07 9 -0.097289689232184 -0.0972874625827 2.22D-06 10 -0.097683251173919 -0.0976830958236 1.55D-07 FOUR STEPS IMPLICIT METHOD FOR THE SOLUTION OF GENERAL … J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 60 REFERENCES Adesanya, A.O., Anake, T.A., Bishop S. A., Osilagun, J.A. 2009. Two steps block method for the solution of general second order initial value problems of ordinary dif-ferential equation. Asset International Jour-nal, University of Agriculture, Abeokuta  Awoyemi, D.O.,  Adesanya, A.O., Ogun-yebi, S.N. 2009. Construction of self Start-ing Numerov method for the solution of initial value problems of general second or-der ordinary differential equation. Int. j. Num. Math, University of Ado Ekiti,  4(2): 269-278 Milne, W.E. 1953. Numerical solution of differential equation, Wiley, New York.  Omar, Z. 1999. Developing parallel block method for solving higher orders ODES directly. Thesis, University putra Malaysia Rosser, J.B. 1967. A Runge Kutta for all season, SIAM, P. 417-452. Sham pine, L.F., Watts, H. 1969. Block implicit one step method, international journal of Math-comp 23: 731-740.     Worland, P.B. 1976. Parallel method for the numerical solution of ordinary differential equation, IEEE trans. Computer C-25: 1045-1048. Yahaya, Y. A., 2007. A note on the con-struction of Numerov method through a quadratic continuous polynomial for the general second order ordinary differential equation. Journal of Nigerian Association of mathematical physics, II: 253-260  Zanariah Abdul, Mohamed Bin, Zurni Omar 2006. 3 point implicit block method for solving ordinary differential equation. Bulletin of the Malaysian Mathematical Sci-ence 2, 29(1):23-31  (Manuscript received: 12th  January, 2009; accepted: 22nd April, 2009) 1OSILAGUN, J. A., 2ADESANYA, A.O.2ANAKE, T. A. AND 2OGHOYON, G. J. J. Nat. Sci. Engr. Tech. 2009, 8(1):52-61 61 

Four Steps Implicit Method for the Solution of General Second Order Ordinary Differential Equations

A note on the construction of Numerov method through a quadratic continuous polynomial for the general second order ordinary differential equation.

Block implicit one step method,

Developing parallel block method for solving higher orders ODES directly. Thesis,

Numerical solution of differential equation,

Parallel method for the numerical solution of ordinary differential equation,

Two steps block method for the solution of general second order initial value problems of ordinary differential equation.

http://eprints.covenantuniversity.edu.ng/2043/1/FOUR%20STEPS%20IMPLICIT%20METHOD%20FOR%20THE.pdf

Four Steps Implicit Method for the Solution of General Second Order Ordinary Differential Equations

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