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A MAXIMAL ORDER ALGORITHM FOR SOLVING INITIAL VALUE PROBLEMS OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

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A maximal order method with continuous variable coefficient is proposed for initial value problems of first order ordinary differential equations. The main objective of thisstudy is to develop some continuous multistep collocation methods with all
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  1 Continental J. Applied Sciences 5:1 - 7, 2010 ISSN: 1597 - 9928©Wilolud Journals, 2010 http://www.wiloludjournal.comA MAXIMAL ORDER ALGORITHM FOR SOLVING INITIAL VALUE PROBLEMS OF FIRST ORDERORDINARY DIFFERENTIAL EQUATIONS.Oluwatusin, E. A.Department of Mathematics, School of Sciences, College of Education, Ikere-Ekiti, Ekiti State, Nigeria,eaoluwatusin@yahoo.comABSTRACTA maximal order method with continuous variable coefficient is proposed for initial valueproblems of first order ordinary differential equations. The main objective of thisstudy is todevelop some continuous multistep collocation methods with all collocation points taken at theselected grid points. The step number k=4 is considered with order of accuracy equals to 2k which is the degree of the basis function. Four Predictors in  y +   k i )1(1 = are also proposed.The derived scheme belongs to the family of linear multistep methods where collocation andinterpolation of differential system and approximate solution was restricted to grid points in  y +   k i )1(1 = and 1)1(1 −= k i respectively. Numerical experiment using the new scheme tosolve both linear and non-linear initial value problems reveal high level of consistency andaccuracy.KEYWORDS and Phrases; Collocation, Interpolation, Grid point, Consistency, Step number,Initial value problemsINTRODUCTIONIn this paper, the numerical solution of the initial value problems in ordinary differential equations of the form 00 /  )(),,()( y x y y x f  x y == (1)is considered where f and y are scalar or vector functions and ),( 00 y x is the given initial starting point.The characteristic of discretization methods is that errors are generated when they are adopted for the solution of Ordinary Differential Equations. Consequently, these errors have to be minimized to ensure the accuracy of themethod. A solution is said to be accurate when the computed value does not deviate significantly from the exactsolution as the iteration progress. The magnitude of these errors referred to as global error, e is in the form of  nn y x ye −= )( (2)where )( n  x y and n  y are exact and computed solutions at n  x respectively. This global error, e is a function of localtruncating error.L ∑ −++ −= k iinin xiyh x yah x y 0 /  )]()(;[]),([ β  (3)And must be negligibly small for high accuracy. Therefore the main objective of this work is to ensure that the localtruncating error (3) is minimized hence the development of maximal order method per even step number k.Many authors and researchers such as Fatunla (1988), Awoyemi (1992), Oluwatusin (2006), Lambert (1973, 1991),Onumanyi, Awoyemi, Jator and Sivisena, (1994). Oladele (1991), Kockler (1994), Jacques and Judd (1987),proposed numerical methods of order of accuracy less than 2k. Awoyemi (2001) points out that some of themathematical models resulting to non-linear Ordinary Differential Equations which might not have analytical  2 Oluwatusin, E. A: Continental J. Applied Sciences 5:1 - 7, 2010solution could be solved by numerical methods of high order of accuracy. It was also discovered according to Ibijolaand Ogunrinde (2006) that the language of engineering is differential equations.This implies that the proposed scheme in this paper can also be used to solve problems in engineering, where highprecise accuracy is a must.DERIVATION OF MAXIMAL ORDER METHODThe derivation of maximal order method is investigated for k=4.Consider an approximate solution to problem (1) in the form ∑ = = k  j j j  xa x y 20 )( (4)And its derivative is ),,()( 201 /   y x f  x ja x y k  j j j == ∑ =− (5)By collocating the differential system (5) at all grid points and interpolating (4) at 3)1(0 , == + i x x in a system of algebraic equations is obtained )6(8765432 8765432 8765432 8765432 8765432 4478467456445434423421 3378367356345334323321 2278267256245234223221 1178167156145134123121 786756453423213388377366355344333322310 2288277266255344233222210 1188177166155144133122110 8877665544332210 +++++++ ++++++++ ++++++++ ++++++++ +++++++++ +++++++++ +++++++++ + =+++++++ =+++++++ =+++++++ =+++++++ =+++++++ =++++++++ =++++++++ =++++++++ =++++++++ nnnnnnn nnnnnnnn nnnnnnnn nnnnnnnn nnnnnnnn nnnnnnnnn nnnnnnnnn nnnnnnnnn nnn  f  xa xa xa xa xa xa xaa  f  xa xa xa xa xa xa xaa  f  xa xa xa xa xa xa xaa  f  xa xa xa xa xa xa xaa  f  xa xa xa xa xa xa xaa  y xa xa xa xa xa xa xa xaa  y xa xa xa xa xa xa xa xaa  y xa xa xa xa xa xa xa xaa  yn xan xan xan xan xan xan xa xaa  Using Gaussian elimination techniques, the values of   j a ’s are given as follows  3 Oluwatusin, E. A: Continental J. Applied Sciences 5:1 - 7, 2010 )}168317368403196028056()32172102514035()306020( )3352010()44()2{( 6)60212601050420701() 23152525231535()2754515() 2155()3{( 24)}56072833656( )918421()126()464{( 120)}81751481788828()819867()25164626410619720720{( 9720)}477648()129 36910531129727281{( 1692}475527921523355072045031360{7200 54233245 8432234 73236225412 33432234 83223 72226523 443223 822761234 65228712341 668234123 771234123 88 h xh xh xhhx xa h xh xhhx xahhx xa hhx xah xa f  f  f  hah xh xhhx xah xhhx xa h xh xah xa f  f  f  hah xhhx xa hhx xah xa f  f  f  f  f  hahhx xah xa hf  f hf  yhf  y y hah xahf  hf hf hf  y y y y hahf hf hf hf hf  y y y y ha nnnn nnnnnnn nnnnnn nnnnn nnnnnnnn nnnnnnnnnnn nnnnnnnnnn nnnnnnnnn nnnnnnnnn +++++ −++++−++− ++−+−+−= ++++−+++− ++−+−−−= +++− ++−+−+−−−= ++−+− −−+−−−= +−−−−+−−+= −−−−+−−+= ++++++++ +++++ ++++++ +++++++ )}428841401408428( )2721210570210521()315303015() 25101510( )266()33(){( 2 6524334256 854233245 7432234 63223 52243122 h xh xh xh xhhx xa h xh xh xhhx xa h xh xhhx xah xhhx xa hhx xah xa f  f  ha nnnnnn nnnnn nnnnnnn nnnnn ++++++ −+++++ −++++−+++− ++−+−−= + n xaan xaan xaan xaan xaan xaa xa f a nnn 7867564534231 8765432 +++++−−= nnn xan xan xan xan xan xan xa xa ya 8877665544332210 −−−−−−−−= (7)  4 Oluwatusin, E. A: Continental J. Applied Sciences 5:1 - 7, 2010Substitute the values  j a ’s into the expression (4) and after algebraic manipulation a continuous scheme is obtainedin the form ∑ ∑ = =++ = k  jk  j jn j jn j f  x y x 00 )()( β α  (8)Where ),(  jn jn jn y x f  f  +++ = by taking , / )( 3 h x xt  n + −= (9)  j α  and  j  β  are determined as follows )47364926586112917967087200( 7200)23201584426661123522600( 300)11107363409302948504( 120)1970380212866723172616900( 900)125814419313236( 2400)1188230148277452180( 2701)1361586414163( 201)8184471236( 161)3435381327188648839673141540( 5401 8765432 08765432 18765432 28765432 38765432 48765432 08765432 18765432 28765432 3 t t t t t t t t  ht t t t t t t t  ht t t t t t t  ht t t t t t t t  ht t t t t t t  ht t t t t t t  t t t t t t t  t t t t t t t  t t t t t t t  −−−−+++= −−−−+++= −−−−++= −−−−−++= ++++++= −−−−++= −−−−++= ++++−−= ++++−−−=  β  β  β  β  β α α α α  (10)Put t=1 implies that 4 + = n  x x and substitute the result in (8) yields a symmetric discrete scheme )163616( 56533335 1234134 nnnnnnnnn f  f  f  f  f  h y y y y ++++=−−+ +++++++ (11)3. Derivation of predictors for the proposed methodsTo Implement the proposed methods, the predictors are required. Hence the derivations of the predictors areconsidered in this section. Taking the approximate solution to problem (1) to be  5 Oluwatusin, E. A: Continental J. Applied Sciences 5:1 - 7, 2010 ∑ −= = 120 )( k  j j j  xa x y (12)The first derivative of (12) is given as ∑ −=− = 1201 /  )( k  j j j  xa x y (13)By collocating (13) at 1)1(0, −== + k  j x x  jn and interpolating (12) at 1)1(0, −== + k  j x x  jn to obtain asystem of non linear equation. By solving for  j α  ‘s and substituting is values into the equation 12 then a continuousscheme for k=4 is derived in the form ∑∑ −=+−=+ += 1010 )()()( k  j jn jk  j jn j f t  yt t  y β α  (14)Where )717176( 2)()8232812( 2)()931514112( 12)()613124( 361)()215394115( 41)()69412( 41)(108728321724736( 361)( 65432 165432 265432 365432 065432 165432 265432 3 t t t t t  ht t t t t t  ht t t t t t t  ht t t t t t t  t t t t t t  t t t t t t  t t t t t t  ++++= ++++= +++++= ++++= ++++= −−−+= −−−−−=  β  β  β α α α α   Evaluating (14) at t=1, that is, 4 + = n  x x , yields a symmetric discrete explicit scheme )3(122828 123134 ++++++ +++++−= nnnnnnn f  f  f h y y y y (16)Similarly, the predictors for 3 + n  y and 2 + n  y are derived to be )25(21193 12123 nnnnnnn f  f  f h y y y y −−+−+= +++++ (17)and
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