Continuous implicit method for tbe solution of genera) second order ordinary differential equations

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1 Journal of the Nigerian Association of Mathematical Physics Volume 5 (November, 009), pp 7-78 J. ofnamp Continuous implicit method for tbe solution of genera) second order ordinary differential equations A. 0. Adesanya, T. A. Anake and 3 G. J. Oghoyon Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria. Abstract Methods of collocation and interpolation were adopted to generate a continuous implicit scheme for the solution of second order ordinary differential equation. Newton polynomial approximation method was used to generate the unknown parameter in tlze corrector. This enables us to solve both initial and boundary value problems. Keywords Collocation, interpolation, continuous, implicit, Newton' s polynomial. corrector..0 Introduction The second order ordinary differential equation of the form y" =f(x,y, y') (. ) subject to y(a) = TJ 0, y (a) = TJ is called initial value problem. When the condition is of the form y(a)=tj 0,y(b)= y fora~x~b it is called a boundary value problem, where.f is a continuous function. Scientific and technological problems often lead to mathematical modeling of real life applications such as motion of projectiles or orbiting bodies, population growth, chemical kinetics and economic growth. Differential equation is often used to model : the problems and most times these equations do not have analytic solution, hence an appror-imate numerical method is. required. to solve the problems. Equations (.) is conventionally solved by first. reducing. it to the system of first order ordinary differential equation and then one applies the various methods available for solving the system of the first -order. This approach is extensively discussed in the literature and we cite few examples among others, [J ], {], [3], (4], [5], [6], (7]. Although this approach has tremendous success yet it has certain drawback. For instance, computer programs associated with the method are often complicated especially when the subroutines to supply the starting values for the methods result. i~ longer computer time and more corriputational work. In addition Vigo -Aguiar and Ramos [8], stated that these methods do not utilize additional information associated with a specific ordinary differential equation, such as oscillatory nature of the solution. ' ~orresponding author: e mail address: torlarjo@yahoo.com, Telephone: , Journal of the Nigerian Associatio of Math ematical Physics Volume 5 (November, 009 ), 7 7R 7

2 Continuous implicit method, A. 0. Adesanya, T. A. Anake and G, j. Oghoyon } ufnamp Block methods for numerical solution of the first order urdinary differential equatiuns have been proposed by several researchers such as in [ J 8, 0. and 7]. Rosser II tj I introduced the 3- point implicit block method based on integration formulae which is basigdl y Newton's cote type. Zanariah et al [6), proposed 3 points implicit block method bjsed on Newton's backward divided difference formula. Considerable atlention hps been devoted to development of method to solve spec ial second order ordinary differential equation of the type y.. = f(x, y ) direc tly without reducing it to system of first order. For instance fl ],[9], f I 0], [I I J and f ] among others. Hairer and Wanner 3] proposed Nystrom type method and stated other conditions for determining the parameter of the method. Chawla and Sharma [ 4] proposed a method due to Runge kutta method. Method of linear multistep method have been considered by Awoycmi and Kayudc 5]. and Knyode (3]. These methods are predictor corrector methods, although the implementation of the nethod in a pc mode yield good accuracy, the procedure is costly to implement. For instance, pc subroutine are very complicated to write, since they require special techniques for supplying the starting values and for varying the step size. which lead to longer time and more human effort. Jatar and Li [ 6], proposed an order S method that was implemented without the need for either predictor or starting values from other methods. Jator 6] proposed an order 6 method based on the same method. Adesanyu et al. [ 7] proposed a I wo step method for the general solution of second order which is self starting and adopt Newton's polynomial to generate the starting v:jiue. Awoyemi et al. 4] recently proposed a self starting Numerov method. This method solves both initial and boundary value solution of ordinary differential equation. Yahaya [ ] constructed a Numerov method from a quadratic continuous polynomial solution. This process led to method applied to both initi;li und boundary value problems. In this work. we propose a block method for three steps. This method adopt Newton s polynomial approximation to generate the starting value nnd solves both initial and boundary value problems..0 Methodology We first state the uniqueness theorem for higher order ordinary differential equation with initinl value problem Th(wrei,i. Let II / "(. (-J)) k( } u =. x,u, u... ;.(l...l( ; ;:rll = (';. k ::: 0, I... u.,.j,.where u are scalars Let R P<:: the region defined by the inequalities x 0 $ x ~ :r 0 +a, /s; -=-cj::; h, j = O,t..., n -l, (.a> 0, h > 0). Suppose f(x, s 0,... ;s _ ) isdefined in Rand in.addition (a) F is non negative and non decreasing in each x, s 0, S _ in R (b) f(x,c 0,.., C _ ) > Ofor x 0 $ x $ x 0 +a (c) ck;:::o,k:::l,,...,n-l Then the initial value problem (I. I) has a unique solution in R. (sec Wend 5)) for details). We consider an approximate solution to (.) in power series k y(x) = Ia//Ji (x) (.) i=o Journal of lite Nigerian Association of Mathematical Physics Volume 5 (November, 009), 7 7X

3 Continuous implicit method, A. 0. Adesanya, T. A. Anake and G, j. Ogho_yon J ofnamp rp == ', a;, j = 0( )k- I are constants to be determined. Consider a linear multistep method of the form where X= [ X -l - l y(x)= L,(x)y,+r +h Lrp,(x).f; +, (.3) r~o, X +r j, k = step length, 'm = the distinct collocation point. t is the interpolation point, for our method, the step length k = 3 i+m-l f/j,.(x) = L 'i+ l.rpi (x) J = 0,,,..., m -I (.4) i~o (.5) y(x +,)=y +>,re[0,,,...,t-l] y"(x)=.f:~ ~,.. r=o, I,m- (.6) To get ;(x) andrp (.r). According to Yahaya [ ], Onumanyi arrived at matrix of the form DC = I, where I is an identity matrix of dimension (I+ m) x (I+ m)..r,, X J.<+i- I II X H + [ _.\u +l!+i- I xn+l D = I... r +- X Jl+l X u+l + l (.7) 0 0 ( t + m - ])( I) m - ~ X _ h o c= a, _o a~,, a~,, _, h"rpl.ll - qji.m- a.o a" ' aj- I h"rpc. _o h"rp... -l (V~) t a /+.0 a/ +. cx,~m r- (/JI +. h' (/Jt+m m- J 3.0 Development value for the unknown Theorem 3. Assuming.thatfE C + Ia.IJ] and xk E[a,b]for k=d. I. n are distinct values. then f(x) = y(x) + R, (x), where y(x) is a polynomial that can be used to approximate f(x) For Newton's polynomial y(x) = a + a (x - x 0 ) + a (x- x 0 )(x- x ) a,(x- x )(x- x )... (x- x,_ ) (3.) f(x)::: y(x). R (X) is the remainder and has the form t tl+l R (x)= (x-x,)(x-x )... (x-x )(x-x) (3.) II (. ) ' - J -'n+l (See Awoyemi et al. l4]) for details. journal of the Nigerian Association of Mathematical Physics Volume 5 (Novcmhcr, HilJ),

4 Continuous implicit method, A. 0. Adesanya, T. A. Anakc and G,.J. Oglwyon./ r~(namp 4.0 Development of three steps method In developing the method with step length k=:3, we consider This gives a, = - q = + xn+l x+z D== {) x,;+, 7 x,:+~ ; x;,+, J XII+ ] 6x, 6xn+l 6x"+" 6xll+.< 4 xll+ l..\.x n+ x II.5 XII +! 5 xu+ 0x ' II J? x x,;+ l x,:+ 0x,~+c x;,+ ' 0x ' ll +.'. (4.) I? rp 0 ( ) = ~ (- 3t 5 + Or - 7) 360 m = 7 (3r' " + t) Y'J 0 rp~ = h - 0 h /fl = -(3t t) '' 360 i, a,=-l,a = (- 3r' + JOt 4 + lor + 60t + 43r) (4.) h ( ~ ) (/} 0 (I) = - - St r I h:_: ( 4 ~ ) rp ( t) = - 5t + 0r - 60r + 0 rp~ (t) = 7 (-5t r' + 30t +I 0t + 43) -.0 rp,'-(0 =~ ri5t 4 +60r' +60t...,s). 360\: (4.3) x- xl/+ d b.. where f::::z Evaluating (4.) at X +3 i.e. when t = an su strtutjilg th~result in (.3) gives yll+.l - 4 Yn+Z + Yn+l = h (f +J + J Of,+ +.J;,+I) ( 4.4) ( 4.4) has order p=4 and error constant cp + = Evaluating (4.) at X i.e. when = - and substituting the result in (.3) gives yn+] - 4 yn+j + Jy = h (f;,+] + /O.f;t+l + J;,) (4.5) ( 4. 5 ) has order p = 4 and error constant cp - =---. +? I 40 Evaluating (4.3) at =- and substituting the result in (.3) gives 360hy" )' Y,+, = h ( -4.;,+ + 9.;, - 44.!:,+, - 7f,) (4.6) 7 ~ ]tmmal oft he Nigerian Association of Malhcmalical PhyJia Volume 5 (November, 009), 7-7N

5 Continuous implicit method, A. 0. Adesanya, T. A. Anake and G, j. Oghoyon J ofnamp Evaluating (4.3) at I =-l and substituting the result in (.3) gives 360hy;,+, -360yll+' +360yll+i = h (7f,+. -S46fn+ +47lfll+ l -YSfJ Evaluating (4.3) at t = 0 and substituting the result in (.3) gives 360hy ;,+' -360 Yn )' + = h (7fn fll+ - J9fll+ l + 8/ ) Evaluating (4.3) at = I, and substitutij)g the result in (.3) gives 360hy ;, )' )' + = h l ( -8JII fn+ ' + 66/ - 7/ ) Solving (4.3 ), (4.4) and (4.5) using matrix inversion method, we obtain the block J 0 0l)'"+IJ r0 0 -Jly_ r J ' gg ~i~ ~ /,,,l/ JO Q ~~0 J;,_,j 0 0 yll+~ I y - ~ = h- - j' + _. () 0 ::_ t' - IS IS JS n+' 36 4S. n-:'. LO 0 I )',,., 0 0 -I Y, ~~ ~~ :~ f,, l O O!~ _.f., (4.7) (4))) (4.lJ) (4.0) Hence, from (4.0) Y+ l = Y + ;~O (lj4fll+ l - 39fll+ + 4{ J ) + hy;, (4.) v = )' + h" (66f. +6f. +6/' +8/' )+hv + II 4S II+ I +' +.\. II (4.) h" f Y + = )' + 0 (34. ll+l + ljn+' + 34jll+. + JJ7j ) + h)',, (4.3) 4. Development of the unknown fork = 3 Evi:lluate the first derivative of (3.), and neglect a 4 i:md higher values of a i.e. Newton's polynomial of order 4, we obtain hy ;, = -:?~y~+48.x/l'"t-l -:)()y,.. t 6'~""3-3y~,.,.,. ')hy;,+ l = -3y;, -JOyll+i + Jgyn+ -9Y:I+. + Yn+4 h} -\l -8)' +g,, - i' ~ n+ -.) n+l ) n~.i - + -l. (4.4). (4.5) (4'.i6) (4. i 7) Making y" + ~ the subject m ( 4. 4) arid substituting inio ( 4. r5)-( 4.6) and solving for )' +.J + and y"+j gives h.... _. v n+l =.-v + -(57 v - I Oy + 3" + 7 ) I ) "" n+ l o+ J n+~ u 7 h... ); = v +-(7 y +) +3v) + ' -- g n+l n+ - (4.8) (4. 9) h.... Yn+. = Yn + S(9yn+l + 6y +' + 3yll+'l + 3yn +n ) (4.0) Journal oftlte Nigerian Association of Mathematical Physics Volume IS (November, 009), 7-7X 75

6 Continuous implicit method, A. 0. Adesanya, T. A. Anake and G,.J. O~hoyon.I tdnamp Make Yn+l the subj ect in (4.5) and then substitute in (4.6)-(4.8) to solve for.:",.:,, and.:, +., give.. ( I J v +- ) +- ) - - ).+-.> h + l )II+ ~ 6 )+.\ 6 )II (4.).. I ( ) ' = +- 3 ) - +- ' - './+ 3./ /z Jn+l 6yll+ 3)+l +3) (4.).. ( I 7. I 3. ) = \' +- v +- v.ill+).ill h Y+ l.,!+. +.\ II (4.3) Comparing (4.0)-(4.3) with the second derivative of (3.) gives ' 504. Y =--v---hf ll+ l 447 II 447 II v ' = 6974, _ 60 h. - >+ I 447 ) II 447./II (4.4) (4.5) If Yn Yn- J44J7 (4.6) 5.0 Numerical example We test the efficiency of our scheme on li near and non linear second mder differential equation. Problem 5;: y.. - x( y ) = 0. v(o) = I, v (0) = -,h = Exact solu ton. y(x) =I+ -In ] (-- +X) -.x (;rid point.. Expected result Calculated result Error Problem 5... y =y ' I ' I ()0.4 6 ~ ' '... j),0050' '.~! J J ; ; : 4.88i40 ~ G9.. : -o~oo7s _ = :. : "6. I 83080, 09~ ~ ~ 0~000 I ,, 0.05,. -~~ ' t.od65oogl38 I ' J ' > ' : ' -~.. LOI J I I y(o) =I, y. (0) =-I, h = 0. I 40 Exact soluti on y(x) =_I_ X. '. 7o.lou mal of the Nigerian Association of Mathematical Physics Volume 5 (Novcmhcr, 009), 7 J - 7/i

7 Continuous implut ;, :... c, A. 0. Adesanya, T. A. Anake and G, J. Oghoyon J o.fnamp Grid point Expected result ll I Calculated result I I I Error I Problem 5.3 y = y + e3x y(o) = 3, y co) = 3,h = o.u4o 4x-3 Exact solution y(x) = exp(-3x) Grid point Expected result ; Calculated result I o ~ (LQ tm [544 I Error ~70~07 Z!-: ,[LJ, J.;n.. References,Lambert ( 973), computational methods in ordinary differential equation,jolm Wiley. New York. (] S. 0. Fatunla (99): Block method for second order ordinary differential equation. International journal... of co~puter mathematics, volume 4 I, issue I &, pg (3] Jennings ( 987): Matrix computation for engineers and scientists, John Wiley and Sons, A Wiley Interscience publication, New York. 4] L. Brujnano, D. Trigiante ( 998): Solving differential problems by multistep initial and boundary value methods. Gordon and Breach science publishers, Amsterdam, Pg [5) P. Onumanyi, S. N Jatar, U. W. Serisena ( 999): Continuous finite difference approximation for solving differential equation. International journal of Computer Mathematics, volume 7, No I, pg [6] S. N. Jator (00): Improvement in Adams- Moulton method for the first order initial value problems. Journal of the Tennessee Academy of science, volume 76, No, pg [7] D. 0. Awoyemi (00 I): A new Smith-order algorithm for general secondary order ordinary. differential equation. International journal of computer mathematics volume 77. pg I 7-4 Journal of the NigerianAssociation of Mathematical Physics Volume 5 (November, 009),

8 Continuous implicit method, A. 0. Adesanya, T. A. Anake and G,.J. Oghoyon J ofnamp pq J. Vigo - Aguiar, H. Ramu.s (0()(,): 'ariabk :;:cp size i:;p!c:m.:n!::ti c~ : or mu!ti -;! :p,n, lh >rl for y.. = f (X, y, y.). Journal of computational a;d applied mathenwtics volume 0 p:gc 4- : I. '=' S. 0. Fatunla (I Y95): A class of block method for second order initial value problems. lntcmati,hl:d joumal of computer mathematics. volume 55 issue I & page I 9- :n. II 0 P. Hcnrici ( %): Discrete variable method for Odes. John Wiley. New York. USA. V. A. Aladeselu (007): Improved family of block method for special second order initial value problems Jou rmll of Nigerian Associat ion'of mathematical physics volume II, pg 53- I 5X. Y. A Yahaya CW07): A not e nn lhc construction of Numerov method through a quadratic continunus polynomial fm the general second order ordinary dillen:ntial equati on. Journa l of Nigerian Association of mathematical physics, volume II. page 53-:~60 I I 3 E. Hairer. G. Wanner, ( llj76): A IIH:ory for Ny strom methods, Numerische mat hcmat ic. volume '\ pal'l' X [4 M. M. Chawla. S. R. Sharma ( 9X5): Families of three stage third order Runge Kut ta-ny.stro lllctlwd for y = f (X, y, y), journal of the Australian mathematical soc iety. volume (. pag..: 375-.lX(l [I 5 D. 0. Awoyemi. S. J Kayode (005): A maximal order collocation method for direct solut ion of initial value problems of general second order mdinary differential equation. Proceedings ()f the runlcn.:nl c organized by the National math~matical centre. Abuja. Niger ia II 6 S. J. Jator, J. Li. (006): A se lf starting linear multistep met hod for a direct solution oi' the general second urder init ial value problems (to appear) 7] A. 0. Adesanya, Anake. T. A.. BishopS. A. (00lJ). Two steps block method for the so lution oi' ~cncral second order initial value problems of ordinary dirferential equation (to appear). I Ill! W. E. Milne ( IY5 3): Numerical so lution of differential equation. Wiley. New York. I I Y I J. B. Rosser (I 967): A Runge Kutta for all season. SlAM, page 4 I [0[ P. B. Worland ( 976): Parallel method for the numeri cal soluti on of ordinary dillcn.:ntial equation. IEEE trans. Computer C-5, page I 045- I 04l\. [] Z. Omar ( I 99): Developing parallel block method for solving higher orders ODES directly. Thesis. University putra Malaysia [:] L. F. Sham pine, H. Walls (969): Block implicit one step method. international journal of Math-comp (3) page ! Kayode, S. J. (005): Some conti nuous method for th e s J iuti on initial value: problems of general second order ordinary differelllial equation. PhD thesis. Federal uni versity of Technology. \kure. Ondo State. Nigeria 4 D. 0. Awoyemi. A.O. Adesanya, S. N. Ogunyebi (009): Construction of sel f Starting Num..:ro, mctlwd for the solution of initial val ue problems of general second order ordinary clii'fcrcntia l cquat on. Int.,i. Num. Math, University of Ado H iti, Vol. 4 No., pp. 69-/l\ 5 V. V. Wend ( 969): Existence and Uniqueness of solut ion of ordinary differential equal ion. Proceed in::_!' ol the American Mathematical. ~o~i~ty, Vol.3, No. I. pp Zanariah AbduL Mohamed Bin. Zurni Omar (006): 3 point implicit block method for solviig (irllii'"iry' differential equation. Bulletin of the Malaysian mathematical science (). 9( I). pp. :-.\ J.... Journal of the Nigerian Association of Mathematical Physics Volume 5 (November, 0()9), 7-7!i

Modified block method for the direct solution of second order ordinary differential equations

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(3),pp 181 188, 2011 http://ijamc.psit.in Modified block method for the direct solution of second order ordinary