 American Journal of Computational Mathematics, 2012, 2, 341-344 http://dx.doi.org/10.4236/ajcm.2012.24047 Published Online December 2012 (http://www.SciRP.org/journal/ajcm) A New Block-Predictor Corrector Algorithm for the Solution of ,, , yfxyy y Adetola O. Adesanya1, Mfon O. Udo2, Adam M. Alkali1 1Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria 2Department of Mathematics and Statistics, Cross River University of Technology, Calabar, Nigeria Email: mfudo4sure@yahoo.com, torlar10@yahoo.com Received May 10, 2012; revised September 16, 2012; accepted September 25, 2012 ABSTRACT We consider direct solution to third order ordinary differential equations in this paper. Method of collection and inter- polation of the power series approximant of single variable is considered to derive a linear multistep method (LMM) with continuous coefficient. Block method was later adopted to generate the independent solution at selected grid points. The properties of the block viz: order, zero stability and stability region are investigated. Our method was tested on third order ordinary differential equation and found to give better result when compared with existing methods. Keywords: Collection; Interpolation; Power Series; Approximant; Linear Multistep; Continuous Coefficient; Block Method 1. Introduction This paper considers the general third order initial value problems of the form  0000 00,, ,,,,.yfxyyyyx yyx yyx y   (1) Conventionally, higher order ordinary differential equ- ations are solved directly by the predictor-corrector me- thod where separate predictors are developed to im- plement the correct and Taylor series expansion adopted to provide the starting values. Predictor-corrector me- thods are extensively studied by [1-5]. These authors proposed linear multistep methods with continuous co- efficient, which have advantage of evaluation at all points within the grid over the proposed method in  The major setbacks of predictor-corrector method are extensively discussed by . Lately, many authors have adopted block method to solve ordinary differential equations because it addresses some of the setbacks of predictor-corrector method discussed by . Among these authors are [8-10]. According to , the general block formula is given by .mn nmyhyhy Y edfbF (2) where is ess vector, is r-vector and is vector, d brrs is the interpolation points and is the collection points. rF is a k-vector whose entry is thj,,njnj njffty  is the order of the differential equation. Given a predictor equation in the form 0mn nyh ydf. (3) YePutting (3) in (2) gives .nn ny hyhy  bFe dfmnyhYe df (4) Equation (4) is called a self starting block-predictor- corrector method because the prediction equation is gotten directly from the block formula as claimed by [11,12]. In this paper, we propose an order six block method with step length of four using the method proposed by  for the solution of third order ordinary differential equation. 2. Methodology 2.1. Derivation of the Continuous Coefficient We consider monomial power series as our basis fun- ction in the form 10.srjjjyxax. (5) The third derivative of (5) gives yx 13312jjjjj axsrj  (6) Copyright © 2012 SciRes. AJCM A. O. ADESANYA ET AL. 342 The solution to (1) is soughted on the partition πN: 012 1nnN within the integration interval [a,b] with constant step length given as axx xxxxb,01.ni nhx xiN hSubstituting (6) into (1) gives  13=3,, ,12sr jjjfxyxyx y xjjj ax .;, (7) Interpolating (5) at collocating (7) gives a system of equations ,113nixj1sr,014nixj0.jjnsjax y (8)  13312sr j.jnrjjjj axf   (9) Solving (8) and (9) for ja’s and substituting back into (5) gives a LMM with continuous coefficients of the form  343110.ttjnjjjyh fnj (10) where ja’s and j’s are given as 2212231=;=21=32;2ttttttttt76 542012 71135 49 26;10080t tttttt 7654 211=421141012652 ;5040tttttt t 765 42212147105532432 ;1680ttttttt 3765 43215040435701758401078452 ;tttt tttt   765 4 341221 77 105105 58.10080ttttttt where 3.nxxth 2.2. Derivation of the Block Method The general block formula proposed by  in the nor- malized form is given by  0.mn nmyhy hy  AY edfbF (11) 2; Evaluating (10) at 1,0,4nxx i;,014;nixi the first and second derivative at and substituting into (10) gives the coefficient of (11) as T113331 14312483675314756251292514,1120 63011201051440 90160 45 720 90 80 45d T111100000000123411110000 ,912 81234111122e 012 12A identity matrix, 107332 1863 2176381176423262516410083155603158540153604540 451038243 324727169811141680215601052403 801530 1510 15435245128 398364 5364221168 31511210536045 84536045404547 198187110080 63011206348030b T.919711030 160720 12045          4 3. Analysis of the Properties of the Block 3.1. Order of the Method We define a linear operator on the block (11) to give :mm mmyx hyhyhy  Ye dfbF(12) Expanding nyxih and nfxjh in Taylor series , (12) gives  0122:pppyxhCyxChyxChyxCh yx (13) The block (11) and associated linear operator are said to have order if p011 20, 0ppCCC C . Copyright © 2012 SciRes. AJCM A. O. ADESANYA ET AL. 343The term 2p is called the error constant and implies that the local truncation error for the block is given by Cnk22 320pp pp ntChyxh  (14) Hence the block (11) has order 6, with error constant 620h21340224,zz91243 321078320 45 448031510080 315916313 8315 16090 160945pCNE 3.2. Zero Stability of the Block The block (11) is said to be zero stable if the roots of the characteristic polynomial satisfies 1, 2,szdet,A1z and the root 1z has multiplicity not exceeding the order of the differential equation. moreover as hz 0,1 ,rz where  is the order of the differential equation, 0dimrAFor the block (11), 12,3r  391z Hence our method is zero stable. 3.3. Convergence A method is said to be convergent if it is zero stable and has order . 1pFrom the theorem above, our method is convergent. 4. Numerical Experiments 4.1. Test Problem We test our schemes with third order initial value pro- blems: Problem 1. Consider a special third order initial value problem 3sinyx 01, 0 0,0201yyy x Exact solution: 23cos22xyxx  This problem was solved by  using self-starting predictor-corrector method for special third order diffe- rential equations where a scheme of order six was pro- posed. Problem 2. Consider a linear third order initial value problem 0yy  01, 01,01,0,1yy yx   Exact solution: 21 cossinyxx x This problem was solved by  where a method of order six was proposed. They adopted predictor corrector method in their implementation. Our result is shown in Table 1. 4.2. Numerical Results The following notations are used in the table. XVAL: Value of the independent variable where nu- merical value is taken; ERC: Exact result at XVAL; NRC: Numerical result of the new result at XVAL; ERR: Magnitude of error of the new result at XVAL. 5. Discussion We have proposed a new block method for solving third order initial value problem in this paper. It should be noted that the method performs better when the step-size is chosen within the stability interval. The Tabl e s 1 and 2 had shown our new method is more efficient in terms of accuracy when compared with the self starting predictor Table 1. Showing result of pr oble m 1, h = 0.01. XVAL ERC NRC NRC ERR IN  0.1 0.990012495834077 0.990012495834077 0.0000+00 9.992007(–16) 0.2 0.960199733523725 0.960199733523724 9.99200(–16) 7.660538(–15) 0.3 0.911009467376818 0.911009467376816 1.55431(–15) 2.287059(–14) 0.4 0.843182982008655 0.843182982008652 3.10862(–15) 5.906386(–14) 0.5 0.757747685671118 0.757747685671113 4.66293(–15) 1.153521(–13) 0.6 0.656006844729035 0.656006844729028 6.88338(–15) 1.982858(–13) 0.7 0.539526561853465 0.539526561853456 9.10382(–15) 3.127498(–13) 0.8 0.410120128041496 0.410120128041484 1.14908(–14) 4.635736(–13) 0.9 0.269829904811992 0.269829904811978 1.42108(–14) 6.542544(–13) 1.0 0.120906917604418 0.120906917604401 1.74582(–14) 8.885253(–13) Copyright © 2012 SciRes. AJCM A. O. ADESANYA ET AL. 344 Table 2. Showing result of pr oble m 2, h = 0.1. XVAL ERC NRC ERR ERR IN  0.1 0.004987516654767 0.004987518195317 1.54055(–09) 1.189947(–11) 0.2 0.019801063624459 0.019801073469968 9.84550(–09) 3.042207(–09) 0.3 0.043999572204435 0.043999595857285 2.36528(–08) 7.779556(–08) 0.4 0.076867491997406 0.076867535270603 4.32732(–08) 7.749556(–07) 0.5 0.117443317649723 0.117443356667842 3.90181(–08) 3.398961(–06) 0.6 0.164557921035623 0.164557928005710 6.97008(–08) 9.501398(–06) 0.7 0.216881160706204 0.216881108673223 5.20329(–08) 1.756558(–06) 0.8 0.272974910431491 0.272974775207245 1.35224(–07) 2.745889(–05) 0.9 0.331350392754953 0.331349917920840 4.74834(–07) 3.888082(–05) 1.0 0.390527531852589 0.390526462491195 1.06936(–06) 5.137153(–05) corrector method proposed by  and . It should be noted that this method performs better when the step size (h) is within the stability interval. REFERENCES  A. O. Adesanya and T. A. 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