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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com


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  • 1. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2297-2301 Three Steps Block Predictor-Block Corrector Method For The Solution Of General Second Order Ordinary Differential Equations *Adesanya, A. Olaide, Odekunle, **M. Remileku, ***Alkali, M. Adamu *Department of Mathematics, Modibbo Adama University of Technology, Yola, Adamawa State, Nigeria **Department of Mathematics, Modibbo Adama University of Technology,Yola, Adamawa State, Nigeria ***Department of Mathematics, Modibbo Adama University of Technology,Yola, Adamawa State, NigeriaAbstract: We consider three steps numerical series to mention few. The major disadvantage of thisintegrator which is derived by collocation and method is that the predictors are reducing orderinterpolation of power series approximation predictors hence it has a lot of effect on the accuracysolution to generate a linear multistep method of the method. Among authors that proposedwhich serves as the corrector. The corrector is predictor corrector method are [6, 7, 12]. Otherthen implemented in block method. We then setbacks of this method are extensively discussed bypropose another numerical integrator which is [7].implemented in block method to serve as a In order to cater for the setbacks of theconstant order predictor to the corrector. Basic predictor-corrector method, scholars adopted blockproperties of the corrector viz, order, zero method for direct solution of higher order ordinarystability and stability interval are investigated. differential equation. This method has the propertiesThe new method was tested on some numerical of being self starting and gives evaluation at selectedexamples and was found to give better grid points without over lapping. It does not requireapproximation than the existing methods. developing predictors or starting values moreover it evaluates fewer functions per step. The major setbackKeyword: Collocation, Interpolation, of this method is that the interpolation points cannotapproximate solution, block method, order, zero exceed the order of the differential equation, hencestability. this method does not exhaust all interpolation pointsAMS Subject classification: 65L05, 65L06, 65D30. therefore methods of lower order are developed. Among these authors are [1, 7, 9, 10, 15, 13, 14, 15,1 Introduction 16]. Scholars later adopted block predictor-block This paper considers method of solving corrector method to cater for some of the setbacks ofgeneral second order initial value problems of block method. They proposed a method which isordinary differential equation of the form capable of exhausting all the possible interpolation point as the corrector and then proposed a constanty  f ( x, y, y ), y k ( x0 )  y0 , k  0,1 k order predictor using block method. The method forms a bridge between the predictor-corrector(1) method and the block method. Among these authorsWhere f is continuous within the interval of are [2, 3, 4]. The major setback of this method is thatintegration. the evaluation at selected points are overlapping Method of reduction of higher order hence the behaviour of the dynamical system atordinary differential equation to systems of first order selected grid points cannot be mentioned.ordinary differential equation has been discussed by In this paper, we proposed a method inmany scholars. These authors suggested that the which the corrector is implemented in block method;method increased the dimension of the resulting hence this method gives evaluation at selected gridsystem of the first order; hence it wastes computer points without overlapping.and human effort. Among such authors are [5, 9, 11].Scholars later adopted predictor–corrector method to 2 Methodologysolve higher order ordinary differential equation 2.1 Development of the block corrector|directly. These authors proposed implicit linear We consider a power series approximate solution ofmultistep methods which serve as the corrector, the the form r  s 1 apredictors proposed to implement this corrector are inreducing order of accuracy. These methods are y ( x)  j xj j 0developed by adopting different approximate solutionsuch as Power series, Chebychev polynomial,Newton polynomial, Lagrange polynomial, Fourier (2) Where 𝑟 and 𝑠 are the number of collocation and interpolation points respectively. 2297 | P a g e
  • 2. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2297-2301 2t  18t 5  55t 4  60t 3  27t  6 The second derivative of (2) gives 1 6 r  s 1 0   y   j ( j  1)a j x j 2 6 j 2 1  3 2t  18t 5  55t 4  60t 3  24t  1 6 (3)  2   2t 6  18t 5  55t 4  60t 3  21t  Substituting (3) into (1) gives 1 r  s 1 6 f ( x, y , y )   j ( j  1)a x j 2 10t 6  93t 5  305t 4  410t 3  180t 2  8t  j 1 j 2 0  360 100t 6  89t 5  2675t 4  2820t 3  936t  (4) 1 Collocating (3) at x n  r , r  0(1)3 and interpolating 1  360 (2) at x n  s , s  0(1)2 gives a system of non linear equation of the form 2  1 360 10t 6  99t 5  335t 4  390t 3  144t  AU  B 3  1 360 3t 5  15t 4  20t 3  8t  (5) 1 xn xn2 xn3 xn 4 xn5 xn  6   x  xn 0 xn1 2 3 xn1 xn1 4 xn1 54 xn1 6 xn1  t , y n  j  y ( xn  jh ) , h 0 x n  2 2 3 xn 2 xn 2 4 xn 2 5 xn 2 xn 2  6  f n j  f ( xn , y( xn  jh), y ( xn  jh)) 4 U  0 0 2 6 xn 12 xn 2 20 xn 3 30 xn  Evaluating (6) at t  4 gives 0 0 2 6 xn1 12 xn1 20 xn1 30 xn1  2 3 4   h2 y n3  3 y n 2  3 y n1  y n   f n3  9 f n 2  9 f n1  f n  0 0 2 6 xn 2 12 xn 2 20 xn 2 30 xn 2  2 3 4 12 0 0 2 6 xn3 12 xn3 20 xn3 30 xn3  2 3 4 (7)   Evaluating the first derivative of (6) at t  0,1 gives a0   yn  a  y  2 h2 y n 2  8 y n1  y n   f n  117 f n1  18 f n2  f n3  9  1  n 1  hyn  3 2 45 a 2   y n2  (8)    , A   a3  , B   f n   17 2 yn2  14 yn1  yn  47 f n  679 f n1  121 f n2  7 f n3  11 h hyn 1  a 4   f n 1  6 6 360     (9)  a5   f n2  Writing (7)-(9) in block form and solving for the a  f   6  n 3  independent solution at selected grid points gives Solving (5) for a j s , using Gaussian A ( 0 )Ym  ey n  cy n  df ( y n )  bF (Ym ) elimination method and substituting back into the (10) approximate solution, gives a continuous linear Ym  yn1 , yn2 , yn3  T multistep method. Where yn  yn1 , yn2 , yn  2 3 y ( x)    j ( x) y n j  h 2   j ( x) f n j T j 0 j 0 (6)  yn  yn1 , yn , yn1 T f ( yn )   f n1 , f n2 , f n  T Where the coefficient of y n  j and f n  j gives F (Ym )   f n1 , f n2 , f n3  T 2298 | P a g e
  • 3. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2297-2301 1 0 0 0 0 1 We then proposed a prediction equation of the form A ( 0 )  0 1 0  , m   e  0 0 1 ,   Ym  ey n   h    F  ( y n ),  0 0 0 1    0 0 1   (12)  17 21  Where 2 is the order of the differential 0 38 38  equation. In this paper, m  2.  14 24  d  c  0 , F  ( yn )   f ( x, y, y ) yn Substituting (10)  19 19  dx 0 33 81  into (11) gives  38 38     2  Ym  eyn  h  dF ( yn )  bh  F  eyn   h   f  f ( y)   851    0  0 0 13680  (13)  127  Writing (10) as d  0 0 ,  855  Ym  e * y n  chyn  d * h 2 f ( y n )  bF (Ym ) * * 0 0 267    1520   (14) Substituting (13) in to (14) gives   1653 93  11  Ym  e * y n  chyn  d * h 2 f ( y n )  bF (Ym ) *  13680 13680 13680  399 111 7  b  (15)  855 855 855  Equation (15) is our block predictor-block  1539 1701 93  corrector method.  1520  1520 1520   4.0 Analysis of basic properties of the method 4.1 Order of the method 2.2 Development of Block predictor We define a linear operator on the block (10) to give [7] developed 3 steps block method of the form (10) ℒ 𝑦 𝑥 :ℎ = which is writing explicitly to give h2 y ( x)  A ( 0 )Ym  ey n  chyn  df ( y n )  bF (Ym ) y n1  y n  hy  n 97 f n  114 f n1  39 f n2  8 f n3  (16) 360 Expanding (16) in Taylor series gives h2 ℒ 𝑦 𝑥 : ℎ = 𝑐0 𝑦 𝑥 + 𝑐1 ℎ𝑦 ′ 𝑥 + 𝑐2 ℎ2 𝑦 ′′ 𝑥 +y n 2  y n  2hyn  28 f n  66 f n1  6 f n 2  2 f n3  ⋯ + 𝑐 𝑝 ℎ 𝑏 (𝑥) 45 Definition: the block (10) and associated linear h2y n3  y n  3hyn  39 f n  108 f n1  27 f n 2  6 f n3  operator are said to have order 𝑝 if 40 c0  c1  c2  ...  c p 1  0 and c p  2  0 . They n1  y n  9 f n  19 f n1  5 f n 2  f n3  h term c p  2 is called the error constant and implies that 24 the local truncation error is ( p  2) p2 p3y n 2  y n   f n  4 f n1  f n 2  h t n k  c p  2 h y xn  O(h ) 3 Hence the block (10) has order five with error  y n3  y n  3 f n  9 f n 1  9 f n 2  3 f n3  h 97 41 9 constant  , , 8  383040 23940 42560   For the derivation and analysis of the block predictor, 4.2 Zero Stability readers are referred to [7]. A block method is said to be zero stable if as h  0 , 2.3 Implementation of the Method the root r j ; j  1(1)k of the first characteristic Given the general block formula proposed by [7] in polynomial  ( R)  0 that is  ( R)  det A R  0 the form k 1 satisfying R  1 ( 0) Ym  ey n  h  df ( y n )  bh  F (Ym ) (11) 2299 | P a g e
  • 4. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2297-2301and for those roots with R  1 must have 11) 08) 1.10 3.74multiplicity equal to unity {see [9] for details} 0. 1.42364593 1.42364893 13(- 47(- 8 0193602 0303739 1 0 0 0 0 1 10) 08)    Hence R 0 1 0  0 0 1  0 0. 1.48470027 1.48470027 1.88 5.95     9 8594052 8782232 18(- 07(- 0 0 1 0 0 1     10) 08) 3.22 9.29R 3  R 2  0 , R 2 R  1  0 R  0,0,1 1. 1.84930614 1.54930614 11(- 40(- 0 4334055 4656174 10) 08)5.0 Numerical ExperimentsWe test our method with second order initial value Table II for problem IIproblems.Problem 1: Consider a non linear initial valueproblem 1y  x( y )  0, y(0)  1 , y (0)  , h  0.01 2 1 2 xExact solution y ( x)  1  log e   2 2 xThis problem was solved by [1] where a block of 0.1order four was proposed for h  and [7] where 32a 3 steps block method was proposed for the samestep size. The result is shown in table IProblem II: We consider a highly oscillatory testproblemy 2 y  0 , where   2 , with initial conditiony(0)  1, y (0)  2, h  0.01Exact solution; y( x)  cos2 x  sin 2 xWe compare our result with [7] where a blockmethod with step length of three was proposed Conclusion:Table I for problem I We have proposed a 3 steps block predictor- Erro block corrector method in this paper. This method Computed Erro x Exact result r in has addressed some of the setbacks of block predictor result r [1] corrector method proposed by [2, 3, 4]. The result has 1.33 4.98 shown clearly that our method gives better 0. 1.05004172 1.05004172 00(- 27(- approximation than the existing method. 1 9278491 9278624 13) 11) 9.94 4.10 References 0. 1.10033534 1.10033534 98(- 43(- [1] Anake, T. A., Awoyemi, D. O., Adesanya, 2 7710756 7732074 13) 10) A. O. (2012): A one step method for the 3.43 1.42 solution of general second order ordinary 0. 1.15114043 1.51140435 96(- 85(- differential equation, Intern. J. of Science 3 5936466 9399064 12) 09) and Tech. 2(4), 159-163 8.68 3.52 [2] Adesanya, A. O., Odekunle, M. R and 0. 1.20273255 1.20273255 03(- 42(- Alkali, M. A., (2012): Order six block 4 0540823 4062762 12) 09) predictor corrector for the solution of 0. 1.25541281 1.25541281 1.83 7.24 y  f ( x, y, y ) , Canadian J. on Science 73(- 35(- 5 1882995 1901369 and Engineering Mathematics, 3(4), 180-185 11) 09) 3.50 1.33 [3] Adesanya, A. O., Odekunle, M. R., James, 0. 1.30951960 1.30951960 A. A. (2012): Starting Hybrid Stomer- 87(- 35(- 6 4203111 4338198 Cowell more accurately by hybrid Adams 11) 08) 0. 1.36544375 1.36544375 6.33 2.28 method for the solution of first order 7 4271396 4334709 13(- 72(- ordinary differential equation, European J. of Scientific Research, 77(4), 580-588 2300 | P a g e
  • 5. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2297-2301[4] Adesanya, A. O., Odekunle, M. R., Numer. Algor. DOI 10.1007/S11075-010- Adeyeye, A. O. (2012): Continuous block 9413-X hybrid predictor corrector method for the solution of y  f ( x, y, y ) , Intern. J. of Mathematics and Soft Computing (In press)[5] Adesanya, A. O., Anake, T. A and Udoh, M. O. (2008): Improved continuous method for direct solution of general second order ordinary differential equation. J. of Nigerian Association of Mathematical Physics, 13, 59-62[6] Adesanya, A. O., Anake, T. A. and Oghonyon, G. J. (2009): Continuous implicit method for the solution of general second order ordinary differential equation, J. of Nig. Assoc. of Math Phys., 15, 71-78[7] Awoyemi, D. O., Adebile, E. A., Adesanya, A. O, and Anake, T. A. (2011): Modified block method for the direct solution of second order ordinary differential equation, International Journal of applied Mathematics and computation 3(3), 181-188[8] Awoyemi, D. O (2001): A new sixth order algorithms for the general second order ordinary differential equation, Intern. J. Comp. Math, 77, 117-124[9] Jator, S. N (2007): A sixth order linear multistep method for the direct solution of y  f ( x, y, y ) Intern. J. pure and applied Mathematics, 40(1), 457-472[10] Jator, S. N and Li, J (2009): A self starting linear multistep method for a direct solution of the general second order initial value problem, Intern. J. Comp. Math, 86(5), 817- 836[11] Kayode, S. J and Awoyemi, D. O. (2010): A multi-derivative collocation method for fifth order ordinary differential equation. J of Mathematics and Statistics, 6(1), 60-63[12] Kayode, S. J. and Adeyeye, A. O. (2011): A 3-steps hybrid method for the direct solution of second order initial value problems, Australian J. of Basic and Applied Sciences, 5(12), 2121-2126[13] Majid, Z. A., Suleiman, M. B and Omar, Z. (2006): 3 points implicit block method for solving ordinary differential equation, Bull Malays Math. Sci. Soc. 29(1), 23-31[14] Omar, Z and Suleiman M. (2003): Parallel R-point implicit block method for solving higher order ordinary differential equation directly, Journal of I.C.T, 3(1), 53-66[15] Salman Abbas (2006): Derivatives of a new block method similar to the block trapezoidal rule for the numerical solution of first order IVP’s Science Echoes, 2, 10-24[16] Siamak Mehrkanoon (2010): A direct variable step block multistep method for solving for general third order ODE’s. J. 2301 | P a g e