Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2943-2948 2943 www.etasr.com Oghonyon et al.: Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations J. G. Oghonyon Department of Mathematics Covenant University Ota, Nigeria godwin.oghonyon@covenantuniversity.edu.ng S. A. Okunuga Department of Mathematics University of Lagos Akoka-Lagos, Nigeria nugabola@yahoo.com K. S. Eke Department of Mathematics, Covenant University Ota, Nigeria kanayo.eke@covenantuniversity.edu.ng O. A. Odetunmibi Department of Mathematics Covenant University Ota, Nigeria wole.odetunmibi@covenantuniversity.edu.ng Abstract—Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor- corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the chief local truncation error] (CLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels. Keywords-Milne’s device; predictor-corrector method; suitable step size; convergence criteria; maximum errors; chief local truncation error] I. INTRODUCTION The extension of predictor-corrector method is important for providing some computational benefits to numerical integration of ordinary differential equations. These computational vantages are enlisted in [1, 2]. This composition is primarily concerned with presenting approximate solution of fourth order ODEs of the form [1-3]: 0 1 2 3 '''' ( , , ', '', '''), ( ) , '( ) , ''( ) , '''( ) y f x y y y y y a y a y a y a          for ≤ ≤ and : × → (1) The numerical solution to (1) is broadly defined as ∑ = ℎ ∑ (2) where the step size is h, = 1, , = 0,1,…, , are unknown quantities which are unequivocally fixed in a way that the expression is of order m as seen in [3, 4]. It is accepted that Rf  is sufficiently differentiable to a certain degree on interval ],[ bax  and meets applicable Lipchitz condition, i.e., thither is a unvarying ≥ 0 such that | ( , ) − ( , | ≤ | − |, ∀ , ∈ . Beneath the given assumption, (1) ascertained existence and quality of singularity outlined on bxa  , likewise looked at to satisfy the Weierstrass theorem [5, 6]. [ , ,…, ] , = [ , , . . . , ] and = [ , , . . . , ], spring up in real world practical applications for state of difficulty in scientific research and applied science such as fluid dynamics and rocket motion [7]. It has been proposed that the established method of solving (1) is by reduction to first-order ODEs. This technique of simplification possesses real severe hindrance which includes waste of human exertion, complication in the use of Mathematica package/computer software for coding and squandering implementation time [2-3, 8-15]. Scholars have developed straight methods for approximating (1) with better results and efficiency. These methods include block method, block predictor-corrector method, block implicit method, block hybrid method, BDF etc.. However, there are advantages and disadvantages for implementing them. Block predictor- corrector method was hinted in [2-3, 10, 12-15]. The backward differentiation formula (BDF) is Gear’s method recognized for Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2943-2948 2944 www.etasr.com Oghonyon et al.: Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations stiff ODEs as seen in [7, 16-19]. The target is to employ Milne’s device to develop a variable step-size block predictor- corrector method [1, 6, 20, 21]. Milne’s device is seen as the numerical estimation of initial value problems (IVPs) and this technic is based on numerous ingredients [2, 17]. These include convergence criteria, predictor-corrector pattern braces of like order, designing suitable step-size and chief local truncation error. In addition, this subject field possesses a great deal of superior qualities as discoursed in [1-3, 6, 20, 21] Definition: − , − ℎ . If r denotes the block size and h is the stepsize, then block size in time is ℎ. Let = 0,1,2,… represent the block number and = , then the − , − method can be written in the following general form: = ∑ + ℎ∑ (3) where = [ ,…, ,…, ] = [ ,…, ,…, ] and are × coefficient matrices [16]. Therefore, from the above definition, a block method has the computational reward that in each practical application, the output is valuated to a greater extent or at more than one point concurrently. The number of points depends on the construction of the block method. Hence, utilizing this method can provide speedier solutions to the problem which can be processed to bring forth the sought-after accuracy [2, 3, 7, 18, 19]. Thus, the motivation of this paper is to suggest Milne’s device of variable step-size block predictor-corrector method for solving non-stiff and mildly stiff ODEs implemented in P(EC)m or P(EC)mE mode. This technique possesses the advantages like designing a suitable step size/changing the step size, specifying the convergence criteria (tolerance level) and error control/minimization likewise addressing the gaps posited above. II. MATERIALS AND METHODS To showcase this process, a variable step-size block predictor-corrector method enforcing the explicit Adams- Bashforth b-step method as a predictor and the implicit Adams- Moulton b-1-step method as a corrector of the same order is prepared [1-3, 21]. This discussion section expends the Newton’s backward difference formula to prepare the block predictor-corrector method. Presuppose f(x) has a continuous ℎ derivative, xn=x0+nh, fn=f(xn), and backward differences are interpreted as ∇ = ∇ − ∇ where ∇ = , then ( ) = + ∇ + ( − )( − − 1) ∇! ++( − )( − − 2)∇! …+ ( − )… −∇( )! + ( − )… − ( )( )! (4) where ( )( ) is the ℎ derivative of measured at some approximate time interval owning , , and . Setting = ( ) and = − 1, (4) turns over as 1 1 1 1 ( ) 1 ( ) ... 0 1 ( 1) ( 1) ( ) 1 m m q q q q q m r r f x f f r r f h f q q                                       where = ( )…( )! and 0 = 0 Interchanging the above in ( ) = ( ) + ( ) , to acquire ( ) = ( ) + ∑ (−1) − ∇ +(−1) ℎ − ( )( ) 1 1 1 0 1 1 ( 1) 0 ( ) ( ) ( 1) ( ) q i m i m i q q y x y x h f r y ds q                      (5) If the terminal figure in (5) is dropped, the unexpended will be addressed with − Adams-Bashforth formula = + ℎ∑ ∇ (6) Establishing the backward differences in terms of the valuates at upholding points as ∇ = (−1) Equation (6) can be written as = + ℎ∑ (7) where = (−1) ∑ − 1 Equation (7) shows in terms of info at , ,…, . Proceeding with (7) produces the block b-step Adams-Bashforth Formula [22]. Likewise, the implicit multistep method-the Adams-Moulton method can be derived by putting = in (4) and subbing into ( ) = ( ) + ( ) , to arrive at ( ) = ( ) + ∑ (−1) − + 1 ∇ +(−1) ℎ − + 1 ( )( ) Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2943-2948 2945 www.etasr.com Oghonyon et al.: Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations Dropping the error term, succumbs the method as = + ℎ∑ ∗∇ (8) Interchanging for ∇ in terms of , , ,…, sets up the form = + ℎ∑ ∗ (9) where ∗ = (−1) ℎ∑ ∗. In good continuation with (9), the block b-1-step Adams- Moulton Method will be brought forth [22]. A. Examining Some Selected Theoretical Properties of Block Method Embracing [5, 6, 9, 10, 20], the consociated block methods (2) and the difference operator can be set as 0 4 0 ( ( ); ] ( ) ''''( ) k i i k i L y x h a y x ih h y x ih         (10) Accepting that y(x) is sufficiently and continuously differentiable on an time interval [a, b] and that y(x) possesses many higher derivatives as needed, then, penning the terms in (10) as a Taylor series formulation of ( ) and ( ) ≡( ) as ( ) = ∑ ( )! ( )( ) and ( ) = ∑ ( )! ( )( ) (11) Replacing (10) and (11) into (2), the next formulation is incurred as [ ( );ℎ] = ( ) + ( ) + ⋯+ ℎ ( )( ) +…+ ℎ ( )( ) + ⋯ (12) Holding onto [5, 6], the block method of (2) maintains order p, if , = 0,1,2,…, = 1,2,… , are presented as: , = 0,1,2,…, = 1,2,… , are given as follows: = + + + ⋯+ , = + 2 + + ⋯+ , = ! ( + + + ⋯+ ) − ( + + + ⋯+), = ! ( + 2 + ⋯+ ) − ( )! ( + 2 +⋯+ ), = 5,4,… So, the block method (2) is stated to sustain order p≥1 and error constants quantity established by the vector, ≠ 0. Agreeing with [6, 20], the block method (2) possesses order p if [ ( );ℎ] = 0ℎ , = = = ⋯ = == = 0, ≠ 0. (13) Consequently, is the error constant quantity and ℎ ( )( ) is the chief local truncation error at the point . B. Stability Analysis For properly analyzing the block method for stability, the block predictor-corrector method is renormalized and scripted as a block method displayed by the matrix finite difference equations as seen in [5, 23]. ( ) = ( ) + ℎ ( ( ) + ( ) ), (14) where = ... , = ... = ... , = ... . The matrices ( ), ( ), ( ( ), ( ) are r by r matrices while , , , are r-vectors outlined above. In accordance with [6, 23], the boundary locus method is employed to ascertain the region of absolute stability of the block method in preparation to find the roots of absolute stability. Thus, subbing the test equation = − and ℎ=ℎ into the block method (14) to arrive at (0) ( 0) 4 4 (1) (1) 4 4 ( ) det[ ( ) ( )] 0 r r A B h A B h         (15) Replacing h=0 in (15), the roots of the inferred equation will either be less than or equal to 1. Therefore, definition [6] of absolutely stability is gratified. In addition, by [16], the boundary of the region of absolute stability can be gotten by stepping in (2) into )( )( )( r r rh    (16) and where cos sinir e i     then after simplification together with valuating (16) within [0o, 180o] the boundary of the region of absolute stability rests on the real axis. Adams- Moulton Method will be brought forth [22]. C. Milne’s Implementation on Block Predictor-Corrector Method Consorting with [1, 6, 20-21], the Milne’s implementation in the P(EC)m or P(EC)mΕ mode turns indispensable for the explicit (predictor) and implicit (corrector) methods if both are discrete and of the same order, and this touchstones make important for the step number of the explicit method to be one step higher than that of the implicit method. Accordingly, the mode P(EC)m or P(EC)mΕ can be formally analyzed as succeeds for μ=1,2,..... P(EC)m: Engineering, Technology & Applied Science Research Vol. 8, No. 3, 2018, 2943-2948 2946 www.etasr.com Oghonyon et al.: Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations fhyy in k i iin k i ikn ]1-[ 1- 0 4][ 1- 0 ]0[ ∑∑           , [ ] [ ] 1 1 [ 1] [ ] 4 [ ] 4 [ 1] 0 0 ( , ) , 0,1,.... 1 u u n k n k n k k k u u n k i n i j n k i n i i i f f x y y a y h f h f u                          (17) P(EC)mΕ: fhyy in k i iin k i ikn ][ 1- 0 4][ 1- 0 ]0[ ∑∑           , [ ] [ ] 1 1 [ 1] [ ] 4 [ ] 4 [ 1] 0 0 ( , ) , 0,1,.... 1 u u n k n k n k k k u u n k i n i j n k i n i i i f f x y y a y h f h f u                          [ ] [ ]( , )n j n k n kf f x y      Mentioning that as m   , the result of approximating with either modality will pitch to those applied by the mode of correcting to convergence. Moreover, predictor and corrector pair based on (2) can be implemented. P(EC)m or P(EC)mΕ modes are specified by (16), where h is the step size. Since the predictor and corrector both have the same order p. *4 4,p pC C  are the error constant quantities of predictor and corrector respectively. The following effects are true: D. Proposition Let us assume that the predictor method gives order p* and the corrector method gives order p engaged in P(EC)m or P(EC)mΕ mode, where p *, p, μ are integers and p*≥1, p≥1, μ≥1. Then, when p*≥p (or p*

p-p*), then the predictor- corrector method gives the same order and the same PLTE as the corrector. If p*