Programming codes of block-Milne's device for solving fourth-order ODEs

Block-Milne’s device is an extension of block-predictor-corrector method and specifically developed to design a worthy step size, resolve the convergence criteria and maximize error. In this study, programming codes of block-Milne’s device (P-CB-MD) for solving fourth order ODEs are considered. Collocation and interpolation with power series as the basic solution are used to devise P-CB-MD. Analysing the P-CB-MD will give rise to the principal local truncation error (PLTE) after determining the order. The P-CB-MD for solving fourth order ODEs is written using Mathematica which can be utilized to evaluate and produce the mathematical results. The P-CB-MD is very useful to demonstrate speed, efficiency and accuracy compare to manual computation applied. Some selected problems were solved and compared with existing methods. This was made realizable with the support of the named computational benefits.

Scholars suggested the established method to work out par (1) by reducing to first-order ODEs. This idea of simplifying to a system of first order ODEs have very strong encumbrance which admits waste of human effort, difficulty in programming/coding and consuming implementation time. Bookmen formulated straight forward method for estimating (1) with improve efficiency and accuracy. Such techniques consist of block method, parallel processing predictorcorrector technique, block implicit method, block hybrid method, Backward Differentiation Formula (BFD) and so on. Nevertheless, each has their benefits and shortcomings for executing them. Interested readers are invited to read Adesanya et al. (2012), Anake and Adoghe (2013), , Awoyemi et al. (2014Awoyemi et al. ( , 2015, Kayode (2008), Kayode et al. (2014), Oghonyon et al. (2015Oghonyon et al. ( ,2016, Olabode (2009), and Olabode and Alabi (2013) for more information.
Definition: z-parallel processing-r point method. Assume r denotes the block size and h is the stepsize, then parallel processing size in time is ℎ. Let = 0,1,2, … describe the parallel processing number and let = , then the − , − technique can be spelt in the succeeding ecumenical category: ( and are × coefficients matrices (Ibrahim et al., 2007).
Thus, from the definition supra, a block method has the numerical benefit that for each virtual application, the result is evaluated to a greater extent or at more than one point simultaneously. The total number of points relies on the formulation of the block method. Therefore, applying these methods can supply faster and more flying results to the problem which can be examined to produce the sought after accuracy Suleiman, 2007, 2008;Mehrkanoon et al., 2010;Mohammed and Tech, 2010;Oghonyon et al., 2015;2016). Thence, the need of this composition is to propose programming codes of block-Milne's device that aid the implementation of fourth order ODEs as well assist to realize the vantages of block-Milne's device like designing a suitable step size, determining the convergence criteria and error control.
The residual of this paper is examined as follows: in Section 2 programming codes of the materials and methods. Section 3 programming codes for implementing block Milnes' device. Section 4 Conclusion as cited (Akinfenwa et al., 2013;Oghonyon et al., 2016).

Programming Codes of the materials and methods
Under this discussion section, the primary goal to be achieved is to formulate the programming codes of block-Milne's device. Block-Milne's device is a combination of the − (predictor) method and − 1 − (corrector) technique of the same order. A combination can be of the form Pars (4) and (5) forms the class of block-Milne's device with , = 0, 1, 2,3,4. Stating that + is the numerical estimate to the analytical results ( + ) i.e. ( + ) ≈ + , and ( + , + ) ≈ + having = 0, 1, 2,3,4. To get par (4) and (5), the basis function approximation is interpolated and collocated at selected intervals. This turns out to become a system of linear equation i.e. Au=b.
Expanding (6) gives birth to the basis function approximation which can presented in programming codes as where 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 and 8 are unknown constants which is required to be determine in a peculiar way. Assume the condition that method (6) agrees with the analytical result at the time interval , − to become the approximate Expecting that the estimating function (7) satisfies problem (1) at the points + , = 0, 1, 2 to formulate the following estimates as ( + ) ≈ + ], = 0, 1, 2,3,4.
Simplifying the expression of (16) and (17)  represents the different the principal local truncation errors. 1 , 2 and 3 are the bounds of the convergence criteria of block-Milne's device. Still, the estimates of the principal local truncation error (18) are employed to determine whether to accept the results of the current step or to repeat the step with a smaller variable step-size. This process is veritably satisfactory on try-out expressed in (18) as quoted in (18) (Ascher and Petzold, 1998;Dormand, 1996;Faires and Burden, 2012;Lambert, 1973;Lambert, 1991;Oghonyon et al., 2015;Oghonyon et al., 2016). The principal local truncation errors par (18) is called the convergence criteria of block-Milne, differently referred to as block-Milne's device (estimate) for adjusting to convergence.
Where +4 * and +4 are the computes of the principal local truncation error of the predictor and corrector method. ̅ + and ̅ + are called the predicted and corrected approximations estimations permitted by the method of order p. Table 1 and Table 2 displays the computational results for calculating the examined problems in the previous section applying P-CB-MD.
Step by step algorithm: A framework of step by step algorithm to design afresh h and valuing maximum errors of P-CB-MD, if the mode is run times, if the mode is run m times.
 Item #1: Prime h.  Item #2: Order of the multiprocessing predictorcorrector approach must be equal.  Item #3: Step number of multiprocessing predictor approach must be unit step more eminent compare to multiprocessing corrector approach.  Item #4: Posit main local truncation errors of both predictor-corrector approach.  Item #5: Set convergence limit.  Item #6: Insert multiprocessing predictorcorrector approach in whatever mathematical programming-language.  Item #7: Employ Taylor's series method to bring forth starting-out economic measures if called for, else disregard item #7 and advance to item #8.  Item #8: Execute P-CB-MD together with the main local truncation errors.  Item #9: If item 8# diverges, reiterate procedure once more and split up h into 2 parts from item #1 or else, continue item #10.  Item 10#: Valuing maximum calculated errors only when convergence has been attained.  Item #11: Write maximum calculated errors.  Item #12: Utilize formula expressed infra to formulate a new step size since converge is reached.

Conclusion
The completed results displayed on Table 1 and Table 2 declared that the P-CB-MD is attained with the support of the convergence limit and a suited/changing step size. Nevertheless, these factors introduce an alternative to decide either to accept or reject result. This terminal result also demonstrate the functioning of the P-CB-MD were discovered to obtain an improve maximum errors than AAFO-SPIBM, AS-SCMM, DBP-CMS, DSIVP, NS-SNM, SSBM, S-SS at all convergence criteria as cited in Awoyemi et al. (2015), Duromola (2016) Thusly, it will be concluded that the P-CB-MD formulated is worthy for solving fourth order ODEs.