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Research Article

Softcode of Multi-Processing Milne’s Device for Estimating First-Order Ordinary Differential Equations

Jimevwo Godwin Oghonyon, Olaide Adetola Adesanya, Hudson. Akewe and Hilary Izuchukwu Okagbue
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Background and Objectives: Softcodes is a form of Mathematica language invented for the successful implementation of MPMD. Technical computing is an aspect of computing for the sole purpose of computation leading to better accuracy. This paper considers softcode of multi-processing Milne’s device for estimating first-order Ordinary Differential Equations (ODEs). Materials and Methods: Multi-Processing Milne’s Device (MPMD) is source from Adams collection of predicting-correcting scheme implemented via interpolation and collocation adopting multinomial finite sequence near resolution. This combination is mathematically assembled in MPMD pattern and analyzed to produce the order of the MPMD thereby setting up the chief local truncation errors. Results: The computational results generated were aided with Softcodes in Mathematica data format and setting the bounds of convergency. Conclusion: The calculated results are compared with subsisting methods to enhance the viability and effectiveness of the MPMD over others.

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Jimevwo Godwin Oghonyon, Olaide Adetola Adesanya, Hudson. Akewe and Hilary Izuchukwu Okagbue, 2018. Softcode of Multi-Processing Milne’s Device for Estimating First-Order Ordinary Differential Equations. Asian Journal of Scientific Research, 11: 553-559.

DOI: 10.3923/ajsr.2018.553.559

Received: April 10, 2018; Accepted: July 26, 2018; Published: September 15, 2018

Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.


Softcode for providing approximate results to Ordinary Differential Equations (ODEs) are very essential in technical computing, since it is greatly utilized to prototype real life applications1-4. Multi-processing Milne’s device for estimating first-order differential equation is of the form Abell and Braselton1, Ken et al.3, Bakoji et al.5 and Adejumo et al.6:


Arising from Eq. 1, there is a need to look for numerical solution enclosed on u∈[c, d] such that c and d are bounded with the assumption that z meets the considerations as seen in Akinfenwa et al.7, Anake et al.8, Anake and Adoghe9, Jain et al.10, Lambert11,12, Sunday et al.13 and Xie and Tian14. Thus, ensures that Eq. 1 possess a specific differential coefficient at every point.

The universal multi-processing Milne’s device is instituted as:


where, and are invariables implying that , Adesanya et al.15.

According to Lambert11, Dormand16 and Faires and Burden17, multi-processing Milne’s device is seen as an alternative to multi-processing predicting-correcting scheme on the account of the numerical vantages it features over others. Generators such as Akinfenwa et al.2, Bakoji et al.5, Anake et al.8, Anake and Adoghe9, Adesanya et al.15, Majid and Suleiman18 and Oghonyon et al.19-21 suggested multi-processing predictor-corrector scheme implemented on first-order ODEs. Multi-processing predicting-correcting scheme derives shortcomings during computation/execution and as such, unable to find a suitable length, resolve bounds of convergency and lack of error maximization.

The motivation of this research study is founded on the concept of generating certain qualities of the multi-processing Milne’s device which are comparable to BDF for implementing stiff ODEs and vibration problem as discussed in Anake et al.8, Anake and Adoghe9, Jain et al.10, Sunday et al.13, Faires and Burden17, Oghonyon et al.19-21, Ascher and Petzold 22, Ngwane and Jator23-25 and Ibrahim et al.26. Again, softcodes of Mathematica codes is projected for implementation1,3.

The main aim of this research work is to develop softcodes of multi-processing Milne’s device for computing first-order ODEs. Furthermore, this originality has been established on various body of literatures as cited Lambert11,12, Dormand16, Faires and Burden17, Oghonyon et al.19-21 and Ascher and Petzold22 for more particulars. This includes some elements like; Adams type, multi-processing predicting-correcting scheme of the like range and chief local truncation errors as remarked above.


Softcode for multi-processing Milne’s device is a collection of multi-processing predicting-correcting scheme of Adams type. This requires Adams-Bashforth-Adams-Moulton (multi-processing predicting-correcting of ilk range) scheme. This involves u-length multi-processing predicting scheme and u-1-length multi-processing correcting scheme of ilk range. This compendium is established as:



Equation 3 and 4 determines the multi-processing predicting-correcting scheme of multi-processing Milne’s device. Remarking g(tm+i)≈gm+i, g(xm+i, gm+i)≈zm+i having j = 0, 1, 2. To attain Eq. 3 and 4, the approximative function is penned below to evaluate the analytical resolution g(t) on clear-cut time intervals of [tn, tn-j] by way of interpolation of the form:


Revising Eq. 5 in softcode format produces the softcode approximate function as:


where, x0, x1, x2 and x3 are parameters required to be settle in a special manner. Presuming that Eq. 6 corresponds with the precise result at approximately selected definite length of time interval tn, tn-j to yield approximation as:


Taking that the approximating function (Eq. 6) gratifies (Eq. 1) at more or less chosen points tn+j, j = 0, 1, 2 to obtain the following approximates as:


Merging Eq. 7 and 8 will generate quadruplet formations which produces At = b:



Figuring out the systems of equation applying Mathematica 9 kernel, softcodes gives xj, j = 0, 1, 2, 3 and putting back values of xj's into Eq. 6 will generates the uninterrupted multi-processing prediction scheme and multi-processing correcting scheme of Milne’s device as:



Assessing the uninterrupted multi-processing prediction scheme and multi-processing correcting scheme of Milne’s device at some favourable grids, tn+j, j = 1, 2, 3 will originate the multi-processing prediction Milne’s device and multi-processing correcting Milne’s device as:


where, β1, β2, β3, μ1, μ2 and μ3 are parametric quantity1,4,11,12,16,17,23-25,27 for more details.

Devising bounds of convergence for multi-processing Milne’s device: To set in motion numeric operation of multi-processing Milne’s device, the r-length multi-processing predicting scheme and r-1-length multi-processing correcting scheme are put to use as multi-processing predicting-correcting scheme owns alike range Locate11,12,16,17,19-22 for more. Uniting Lambert11,12, Dormand16, Faires and Burden17, Oghonyon et al.19-21 and Ascher and Petzold22, it is workable to find approximative chief local truncation error of multi-processing predicting-correcting scheme in absentia of higher order differential coefficients, g(t). What is more, p1 = c1 where, p1 and c1 represents range of multi-processing predicting and correcting schemes. Straightaway, scheme of range p1, taking apart multi-processing predicting r-length gives rise to the chief principal local truncation errors:


Likewise, looking into multi-processing correcting scheme r-1-step brings forth chief local truncation errors as:


where, continues as classified quantity of length h1 and g(t) behave as analytic resolution to higher derived function conforming to the initial stipulation g(tn)≈gn. Look into Lambert11,12, Dormand16, Faires and Burden17, Oghonyon et al.19-21 and Ascher and Petzold22 more items.

Further advancement for less precondition measures of length h1 is reached g(4) (tn)≈g(4) (tn) and the potency of multi-processing Milne’s device trusts instantly on this presumption stated over.

Reducing in advance the chief the principal local truncation errors of Eq. 14 and 15 over besides dismissing considerations of range O(hp1+5). Thus, introduces no concern achieving the numerical formulation of chief local truncation errors of the multi-processing Milne’s device:


Referring the avouchment that and are named predicting and correcting estimations founded thru multi-processing Milne’s device of order p1, even though and are each separately called chief local truncation errors. τ1, τ2 and τ3 are bounds of convergency of the multi-processing Milne’s device.

Advancing forward, these approximates of the chief local truncation error (Eq. 16) is utilized to make decision on acceptance or rejection thereby iterating with less or smaller varying length. The length is sustain free-based on a try out laid down by Eq. 16 11,12,16,17,19-22 for more details. The chief local truncation errors (Eq. 16) is the bounds of convergence of the multi-processing Milne’s, device denoted differently as multi-processing Milne’s device for adjusting to convergence.

Numerical problems: Two problems tested are worked with MPMD. The bounds of convergency considered includes; 10–4, 10–6, 10–8, 10–10, 10–11 and 10–14. Find Sunday et al.13, Rufai et al.28 and Sunday et al.29 for more actions. A computer programming codes on MPMD is written utilizing Mathematica 9 kernel. The act of accomplishment is carried out in a multi-processing manner via MPMD (Appendix).

Test problem 1: Consider the nonlinear IVP, g'(t) = -10(g(t)-1)^2, g(0) = 2.

Analytical result:

Test problem 2: Consider Prothero-Robinson periodic vibration ODE, g'(t) = L(g(t)-sin(t))+cos (t), L = -1, g(0) = 0.

Analytical result: g(t) = sin (t).


Under this section, the computational output shows the execution of MPMD for solving first-order ODEs. The final output supplied were obtained with the aid of Mathematica 9 Kernel 64 on Microsoft windows (64 bit) to demonstrate the efficiency and accuracy of the first-order ODEs13,28,29.

Table 1 demonstrates the numerical results of problems 1 and 2 using MPMD equated with existing methods.

Table 1 presents a summary of the result displayed and items considered. This includes; method utilized, computed max errors and bounds of convergency. Again, shows the comparison with other existing results and justifies MPMD as a preferable proficiency in terms of the computed max errors:

Table 1:Summary of results

The signifiers mentioned on Table 1 are stated below:

MPMD: Computed max errors in MPMD (multi-processing Milne’s device) for time-tested problems 1 and 2
Mutilized: Method utilized
Maxerrors: Magnitude of computed max errors in MPMD
Bcov: Bounds of convergency
1/6 HBM: Computed max errors in one-sixth HBM (1/6 hybrid block method) for time-tested problem 1 28
ERR (BI): Computed max errors in ERR (BI) (block integrator) for time-tested tested problem 1 and 2 13
ERR: Computed max errors in ERR (quarter-step method of 10–4) for tested problem 1 29

Softcodes algorithm rule: A well written algorithmic rule that will execute MMD and assess the computed max errors of MPMD in the family of P(EC)j or P(EC)j E style, conditionally, when the style is implemented as many times to ensure convergence. Check out23:

Step 1: Take length for h
Step 2: The MPMD of predicting-correcting scheme must have alike range
Step 3: The length of predicting must have higher length than correcting scheme
Step 4: Estimate the chief local truncation errors of the MPMD only when CLTE is reached
Step 5: Fix the bounds of convergency
Step 6: Generate the softcodes of MPMD utilizing Mathematica 9 kernel
Step 7: Adopt single step technique to kick start the procedure if necessary, otherwise avoid 7 and go to step 8
Step 8: Perform the MPMD in the family of P(EC)j or P(EC)j E style as j increases
Step 9: When step 8 did not attain convergency, ingeminate the process once again and half the length (h) from step 1 or otherwise, go on to step 10
Step 10: Calculate the computed max errors when convergency is fulfilled
Step 11: Publish computed max errors
Step 12: Use this equation below to devise a new length only when convergency is attained


The computed results displayed MPMD is reached utilizing the bounds of convergency. This bounds of convergency examine the acceptance or rejection of the looping with a smaller length. The mathematical outputs establish the performance of MPMD is remarked to showcase a more acceptable computed max error at all bounds of convergency. This is made possible by seeking a suitable/changing length, determining the bounds of convergency as compare to subsisting schemes implemented without these features. This proficiency for a better result is executed at all examined bounds of convergency such as 10–4, 10–6, 10–8, 10–10, 10–11 and 10–14. Thence, it will be concluded that MPMD is worthy for estimating ODEs. Furthermore, MPMD is better and preferred to schemes such as block predictor-corrector methods, block implicit method, block hybrid method because their applications are based on fixed step size, no bounds of convergency and always implemented in predictor-corrector method. Continuous research can be carried out to increase the order of MPMD for examining performance.


The significant of this study is as follows:

A new basis function approximation is designed in form of Softcodes for yielding interpolation and collocation estimates

The scientific community will benefit by using Softcodes in Mathematica format, encrypted for the successful implementation of MPMD
The accuracy of MPMD is validated on nonlinear IVP and vibration problem
MPMD advances the utilization of the chief local truncation error outside showing the order
The MPMD is considered as an option to Backward Differentiation Formula (BDF) on account of some similar advantages it possesses


The authors would like to appreciate Covenant University for financing this research work.

Appendix: The softcodes of problem 1 and 2 implemented via MPMD is given below

1:  Abell, M.L. and J.P. Braselton, 2009. Mathematica By Example. 4th Edn., Elsevier, USA., ISBN: 9780080921693, Pages: 576.

2:  Akinfenwa, O., S. Jator and N. Yoa, 2011. An eighth order backward differentiation formula with continuous coefficients for stiff ordinary differential equations. Int. J. Math. Comput. Sci., 7: 160-165.
Direct Link  |  

3:  Ken, Y.L., I.F. Ismail and M. Suleiman, 2011. Block Methods for Special Second Order ODEs. Lambert Academic Publishing, Germany, ISBN-13: 978-3844384130, Pages: 196.

4:  Ngwane, F.F. and S.N. Jator, 2013. Solving oscillatory problems using a block hybrid trigonometrically fitted method with two off-step points. Electron. J. Differ. Equations, 20: 119-132.
Direct Link  |  

5:  Bakoji, A.M., A.M. Bukar and M.I. Bello, 2014. Formulation of predictor-corrector methods from 2-step hybrid Adams methods for the solution of initial value problems of ordinary differential equations. Int. J. Eng. Applied Sci., 5: 9-13.
Direct Link  |  

6:  Adejumo, G., M.T. Abioye and T.A. Anake, 2014. Adoption of computer assisted language learning software among Nigerian secondary school. Proceedings of the International Conference on Education and New Learning Technologies, July 7-9, 2014, Barcelona, Spain, pp: 950-955.

7:  Akinfenwa, O.A., S.N. Jator and N.M. Yao, 2013. Continuous block backward differentiation formula for solving stiff ordinary differential equations. Comput. Math. Applic., 65: 996-1005.
CrossRef  |  Direct Link  |  

8:  Anake, T.A., D.O. Awoyemi and A.O. Adesanya, 2012. One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations. IAENG Int. J. Applied Math., 42: 224-228.
Direct Link  |  

9:  Anake, T.A. and L.O. Adoghe, 2013. A four point block integration method for the solutions of IVP in ODE. Aust. J. Basic Applied Sci., 7: 467-473.
Direct Link  |  

10:  Jain, M.K., S.R.K. Iyengar and R.K. Jain, 2007. Numerical Methods for Scientific and Engineering Computation. 5th Edn., New Age International (Pvt.) Ltd., New Delhi, India, ISBN-13: 978-8122420012, Pages: 816.

11:  Lambert, J.D., 1973. Computational Methods in Ordinary Differential Equations. John Wiley and Sons, New York, USA., ISBN: 9780471511946, pp: 87-88.

12:  Lambert, J.D., 1991. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. 1st Edn., John Wiley and Sons, New York, USA., ISBN: 9780471929901, pp: 103-105.

13:  Sunday, J., M.R. Odekunle, A.A. James and A.O. Adesanya, 2014. Numerical solution of stiff and oscillatory differential equations using a block integrator. Br. J. Math. Comput. Sci., 4: 2471-2481.
Direct Link  |  

14:  Xie, L. and H. Tian, 2014. Continuous parallel block methods and their applications. Applied Math. Comput., 241: 356-370.
CrossRef  |  Direct Link  |  

15:  Adesanya, A.O., A.A. James and J. Sunday, 2014. Hybrid block predictor-hybrid block corrector for the solution of first-order ordinary differential equations. Eng. Math. Lett., 13: 1-12.
Direct Link  |  

16:  Dormand, J.R., 1996. Numerical Methods for Differential Equations: A Computational Approach. CRC Press, Boca Raton, FL., ISBN: 9780849394331, pp: 10, 142-143, 231.

17:  Faires, J.D. and R.L. Burden, 2012. Initial-value problems for ODEs, variable step-size multistep methods. Dublin City University, Dublin, Republic of Ireland, pp: 2-32.

18:  Majid, Z.A. and M. Suleiman, 2011. Predictor-corrector block iteration method for solving ordinary differential equations. Sains Malays., 40: 659-664.
Direct Link  |  

19:  Oghonyon, J.G., S.A. Okunuga and O.O. Agboola, 2015. K-step block predictor-corrector methods for solving first order ordinary differential equations. Res. J. Applied Sci., 10: 779-785.
Direct Link  |  

20:  Oghonyon, J.G., S.A. Okunuga and S.A. Iyase, 2016. Milne‚Äôs implementation on block predictor-corrector methods. J. Applied Sci., 16: 236-241.
CrossRef  |  Direct Link  |  

21:  Oghonyon, J.G., S.A. Okunuga and S.A. Bishop, 2017. A variable-step-size block predictor-corrector method for ordinary differential equations. Asian J. Applied Sci., 10: 96-101.
CrossRef  |  Direct Link  |  

22:  Ascher, U.M. and L.R. Petzold, 1998. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, USA., ISBN-13: 9780898714128, Pages: 314.

23:  Ngwane, F.F. and S.N. Jator, 2014. Trigonometrically-fitted second derivative method for oscillatory problems. SpringerPlus, Vol. 3. 10.1186/2193-1801-3-304

24:  Ngwane, F.F. and S.N. Jator, 2015. Solving the telegraph and oscillatory differential equations by a block hybrid trigonometrically fitted algorithm. Int. J. Differ. Equations, Vol. 2015. 10.1155/2015/347864

25:  Ngwane, F.F. and S.N. Jator, 2017. A Trigonometrically fitted block method for solving oscillatory second-order initial value problems and hamiltonian systems. Int. J. Differ. Equations, Vol. 2017. 10.1155/2017/9293530

26:  Ibrahim, Z.B., K. Othman and M. Suleiman, 2007. Variable step block backward differentiation formula for solving first-order stiff ODEs. Proceedings of the World Congress on Engineering, July 2-4, 2007, London, UK., pp: 2-6.

27:  Ngwane, F.F. and S.N. Jator, 2013. Block hybrid method using trigonometric basis for initial value problems with oscillating solutions. Numer. Algorithm, 63: 713-725.
CrossRef  |  Direct Link  |  

28:  Rufai, M.A., M.K. Duromola and A.A. Ganiyu, 2016. Derivation of one-sixth hybrid block method for solving general first order ordinary differential equations. IOSR J. Math., 12: 20-27.
Direct Link  |  

29:  Sunday, J., D. Yusuf and J.N. Andest, 2016. Integration of first-order modeled differential equations using a quarter-step method. Adv. Res., 7: 1-8.
CrossRef  |  Direct Link  |  

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