
Research Article


Softcode of MultiProcessing Milne’s Device for Estimating FirstOrder Ordinary Differential Equations


Jimevwo Godwin Oghonyon,
Olaide Adetola Adesanya,
Hudson. Akewe
and
Hilary Izuchukwu Okagbue


ABSTRACT

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 multiprocessing Milne’s device for estimating firstorder Ordinary Differential Equations (ODEs). Materials and Methods: MultiProcessing Milne’s Device (MPMD) is source from Adams collection of predictingcorrecting 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.





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.


INTRODUCTION
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 applications^{14}. Multiprocessing Milne’s device for estimating firstorder differential equation is of the form Abell and Braselton^{1}, 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 Adoghe^{9}, Jain et al.^{10}, Lambert^{11,12}, Sunday et al.^{13} and Xie and Tian^{14}. Thus, ensures that Eq. 1 possess a specific differential coefficient at every point.
The universal multiprocessing Milne’s device is instituted as:
where, and are invariables implying that , Adesanya et al.^{15}.
According to Lambert^{11}, Dormand^{16} and Faires and Burden^{17}, multiprocessing Milne’s device is seen as an alternative to multiprocessing predictingcorrecting 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 Adoghe^{9}, Adesanya et al.^{15}, Majid and Suleiman^{18} and Oghonyon et al.^{1921} suggested multiprocessing predictorcorrector scheme implemented on firstorder ODEs. Multiprocessing predictingcorrecting 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 multiprocessing Milne’s device which are comparable to BDF for implementing stiff ODEs and vibration problem as discussed in Anake et al.^{8}, Anake and Adoghe^{9}, Jain et al.^{10}, Sunday et al.^{13}, Faires and Burden^{17}, Oghonyon et al.^{1921}, Ascher and Petzold ^{22}, Ngwane and Jator^{2325} and Ibrahim et al.^{26}. Again, softcodes of Mathematica codes is projected for implementation^{1,3}.
The main aim of this research work is to develop softcodes of multiprocessing Milne’s device for computing firstorder ODEs. Furthermore, this originality has been established on various body of literatures as cited Lambert^{11,12}, Dormand^{16}, Faires and Burden^{17}, Oghonyon et al.^{1921} and Ascher and Petzold^{22} for more particulars. This includes some elements like; Adams type, multiprocessing predictingcorrecting scheme of the like range and chief local truncation errors as remarked above.
MATERIALS AND METHODS Softcode for multiprocessing Milne’s device is a collection of multiprocessing predictingcorrecting scheme of Adams type. This requires AdamsBashforthAdamsMoulton (multiprocessing predictingcorrecting of ilk range) scheme. This involves ulength multiprocessing predicting scheme and u1length multiprocessing correcting scheme of ilk range. This compendium is established as:
Equation 3 and 4 determines the multiprocessing predictingcorrecting scheme of multiprocessing Milne’s device. Remarking g(t_{m+i})≈g_{m+i}, g(x_{m+i}, g_{m+i})≈z_{m+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 clearcut time intervals of [t_{n}, t_{nj}] by way of interpolation of the form:
Revising Eq. 5 in softcode format produces the softcode approximate function as: where, x_{0}, x_{1}, x_{2} and x_{3} 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 t_{n}, t_{nj} to yield approximation as: Taking that the approximating function (Eq. 6) gratifies (Eq. 1) at more or less chosen points t_{n+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 x_{j}, j = 0, 1, 2, 3 and putting back values of x_{j}'s into Eq. 6 will generates the uninterrupted multiprocessing prediction scheme and multiprocessing correcting scheme of Milne’s device as:
Assessing the uninterrupted multiprocessing prediction scheme and multiprocessing correcting scheme of Milne’s device at some favourable grids, t_{n+j}, j = 1, 2, 3 will originate the multiprocessing prediction Milne’s device and multiprocessing correcting Milne’s device as:
where, β_{1}, β_{2}, β_{3}, μ_{1}, μ_{2} and μ_{3} are parametric quantity^{1,4,11,12,16,17,2325,27} for more details.
Devising bounds of convergence for multiprocessing Milne’s device: To set in motion numeric operation of multiprocessing Milne’s device, the rlength multiprocessing predicting scheme and r1length multiprocessing correcting scheme are put to use as multiprocessing predictingcorrecting scheme owns alike range Locate^{11,12,16,17,1922 } for more. Uniting Lambert^{11,12}, Dormand^{16}, Faires and Burden^{17}, Oghonyon et al.^{1921} and Ascher and Petzold^{22}, it is workable to find approximative chief local truncation error of multiprocessing predictingcorrecting scheme in absentia of higher order differential coefficients, g(t). What is more, p_{1} = c_{1} where, p_{1} and c_{1} represents range of multiprocessing predicting and correcting schemes. Straightaway, scheme of range p_{1}, taking apart multiprocessing predicting rlength gives rise to the chief principal local truncation errors:
Likewise, looking into multiprocessing correcting scheme r1step brings forth chief local truncation errors as:
where, continues as classified quantity of length h_{1} and g(t) behave as analytic resolution to higher derived function conforming to the initial stipulation g(t_{n})≈g_{n}. Look into Lambert^{11,12}, Dormand^{16}, Faires and Burden^{17}, Oghonyon et al.^{1921} and Ascher and Petzold^{22} more items.
Further advancement for less precondition measures of length h_{1} is reached g^{(4)} (t_{n})≈g^{(4)} (t_{n}) and the potency of multiprocessing 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(h^{p1+5}). Thus, introduces no concern achieving the numerical formulation of chief local truncation errors of the multiprocessing Milne’s device:
Referring the avouchment that and are named predicting and correcting estimations founded thru multiprocessing Milne’s device of order p_{1}, even though and are each separately called chief local truncation errors. τ_{1}, τ_{2} and τ_{3} are bounds of convergency of the multiprocessing 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 freebased on a try out laid down by Eq. 16 ^{11,12,16,17,1922 } for more details. The chief local truncation errors (Eq. 16) is the bounds of convergence of the multiprocessing Milne’s, device denoted differently as multiprocessing 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 multiprocessing 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 ProtheroRobinson periodic vibration ODE, g'(t) = L(g(t)sin(t))+cos (t), L = 1, g(0) = 0. Analytical result: g(t) = sin (t). RESULTS AND DISCUSSION Under this section, the computational output shows the execution of MPMD for solving firstorder 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 firstorder ODEs^{13,28,29}.
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 (multiprocessing Milne’s device) for timetested problems 1 and 2 
M_{utilized}: 
Method utilized 
Max_{errors}: 
Magnitude of computed max errors in MPMD 
B_{cov}: 
Bounds of convergency 
1/6 HBM: 
Computed max errors in onesixth HBM (1/6 hybrid block method) for timetested problem 1^{ 28} 
ERR (BI): 
Computed max errors in ERR (BI) (block integrator) for timetested tested problem 1 and 2 ^{13} 
ERR: 
Computed max errors in ERR (quarterstep 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 out^{23}:
Step 1: 
Take length for h 
Step 2: 
The MPMD of predictingcorrecting 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 
CONCLUSION 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 predictorcorrector methods, block implicit method, block hybrid method because their applications are based on fixed step size, no bounds of convergency and always implemented in predictorcorrector method. Continuous research can be carried out to increase the order of MPMD for examining performance. SIGNIFICANT STATEMENT 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 
ACKNOWLEDGMENT 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 


REFERENCES 
Abell, M.L. and J.P. Braselton, 2009. Mathematica By Example. 4th Edn., Elsevier, USA., ISBN: 9780080921693, Pages: 576.
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 79, 2014, Barcelona, Spain, pp: 950955.
Adesanya, A.O., A.A. James and J. Sunday, 2014. Hybrid block predictorhybrid block corrector for the solution of firstorder ordinary differential equations. Eng. Math. Lett., 13: 112. Direct Link 
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: 160165. Direct Link 
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: 9961005. CrossRef  Direct Link 
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: 467473. Direct Link 
Anake, T.A., D.O. Awoyemi and A.O. Adesanya, 2012. Onestep implicit hybrid block method for the direct solution of general second order ordinary differential equations. IAENG Int. J. Applied Math., 42: 224228. Direct Link 
Ascher, U.M. and L.R. Petzold, 1998. Computer Methods for Ordinary Differential Equations and DifferentialAlgebraic Equations. SIAM, Philadelphia, USA., ISBN13: 9780898714128, Pages: 314.
Bakoji, A.M., A.M. Bukar and M.I. Bello, 2014. Formulation of predictorcorrector methods from 2step hybrid Adams methods for the solution of initial value problems of ordinary differential equations. Int. J. Eng. Applied Sci., 5: 913. Direct Link 
Dormand, J.R., 1996. Numerical Methods for Differential Equations: A Computational Approach. CRC Press, Boca Raton, FL., ISBN: 9780849394331, pp: 10, 142143, 231.
Faires, J.D. and R.L. Burden, 2012. Initialvalue problems for ODEs, variable stepsize multistep methods. Dublin City University, Dublin, Republic of Ireland, pp: 232.
Ibrahim, Z.B., K. Othman and M. Suleiman, 2007. Variable step block backward differentiation formula for solving firstorder stiff ODEs. Proceedings of the World Congress on Engineering, July 24, 2007, London, UK., pp: 26.
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, ISBN13: 9788122420012, Pages: 816.
Ken, Y.L., I.F. Ismail and M. Suleiman, 2011. Block Methods for Special Second Order ODEs. Lambert Academic Publishing, Germany, ISBN13: 9783844384130, Pages: 196.
Lambert, J.D., 1973. Computational Methods in Ordinary Differential Equations. John Wiley and Sons, New York, USA., ISBN: 9780471511946, pp: 8788.
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: 103105.
Majid, Z.A. and M. Suleiman, 2011. Predictorcorrector block iteration method for solving ordinary differential equations. Sains Malays., 40: 659664. Direct Link 
Ngwane, F.F. and S.N. Jator, 2013. Block hybrid method using trigonometric basis for initial value problems with oscillating solutions. Numer. Algorithm, 63: 713725. CrossRef  Direct Link 
Ngwane, F.F. and S.N. Jator, 2013. Solving oscillatory problems using a block hybrid trigonometrically fitted method with two offstep points. Electron. J. Differ. Equations, 20: 119132. Direct Link 
Ngwane, F.F. and S.N. Jator, 2014. Trigonometricallyfitted second derivative method for oscillatory problems. SpringerPlus, Vol. 3. 10.1186/219318013304
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
Ngwane, F.F. and S.N. Jator, 2017. A Trigonometrically fitted block method for solving oscillatory secondorder initial value problems and hamiltonian systems. Int. J. Differ. Equations, Vol. 2017. 10.1155/2017/9293530
Oghonyon, J.G., S.A. Okunuga and O.O. Agboola, 2015. Kstep block predictorcorrector methods for solving first order ordinary differential equations. Res. J. Applied Sci., 10: 779785. Direct Link 
Oghonyon, J.G., S.A. Okunuga and S.A. Bishop, 2017. A variablestepsize block predictorcorrector method for ordinary differential equations. Asian J. Applied Sci., 10: 96101. CrossRef  Direct Link 
Oghonyon, J.G., S.A. Okunuga and S.A. Iyase, 2016. Milne’s implementation on block predictorcorrector methods. J. Applied Sci., 16: 236241. CrossRef  Direct Link 
Rufai, M.A., M.K. Duromola and A.A. Ganiyu, 2016. Derivation of onesixth hybrid block method for solving general first order ordinary differential equations. IOSR J. Math., 12: 2027. Direct Link 
Sunday, J., D. Yusuf and J.N. Andest, 2016. Integration of firstorder modeled differential equations using a quarterstep method. Adv. Res., 7: 18. CrossRef  Direct Link 
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: 24712481. Direct Link 
Xie, L. and H. Tian, 2014. Continuous parallel block methods and their applications. Applied Math. Comput., 241: 356370. CrossRef  Direct Link 



