Five Steps Block Predictor-Block Corrector Method for the Solution of *y''* = *f* (*x*,*y*,*y'*)

Author(s)
Mathew Remilekun Odekunle,
Michael Otokpa Egwurube,
Adetola Olaide Adesanya,
Mfon Okon Udo

Affiliation(s)

Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria.

Department of Mathematics and Statistics, Cross River University of Technology, Calabar, Nigeria.

Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria.

Department of Mathematics and Statistics, Cross River University of Technology, Calabar, Nigeria.

ABSTRACT

Theory has it that
increasing the step length improves the accuracy of a method. In order to
affirm this we increased the step length of the concept in [1] by one to get *k* = 5. The technique of collocation and interpolation of the power
series approximate solution at some selected grid points is considered so as to
generate continuous linear multistep methods with constant step sizes. Two,
three and four interpolation points are considered to generate the continuous
predictor-corrector methods which are implemented in block method respectively.
The proposed methods when tested on some numerical examples performed more
efficiently than those of [1]. Interestingly the concept of self starting [2] and that of
constant order are reaffirmed in our new methods.

KEYWORDS

Step Length, Power Series, Block Predictor, Block Corrector, Constant Order, Step Size, Grid Points, Self Starting, Efficiency

Step Length, Power Series, Block Predictor, Block Corrector, Constant Order, Step Size, Grid Points, Self Starting, Efficiency

Cite this paper

Odekunle, M. , Egwurube, M. , Adesanya, A. and Udo, M. (2014) Five Steps Block Predictor-Block Corrector Method for the Solution of*y''* = *f* (*x*,*y*,*y'*). *Applied Mathematics*, **5**, 1252-1266. doi: 10.4236/am.2014.58117.

Odekunle, M. , Egwurube, M. , Adesanya, A. and Udo, M. (2014) Five Steps Block Predictor-Block Corrector Method for the Solution of

References

[1] Odekunle, M.R., Egwurube, M.O., Adesanya, A.O. and Udo, M.O. (2014) Body Math Four Steps Block PredictorBlock Corrector. Method for the Solution of Journal of Advances in Mathematics, 5, 746-755.

[2] Jator, S.N. and Li, J. (2009) A Self Starting Linear Multistep Method for the Direct Solution of General Second Order Initial Value Problems. International Journal of Computer Mathematics, 86, 817-836.

http://dx.doi.org/10.1080/00207160701708250

[3] Adesanya, A.O., Odekunle, M.R. and Adeyeye, A.O. (2012) Continuous Block Hybrid-Predictor-Corrector Method for the Solution of International Journal of Mathematics and Soft computing, 2, 35-42.

[4] Adesanya, A.O., Odekunle, M.R. and Udo, M.O. (2013) Four Steps Continuous Method for the Solution of American Journal of Computational Mathematics, 3, 169-174.

[5] Awoyemi, D.O. and Kayode, S.J. (2005) A Maximal Order Collocation Method for Direct Solution of Initial Value Problems of General Second Order Ordinary Differential Equation. Proceedings of the Conference Organised by the National Mathematical Centre, Abuja.

[6] Jator, S.N. (2007) A Sixth Order Linear Multistep Method for Direct Solution of International Journal of Pure and Applied Mathematics, 40, 457-472.

[7] Awoyemi, D.O. (2001) A New Sixth Order Algorithm for General Second Order Ordinary Differential Equation. International Journal of Computer Mathematics, 77, 117-124.

[8] Awoyemi, D.O., Adebile, E.A., Adesanya, A.O. and Anake, T.A. (2011) Modifid Block Method for the Direct Solution of Second Order Ordinary Differential Equation. International Journal of Applied Mathematics and Computation, 3, 181-188.

[9] Lambert, J.D. (1973) Computational Methods in ODES. John Wiley and Sons, New York.

[10] Adesanya, A.O., Anake, T.A. and Udo, M.O. (2008) Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equation. Journal of the Nigerian Association of Mathematical Physics, 13, 59-62.

[11] Udo, M.O., Olayi, G.A. and Ademiluyi, R.A. (2007) Linear Multistep Method for Solution of Second Order Initial Value Problems of Ordinary Differential Equations: A Truncation Error Approach. Global Journal of Mathematical Sciences, 6, 119-126.

[12] Zarina, B.I., Mohamed, S. and Iskanla, I.O. (2009) Direct Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equation. Journal of Mathematics and Computation Sciences, 3, 120-122.

[13] James, A.A., Adesanya, A.O. and Sunday, J. (2013) Continuous Block Method for the Solution of Second Order Initial Value Problems of Ordinary Differential Equations. Journal of Mathematics and Computation Sciences, 83, 405-416.

[14] Awoyemi, D.O. and Idowu, M.O. (2005) A Class of Hybrid Collocation Method for Third Order Ordinary Differential Equation. International Journal of Computer Mathematics, 82, 1287-1293.

http://dx.doi.org/10.1080/00207160500112902

[15] Awoyemi, D.O., Udo, M.O. and Adesanya, A.O. (2006) Non-Symmetric Collocation Method for Direct Solution of General Second Order Initial Value Problems of Ordinary Differential Equations. Journal of Natural and Applied Sciences, 7, 31-37.

[16] Awoyemi, D.O. (2003) A p-Stable Linear Multistep Method for Solving Third Order Ordinary Differential Equation. International Journal of Computer Mathematics, 80, 85-99.

http://dx.doi.org/10.1080/0020716031000079572

[17] Yahaya, Y.A. and Badmus, A.M. (2009) A Class of Collocation Methods for General Second Order Differential Equation. African Journal of Mathematics and Computer Research, 2, 69-71.

[1] Odekunle, M.R., Egwurube, M.O., Adesanya, A.O. and Udo, M.O. (2014) Body Math Four Steps Block PredictorBlock Corrector. Method for the Solution of Journal of Advances in Mathematics, 5, 746-755.

[2] Jator, S.N. and Li, J. (2009) A Self Starting Linear Multistep Method for the Direct Solution of General Second Order Initial Value Problems. International Journal of Computer Mathematics, 86, 817-836.

http://dx.doi.org/10.1080/00207160701708250

[3] Adesanya, A.O., Odekunle, M.R. and Adeyeye, A.O. (2012) Continuous Block Hybrid-Predictor-Corrector Method for the Solution of International Journal of Mathematics and Soft computing, 2, 35-42.

[4] Adesanya, A.O., Odekunle, M.R. and Udo, M.O. (2013) Four Steps Continuous Method for the Solution of American Journal of Computational Mathematics, 3, 169-174.

[5] Awoyemi, D.O. and Kayode, S.J. (2005) A Maximal Order Collocation Method for Direct Solution of Initial Value Problems of General Second Order Ordinary Differential Equation. Proceedings of the Conference Organised by the National Mathematical Centre, Abuja.

[6] Jator, S.N. (2007) A Sixth Order Linear Multistep Method for Direct Solution of International Journal of Pure and Applied Mathematics, 40, 457-472.

[7] Awoyemi, D.O. (2001) A New Sixth Order Algorithm for General Second Order Ordinary Differential Equation. International Journal of Computer Mathematics, 77, 117-124.

[8] Awoyemi, D.O., Adebile, E.A., Adesanya, A.O. and Anake, T.A. (2011) Modifid Block Method for the Direct Solution of Second Order Ordinary Differential Equation. International Journal of Applied Mathematics and Computation, 3, 181-188.

[9] Lambert, J.D. (1973) Computational Methods in ODES. John Wiley and Sons, New York.

[10] Adesanya, A.O., Anake, T.A. and Udo, M.O. (2008) Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equation. Journal of the Nigerian Association of Mathematical Physics, 13, 59-62.

[11] Udo, M.O., Olayi, G.A. and Ademiluyi, R.A. (2007) Linear Multistep Method for Solution of Second Order Initial Value Problems of Ordinary Differential Equations: A Truncation Error Approach. Global Journal of Mathematical Sciences, 6, 119-126.

[12] Zarina, B.I., Mohamed, S. and Iskanla, I.O. (2009) Direct Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equation. Journal of Mathematics and Computation Sciences, 3, 120-122.

[13] James, A.A., Adesanya, A.O. and Sunday, J. (2013) Continuous Block Method for the Solution of Second Order Initial Value Problems of Ordinary Differential Equations. Journal of Mathematics and Computation Sciences, 83, 405-416.

[14] Awoyemi, D.O. and Idowu, M.O. (2005) A Class of Hybrid Collocation Method for Third Order Ordinary Differential Equation. International Journal of Computer Mathematics, 82, 1287-1293.

http://dx.doi.org/10.1080/00207160500112902

[15] Awoyemi, D.O., Udo, M.O. and Adesanya, A.O. (2006) Non-Symmetric Collocation Method for Direct Solution of General Second Order Initial Value Problems of Ordinary Differential Equations. Journal of Natural and Applied Sciences, 7, 31-37.

[16] Awoyemi, D.O. (2003) A p-Stable Linear Multistep Method for Solving Third Order Ordinary Differential Equation. International Journal of Computer Mathematics, 80, 85-99.

http://dx.doi.org/10.1080/0020716031000079572

[17] Yahaya, Y.A. and Badmus, A.M. (2009) A Class of Collocation Methods for General Second Order Differential Equation. African Journal of Mathematics and Computer Research, 2, 69-71.