Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

Friday Oghenerukevwe Obarhua; Sunday Jacob Kayode

doi:doi:10.11648/j.pamj.20200901.14

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Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

Friday Oghenerukevwe Obarhua, Sunday Jacob Kayode

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 1)

Received: 23 December 2019 Accepted: 9 January 2020 Published: 13 February 2020

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Abstract

This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 1)
DOI	10.11648/j.pamj.20200901.14
Page(s)	26-31
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Continuous, Duffing Equation, Explicit, Symmetric, Zero-stable, Approximate Solution, Power Series

References

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[2]	Areo, E. A. and Rufai. M. A., (2016b): An Efficient One-Eight Step Hybrid Block Method for Solving second Order Initial Value Problems of ODEs. Intern. J. of Diff. Eqns. & Appli. 15 (2): 117-139
[3]	Sahi, R. K., Jator, S. N., & Khan, N. A., (2013): Continuous Fourth Derivative Method for Third Order Boundary Value Problems, Intern. Journal of Pure and Applied Mathematics, 85 (2), 907923.
[4]	Adesanya, A. O. Abdulqadri, B, and Ibrahim, Y. S., (2014): Hybrid One Step Block Method for the Solution of Third Order Initial Value Problems of Ordinary Differential Equations, Intern. J. of Appli. Maths. & Comput., 6 (1), 10-16.
[5]	Adeyeye, O. and Omar, Z., (2018): New Self-Starting Approach for Solving Special Third Order Initial Value Problems. Int. J. of Pure and Applied Mathematics. Vol. 118, No. 3, 511-517.
[6]	Liu C. S and Jhao W. S (2014): The Power Series Method for a Long Term Solution of Duffing Oscillator, Communications in Numerical Analysis, doi: 105899/2014/can- 00214, 1-14.
[7]	Yusuf, D. J., Ismail, F. and Senu, N., (2015): Zero-Dissipative Trigonometrically Fitted Hybrid Method for Numerical Solution of Oscillatory Problems, Sians Malaysiana, 44 (3), 473-482.
[8]	Olabode, B. T. and Momoh, A. L. (2016): Continuous Hybrid Multistep Methods with Legendre Basis Function for Direct Treatment of Second Order Stiff ODEs, American J. of Computal. & Applied Maths., 6 (2): 38-49.
[9]	Ibrahim I. H. (2017): Hybrid Multistep Method for Solving Second Order Initial Value Problems, Journal of the Egyptian Mathematical Society, 25: 355-362.
[10]	Allogmany, R., Ismail, F. and Ibrahim, Z. B., (2019): Implicit Two-Point Block Method with Third and Fourth Derivatives for Solving General Second Order ODEs, Mathematics and Statistics, 7 (4): 116-123.
[11]	Turki, M. Y., Ismail, F. and Ibrahim, Z. B., (2016): Second Derivative Multistep Method for Solving First-Order Ordinary Differential Equations. In AIP Conference Proceedings (vol. 1739, No. 1, pp 020054), AIP Publishing.
[12]	Singh, G. and Ramos, H. (2019): An Optimized Two-Step Hybrid Block Method Formulated in Variable Step-Size Mode for Integrating y″=f(x, y, y′) Numerically. Numerical Mathematics- Theory Methods and Applications, 12 (2), 640-660.
[13]	Kayode, S. J. and Obarhua, F. O., (2013): Continuous y-function Hybrid Methods for Direct Solution of Differential Equations. Intern. J. of Diff. Eq. 12 (1), 37- 48.
[14]	Kayode, S. J. and Obarhua, F. O., (2015): 3-Step y-function Hybrid Methods for Direct Numerical Integration of Second Order IVPs in ODEs. Theo. Math. & Appl., 5 (1), 39-51.
[15]	Majid, Z. A., Azmi, N. A. and Suleiman, M., (2009): Solving Second Order Ordinary Differential Equations Using Two Point Four Step Direct Implicit Block Method, European Journal of Scientific Research, 31 (1), pp. 29-3.

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APA Style

Friday Oghenerukevwe Obarhua, Sunday Jacob Kayode. (2020). Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(1), 26-31. https://doi.org/10.11648/j.pamj.20200901.14

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ACS Style

Friday Oghenerukevwe Obarhua; Sunday Jacob Kayode. Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(1), 26-31. doi: 10.11648/j.pamj.20200901.14

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AMA Style

Friday Oghenerukevwe Obarhua, Sunday Jacob Kayode. Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure Appl Math J. 2020;9(1):26-31. doi: 10.11648/j.pamj.20200901.14

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@article{10.11648/j.pamj.20200901.14,
  author = {Friday Oghenerukevwe Obarhua and Sunday Jacob Kayode},
  title = {Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {1},
  pages = {26-31},
  doi = {10.11648/j.pamj.20200901.14},
  url = {https://doi.org/10.11648/j.pamj.20200901.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200901.14},
  abstract = {This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.},
 year = {2020}
}

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TY  - JOUR
T1  - Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations
AU  - Friday Oghenerukevwe Obarhua
AU  - Sunday Jacob Kayode
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200901.14
DO  - 10.11648/j.pamj.20200901.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200901.14
AB  - This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.
VL  - 9
IS  - 1
ER  -

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Author Information

Friday Oghenerukevwe Obarhua

Department of Mathematical Sciences, School of Sciences, The Federal University of Technology, Akure, Nigeria
Sunday Jacob Kayode

Department of Mathematical Sciences, School of Sciences, The Federal University of Technology, Akure, Nigeria

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Plain Text BibTeX RIS

APA Style

Friday Oghenerukevwe Obarhua, Sunday Jacob Kayode. (2020). Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(1), 26-31. https://doi.org/10.11648/j.pamj.20200901.14

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ACS Style

Friday Oghenerukevwe Obarhua; Sunday Jacob Kayode. Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(1), 26-31. doi: 10.11648/j.pamj.20200901.14

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AMA Style

Friday Oghenerukevwe Obarhua, Sunday Jacob Kayode. Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations. Pure Appl Math J. 2020;9(1):26-31. doi: 10.11648/j.pamj.20200901.14

Copy | Download

@article{10.11648/j.pamj.20200901.14,
  author = {Friday Oghenerukevwe Obarhua and Sunday Jacob Kayode},
  title = {Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {1},
  pages = {26-31},
  doi = {10.11648/j.pamj.20200901.14},
  url = {https://doi.org/10.11648/j.pamj.20200901.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200901.14},
  abstract = {This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations
AU  - Friday Oghenerukevwe Obarhua
AU  - Sunday Jacob Kayode
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200901.14
DO  - 10.11648/j.pamj.20200901.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 31
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200901.14
AB  - This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.
VL  - 9
IS  - 1
ER  -

Copy | Download