On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang; Soomiyol Mrumun Comfort

doi:doi:10.11648/j.pamj.20170605.11

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On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Soomiyol Mrumun Comfort

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 5)

Received: 7 September 2017 Accepted: 18 September 2017 Published: 11 October 2017

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Abstract

This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.

Published in	Pure and Applied Mathematics Journal (Volume 6, Issue 5)
DOI	10.11648/j.pamj.20170605.11
Page(s)	137-143
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), 2017. Published by Science Publishing Group

Keywords

Hybrid Method, Collocation, Interpolation, Shifted Legendre Polynomials Approximation, Continuous Block Method, Order, Consistency, Zero Stability, Convergent

References

[1]	Adee, S. O., Onumanyi, P., Sirisena, U. W. andYahaya, Y. A. (2005) Note on Starting the Numerov Method More Accurately by a Hybrid Formula of Order Four for Initial Value Problems. Journal of Computational and Applied Mathematics, 175: 369-373.
[2]	Ademiluyi, R. A. (1987) New Hybrid Methods for Systems of Stiff Equations, Benin City, Nigeria: PhD Thesis, University of Benin.
[3]	Akinfenwa, O. A., Jator, S. N. and Yao, N. M. (2011) Linear Multistep Hybrid Methods with Continuous Coefficients for Solving Stiff Ordinary Differential Equations, Journal of Modern Mathematics and Statistics, 5(2): 47-57.
[4]	Anake, T. A. (2011). Continuous Implicit Hybrid One-Step Methods for the Solution of Initial Value Problems of Second Order Ordinary Differential Equations. Ogun State, Nigeria. PhD Thesis, Covenant University, Ota.
[5]	Awari, Y. S., Abada, A. A., Emma, P. M. and Kamoh, N. M. (2013) Application of Two Step Continuous Hybrid Butcher’s Method in Block Form for the Solution of First Order Initial Value Problems. Natural and Applied Sciences, 4(4): 209-218.
[6]	Fatunla, S. O. (1988) Numerical Methods for Initial Value Problems for Ordinary Differential Equations USA, Academy press, Boston 295.
[7]	Henrici, P. (1962) Discrete Variable Methods for ODE`s, New York, USA, John Wiley and Sons, (1962).
[8]	Lambert, J. D. (1991) Numerical Methods for Ordinary Differential Equations, New York: John Wiley and Sons pp 293.
[9]	Mohammed, U. and Adeniyi, R. B (2014) A Three Step Implicit Hybrid Linear Multistep Method for the Solution of Third Order Ordinary Differential Equations. General Mathematics Notes, 25(1): 62-74.
[10]	Odejide, S. A. and Adenira, A. O. (2012) A Hybrid Linear Collocation Multistep Scheme For Solving First Order Initial Value Problems. Journal of the Nigerian Mathematical Society 31: 229-241.
[11]	Serisina, U. W., Kumleng, G. M. and Yahaya, Y. A. (2004) A New Butcher Type Two-Step Block Hybrid Multistep Method for Accurate and Efficient Parallel Solution of Ordinary Differential Equations, Abacus Mathematics Series. 31: 1-7.
[12]	Yakusak, N. S., Emmanuel, S. and Ogunniran, M. O. (2015) Uniform Order Legendre Approach for Continuous Hybrid Block Methods for the Solution of First Order Ordinary Differential Equations. IOSR Journal of Mathematics, 11(1): 09-14.
[13]	Zill, D. G. and Warren, W. S. (2013) Differential Equations with Boundary- Value Problems, Cengage Learning: Eight edition books/Cole.

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APA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Soomiyol Mrumun Comfort. (2017). On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure and Applied Mathematics Journal, 6(5), 137-143. https://doi.org/10.11648/j.pamj.20170605.11

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ACS Style

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang; Soomiyol Mrumun Comfort. On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure Appl. Math. J. 2017, 6(5), 137-143. doi: 10.11648/j.pamj.20170605.11

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AMA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Soomiyol Mrumun Comfort. On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure Appl Math J. 2017;6(5):137-143. doi: 10.11648/j.pamj.20170605.11

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@article{10.11648/j.pamj.20170605.11,
  author = {Kamoh Nathaniel Mahwash and Gyemang Dauda Gyang and Soomiyol Mrumun Comfort},
  title = {On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {5},
  pages = {137-143},
  doi = {10.11648/j.pamj.20170605.11},
  url = {https://doi.org/10.11648/j.pamj.20170605.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170605.11},
  abstract = {This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.},
 year = {2017}
}

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TY  - JOUR
T1  - On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations
AU  - Kamoh Nathaniel Mahwash
AU  - Gyemang Dauda Gyang
AU  - Soomiyol Mrumun Comfort
Y1  - 2017/10/11
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170605.11
DO  - 10.11648/j.pamj.20170605.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 137
EP  - 143
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170605.11
AB  - This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.
VL  - 6
IS  - 5
ER  -

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Author Information

Kamoh Nathaniel Mahwash

Department of Mathematics/Statistics, Bingham University, Karu, Nigeria
Gyemang Dauda Gyang

Department of Mathematics and Computer Science, Benue State University, Makurdi, Nigeria
Soomiyol Mrumun Comfort

Department of Mathematics/Statistics, Plateau State Polytechnic, BarkinLadi, Nigeria

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Plain Text BibTeX RIS

APA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Soomiyol Mrumun Comfort. (2017). On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure and Applied Mathematics Journal, 6(5), 137-143. https://doi.org/10.11648/j.pamj.20170605.11

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ACS Style

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang; Soomiyol Mrumun Comfort. On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure Appl. Math. J. 2017, 6(5), 137-143. doi: 10.11648/j.pamj.20170605.11

Copy | Download

AMA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Soomiyol Mrumun Comfort. On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. Pure Appl Math J. 2017;6(5):137-143. doi: 10.11648/j.pamj.20170605.11

Copy | Download

@article{10.11648/j.pamj.20170605.11,
  author = {Kamoh Nathaniel Mahwash and Gyemang Dauda Gyang and Soomiyol Mrumun Comfort},
  title = {On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {5},
  pages = {137-143},
  doi = {10.11648/j.pamj.20170605.11},
  url = {https://doi.org/10.11648/j.pamj.20170605.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170605.11},
  abstract = {This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations
AU  - Kamoh Nathaniel Mahwash
AU  - Gyemang Dauda Gyang
AU  - Soomiyol Mrumun Comfort
Y1  - 2017/10/11
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170605.11
DO  - 10.11648/j.pamj.20170605.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 137
EP  - 143
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170605.11
AB  - This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.
VL  - 6
IS  - 5
ER  -

Copy | Download