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
Volume 6, Issue 5, October 2017, Pages: 137-143
Received: Sep. 7, 2017; Accepted: Sep. 18, 2017; Published: Oct. 11, 2017
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Authors
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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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.
Keywords
Hybrid Method, Collocation, Interpolation, Shifted Legendre Polynomials Approximation, Continuous Block Method, Order, Consistency, Zero Stability, Convergent
To cite this article
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 and Applied Mathematics Journal. Vol. 6, No. 5, 2017, pp. 137-143. doi: 10.11648/j.pamj.20170605.11
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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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