Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae

Yohanna Sani Awari; Micah Geoffrey Kumleng

doi:doi:10.11648/j.ajmcm.20170204.13

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Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae

Yohanna Sani Awari, Micah Geoffrey Kumleng

Published in American Journal of Mathematical and Computer Modelling (Volume 2, Issue 4)

Received: 18 September 2017 Accepted: 9 October 2017 Published: 8 November 2017

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Abstract

The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulae (BDF). In this paper, we focus on the derivation of the fourth, sixth and eighth order extended trapezoidal rule of first kind (ETRs) formulae through Hermite polynomial as basis function which we named FETR, SETR and EETR respectively. We then interpolate and collocate at some points of interest to generate the desire method. The stability analysis on our methods suggests that they are not only convergent but possess regions suitable for the solution of stiff ordinary differential equations (ODEs). The methods were very efficient when implemented in block form, they tend to perform better over existing methods.

Published in	American Journal of Mathematical and Computer Modelling (Volume 2, Issue 4)
DOI	10.11648/j.ajmcm.20170204.13
Page(s)	103-116
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

Stiffness, Hermite Polynomial, ETRs, A-Stability, Ordinary Differential Equations

References

[1]	Awari, Y. S. (2017). A-Stable Block ETRs/ETR2s Methods for Stiff ODEs, MAYFEB Journal of Mathematics Vol. 4: Pages 33-52.
[2]	Awari, Y. S., Garba, E. J. D., Kumleng, M. G. and Tsaku, N. (2016). Efficient implementation of Some Block Extended Trapezoidal rule of first kind (ETRs) class of method Applied to First Order Initial Value Problems of Differential Equations, Global Journal of Mathematics, Vol. 7 (2).
[3]	Biala, T. A., Jator, S. N., Adeniyi, R. B. and Ndukum, P. L. (2015). Block Hybrid Simpson’s Method with Two Off-grid Points for Stiff Systems International Journal of Nonlinear Science Vol. 20 (1), pp. 3-10.
[4]	Happy K., Shelly, Arora and Nagaich, R. K. (2013). Solution of Non Linear Singular Perturbation Equation Using Hermite Collocation Method, Applied Mathematical Sciences, Vol. 7, 2013, no. 109, 5397–5408.
[5]	Khadijah, M. Abualnaja, (2015). A Block Procedure with Linear Multistep Methods Using Legendre Polynomials for Solving ODEs, Applied Mathematics 6:717-723.
[6]	Miletics, E. and Moln´arka, G. (2009). Taylor Series Method with Numerical Derivatives for Numerical Solution of ODE Initial Value Problems, HU ISSN 1418-7108: HEJ Manuscript no.: ANM-030110-B.
[7]	Okuonghae, R. I. and Ikhile, M. N. O. (2011). A(α)-Stable Linear Multistep Methods. for Stiff IVPs in ODEs Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 50 (1), 73–90.
[8]	Skwame, Y., Sunday, J., Ibijola, E. A. (2012). L-Stable Block Hybrid Simpson’s Methods for Numerical Solution of Initial Value Problems in Stiff Ordinary Differential Equations, International Journal of Pure and Applied Sciences and Technology 11(2):45-54.
[9]	Tahmasbi, A. (2008). Numerical Solutions for Stiff Ordinary Differential Equation Systems, International Mathematical Forum, 3, 2008, no. 15, 703 – 711.
[10]	Yao, N. M., Akinfenwa, O. A., Jator, S. N. (2011). A linear multistep hybrid Method with continuous coefficients for solving stiff ordinary differential equations, Int. J. Comp. Maths. 5(2): 47-53.

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APA Style

Yohanna Sani Awari, Micah Geoffrey Kumleng. (2017). Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. American Journal of Mathematical and Computer Modelling, 2(4), 103-116. https://doi.org/10.11648/j.ajmcm.20170204.13

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ACS Style

Yohanna Sani Awari; Micah Geoffrey Kumleng. Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. Am. J. Math. Comput. Model. 2017, 2(4), 103-116. doi: 10.11648/j.ajmcm.20170204.13

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AMA Style

Yohanna Sani Awari, Micah Geoffrey Kumleng. Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. Am J Math Comput Model. 2017;2(4):103-116. doi: 10.11648/j.ajmcm.20170204.13

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@article{10.11648/j.ajmcm.20170204.13,
  author = {Yohanna Sani Awari and Micah Geoffrey Kumleng},
  title = {Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {4},
  pages = {103-116},
  doi = {10.11648/j.ajmcm.20170204.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170204.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170204.13},
  abstract = {The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulae (BDF). In this paper, we focus on the derivation of the fourth, sixth and eighth order extended trapezoidal rule of first kind (ETRs) formulae through Hermite polynomial as basis function which we named FETR, SETR and EETR respectively. We then interpolate and collocate at some points of interest to generate the desire method. The stability analysis on our methods suggests that they are not only convergent but possess regions suitable for the solution of stiff ordinary differential equations (ODEs). The methods were very efficient when implemented in block form, they tend to perform better over existing methods.},
 year = {2017}
}

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TY  - JOUR
T1  - Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae
AU  - Yohanna Sani Awari
AU  - Micah Geoffrey Kumleng
Y1  - 2017/11/08
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170204.13
DO  - 10.11648/j.ajmcm.20170204.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
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AB  - The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulae (BDF). In this paper, we focus on the derivation of the fourth, sixth and eighth order extended trapezoidal rule of first kind (ETRs) formulae through Hermite polynomial as basis function which we named FETR, SETR and EETR respectively. We then interpolate and collocate at some points of interest to generate the desire method. The stability analysis on our methods suggests that they are not only convergent but possess regions suitable for the solution of stiff ordinary differential equations (ODEs). The methods were very efficient when implemented in block form, they tend to perform better over existing methods.
VL  - 2
IS  - 4
ER  -

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Author Information

Yohanna Sani Awari

Department of Mathematical Sciences, Taraba State University, Jalingo, Nigeria
Micah Geoffrey Kumleng

Department of Mathematics, University of Jos, Jos, Nigeria

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APA Style

Yohanna Sani Awari, Micah Geoffrey Kumleng. (2017). Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. American Journal of Mathematical and Computer Modelling, 2(4), 103-116. https://doi.org/10.11648/j.ajmcm.20170204.13

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ACS Style

Yohanna Sani Awari; Micah Geoffrey Kumleng. Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. Am. J. Math. Comput. Model. 2017, 2(4), 103-116. doi: 10.11648/j.ajmcm.20170204.13

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AMA Style

Yohanna Sani Awari, Micah Geoffrey Kumleng. Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae. Am J Math Comput Model. 2017;2(4):103-116. doi: 10.11648/j.ajmcm.20170204.13

Copy | Download

@article{10.11648/j.ajmcm.20170204.13,
  author = {Yohanna Sani Awari and Micah Geoffrey Kumleng},
  title = {Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {4},
  pages = {103-116},
  doi = {10.11648/j.ajmcm.20170204.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170204.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170204.13},
  abstract = {The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulae (BDF). In this paper, we focus on the derivation of the fourth, sixth and eighth order extended trapezoidal rule of first kind (ETRs) formulae through Hermite polynomial as basis function which we named FETR, SETR and EETR respectively. We then interpolate and collocate at some points of interest to generate the desire method. The stability analysis on our methods suggests that they are not only convergent but possess regions suitable for the solution of stiff ordinary differential equations (ODEs). The methods were very efficient when implemented in block form, they tend to perform better over existing methods.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae
AU  - Yohanna Sani Awari
AU  - Micah Geoffrey Kumleng
Y1  - 2017/11/08
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170204.13
DO  - 10.11648/j.ajmcm.20170204.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 103
EP  - 116
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170204.13
AB  - The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulae (BDF). In this paper, we focus on the derivation of the fourth, sixth and eighth order extended trapezoidal rule of first kind (ETRs) formulae through Hermite polynomial as basis function which we named FETR, SETR and EETR respectively. We then interpolate and collocate at some points of interest to generate the desire method. The stability analysis on our methods suggests that they are not only convergent but possess regions suitable for the solution of stiff ordinary differential equations (ODEs). The methods were very efficient when implemented in block form, they tend to perform better over existing methods.
VL  - 2
IS  - 4
ER  -

Copy | Download