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Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae
American Journal of Mathematical and Computer Modelling
Volume 2, Issue 4, November 2017, Pages: 103-116
Received: Sep. 18, 2017; Accepted: Oct. 9, 2017; Published: Nov. 8, 2017
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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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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.
Stiffness, Hermite Polynomial, ETRs, A-Stability, Ordinary Differential Equations
To cite this article
Yohanna Sani Awari, Micah Geoffrey Kumleng, Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae, American Journal of Mathematical and Computer Modelling. Vol. 2, No. 4, 2017, pp. 103-116. doi: 10.11648/j.ajmcm.20170204.13
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Awari, Y. S. (2017). A-Stable Block ETRs/ETR2s Methods for Stiff ODEs, MAYFEB Journal of Mathematics Vol. 4: Pages 33-52.
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).
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.
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.
Khadijah, M. Abualnaja, (2015). A Block Procedure with Linear Multistep Methods Using Legendre Polynomials for Solving ODEs, Applied Mathematics 6:717-723.
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.
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.
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.
Tahmasbi, A. (2008). Numerical Solutions for Stiff Ordinary Differential Equation Systems, International Mathematical Forum, 3, 2008, no. 15, 703 – 711.
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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