About this Journal Submit a Manuscript Table of Contents 
Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 989381, 11 pages
http://dx.doi.org/10.1155/2013/989381
Research Article

A Numerical Algorithm for Solving Stiff Ordinary Differential Equations

S. A. M. Yatim,1 Z. B. Ibrahim,2 K. I. Othman,3 and M. B. Suleiman2

1School of Distance Education, Universiti Sains Malaysia, 11800 Penang, Malaysia
2Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Selangor, 43400 Serdang, Malaysia
3Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Selangor, 40450 Shah Alam, Malaysia

Received 6 August 2012; Revised 24 October 2012; Accepted 7 November 2012

Academic Editor: J. Rodellar

Copyright © 2013 S. A. M. Yatim et al. This 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.

Linked References

  1. W. H. Enright, T. E. Hull, and B. Lindberg, “Comparing numerical methods for stiff systems of O.D.E:s,” BIT, vol. 15, no. 1, pp. 10–48, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. G. D. Byrne, A. C. Hindmarsh, K. R. Jackson, and H. G. Brown, “A comparison of two ode codes: gear and episode,” Computers and Chemical Engineering, vol. 1, no. 2, pp. 125–131, 1977. View at Scopus
  3. R. Sacks-Davis, “Fixed leading coefficient implementation of sd-formulas for stiff ODES,” ACM Transactions on Mathematical Software, vol. 6, no. 4, pp. 540–562, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. A. S. Mahmood, L. Casasús, and W. Al-Hayani, “The decomposition method for stiff systems of ordinary differential equations,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 964–975, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. J. Ibáñez, V. Hernández, E. Arias, and P. A. Ruiz, “Solving initial value problems for ordinary differential equations by two approaches: BDF and piecewise-linearized methods,” Computer Physics Communications, vol. 180, no. 5, pp. 712–723, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. H. Aminikhah and J. Biazar, “A new analytical method for system of ODEs,” Numerical Methods for Partial Differential Equations, vol. 26, no. 5, pp. 1115–1124, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. S. Abelman and K. C. Patidar, “Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations,” Computers and Mathematics with Applications, vol. 55, no. 4, pp. 733–744, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. L. F. Shampine and M. W. Reichelt, “The MATLAB ode suite,” SIAM Journal on Scientific Computing, vol. 18, no. 1, pp. 1–22, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. L. F. Shampine, M. W. Reichelt, and J. A. Kierzenka, “Solving index-I DAEs in MATLAB and simulink,” SIAM Review, vol. 41, no. 3, pp. 538–552, 1999. View at Publisher · View at Google Scholar · View at Scopus
  10. T. Kato and K. Ikeuchi, “Variable order and variable step-size integration method for transient analysis programs,” IEEE Transactions on Power Systems, vol. 6, no. 2, pp. 206–213, 1991. View at Publisher · View at Google Scholar · View at Scopus
  11. Z. B. Ibrahim, M. Suleiman, and K. I. Othman, “Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations,” European Journal of Scientific Research, vol. 21, no. 3, pp. 508–520, 2008. View at Scopus
  12. Z. B. Ibrahim, K. I. Othman, and M. B. Suleiman, “Variable step size block backward differentiation formula for solving stiff odes,” in Proceedings of the World Congress on Engineering, vol. 2, pp. 785–789, London, UK, 2007.
  13. S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, and F. Ismail, “Fifth order variable step block backward differentiation formulae for solving stiff ODEs,” Proceedings of World Academy of Science, Engineering and Technology, vol. 62, pp. 998–1000, 2010. View at Scopus
  14. S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, and M. B. Suleiman, “Quantitative comparison of numerical method for solving stiff ordinary differential equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 193691, 12 pages, 2011. View at Publisher · View at Google Scholar
  15. J. D. Lambert, Computational Methods in Ordinary Differential Equation, John Wiley & Sons, New York, NY, USA, 1991.
  16. K. R. Jackson, “The numerical solution of large systems of stiff IVPs for ODEs,” Applied Numerical Mathematics, vol. 20, no. 1-2, pp. 5–20, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. G. Hall and J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, UK, 1976.
  18. P. Kaps and G. Wanner, “A study of Rosenbrock-type methods of high order,” Numerische Mathematik, vol. 38, no. 2, pp. 279–298, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus