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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 579431, 10 pages
http://dx.doi.org/10.1155/2012/579431
Research Article

Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method

1Department of Computer Engineering, Faculty of Technology, Gazi University, Teknikokullar, 06500 Ankara, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayıs University, 55139 Samsun, Turkey
3Department of Mathematics, Faculty of Arts and Sciences, TOBB University of Economics and Technology, Söğütözü, 06530 Ankara, Turkey

Received 13 March 2012; Accepted 24 March 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 Nurettin Doğan 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. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, NY, USA, 1978. View at Zentralblatt MATH
  2. J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, vol. 34 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1981. View at Zentralblatt MATH
  3. R. E. O'Malley Jr., Introduction to Singular Perturbations, Applied Mathematics and Mechanics, Vol. 14, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH
  4. C. Liu, “The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1506–1521, 2012. View at Google Scholar
  5. Y. Wang, L. Su, X. Cao, and X. Li, “Using reproducing kernel for solving a class of singularly perturbed problems,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 421–430, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. K. Kadalbajoo and P. Arora, “B-splines with artificial viscosity for solving singularly perturbed boundary value problems,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 654–666, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. K. Kadalbajoo and D. Kumar, “Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme,” Computers & Mathematics with Applications, vol. 57, no. 7, pp. 1147–1156, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. R. K. Mohanty and U. Arora, “A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 531–544, 2006. View at Publisher · View at Google Scholar
  9. R. K. Mohanty and N. Jha, “A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems,” Applied Mathematics and Computation, vol. 168, no. 1, pp. 704–716, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Evrenosoglu and S. Somali, “Least squares methods for solving singularly perturbed two-point boundary value problems using Bézier control points,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1029–1032, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. G. E. Pukhov, Differential transformations and mathematical modelling of physical processes, Naukova Dumka, Kiev, Ukraine, 1986.
  12. A. Gökdoğan, M. Merdan, and A. Yildirim, “The modified algorithm for the differential transform method to solution of Genesio systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 45–51, 2012. View at Publisher · View at Google Scholar
  13. A. K. Alomari, “A new analytic solution for fractional chaotic dynamical systems using the differential transform method,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2528–2534, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. Thongmoon and S. Pusjuso, “The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 425–431, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S.-H. Chang and I.-L. Chang, “A new algorithm for calculating one-dimensional differential transform of nonlinear functions,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 799–805, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. H. Liu and Y. Song, “Differential transform method applied to high index differential-algebraic equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 748–753, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. H. Liu and Y. Song, “Differential transform method applied to high index differential-algebraic equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 748–753, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. N. Doǧan, V. S. Ertürk, S. Momani, Ö. Akin, and A. Yildirim, “Differential transform method for solving singularly perturbed Volterra integral equations,” Journal of King Saud University - Science, vol. 23, pp. 223–228, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. A. S. V. Ravi Kanth and K. Aruna, “Solution of singular two-point boundary value problems using differential transformation method,” Physics Letters A, vol. 372, no. 26, pp. 4671–4673, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. V. S. Ertürk and S. Momani, “Comparing numerical methods for solving fourth-order boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1963–1968, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. Sari, “Differential quadrature method for singularly perturbed two-point boundary value problems,” Journal of Applied Sciences, vol. 8, no. 6, pp. 1091–1096, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. M. Mokarram Shahraki and S. Mohammad Hosseini, “Comparison of a higher order method and the simple upwind and non-monotone methods for singularly perturbed boundary value problems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 460–473, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. K. Kadalbajoo and K. C. Patidar, “Numerical solution of singularly perturbed two-point boundary value problems by spline in tension,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 299–320, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. Lorenz, “Combinations of initial and boundary value methods for a class of singular perturbation problems,” in Numerical Analysis of Singular Perturbation Problems, pp. 295–315, Academic Press, London, UK, 1979. View at Google Scholar · View at Zentralblatt MATH