Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 794716, 6 pages
http://dx.doi.org/10.1155/2014/794716
Research Article

Solving Singularly Perturbed Multipantograph Delay Equations Based on the Reproducing Kernel Method

Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China

Received 13 November 2013; Accepted 20 January 2014; Published 27 February 2014

Academic Editor: Valery Y. Glizer

Copyright © 2014 F. Z. Geng and S. P. Qian. 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. G. M. Amiraliyev and F. Erdogan, “Uniform numerical method for singularly perturbed delay differential equations,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1251–1259, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. I. G. Amiraliyeva, F. Erdogan, and G. M. Amiraliyev, “A uniform numerical method for dealing with a singularly perturbed delay initial value problem,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1221–1225, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. M. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of singularly perturbed delay differential equations with layer behavior,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 11–28, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. M. K. Kadalbajoo and V. P. Ramesh, “Hybrid method for numerical solution of singularly perturbed delay differential equations,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 797–814, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. M. K. Kadalbajoo and K. K. Sharma, “A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 692–707, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. M. K. Kadalbajoo and D. Kumar, “Fitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 90–98, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. G. M. Amiraliyev and E. Cimen, “Numerical method for a singularly perturbed convection-diffusion problem with delay,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2351–2359, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. P. Rai and K. K. Sharma, “Numerical analysis of singularly perturbed delay differential turning point problem,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3483–3498, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009. View at MathSciNet
  10. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic, Boston, Mass, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  11. F. Z. Geng and M. Cui, “Solving a nonlinear system of second order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1167–1181, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. F. Z. Geng and M. Cui, “New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 165–172, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. F. Z. Geng, “A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4211–4215, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. F. Z. Geng, S. P. Qian, and S. Li, “A numerical method for singularly perturbed turning point problems with an interior layer,” Journal of Computational and Applied Mathematics, vol. 255, pp. 97–105, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Wu and X. Li, “Iterative reproducing kernel method for nonlinear oscillator with discontinuity,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1301–1304, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. W. Wang, B. Han, and M. Yamamoto, “Inverse heat problem of determining time-dependent source parameter in reproducing kernel space,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 875–887, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Wang, M. Du, F. Tan, Z. Li, and T. Nie, “Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5918–5925, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Inc, A. Akgül, and A. Kiliçman, “A novel method for solving KdV equation based on reproducing kernel Hilbert space method,” Abstract and Applied Analysis, vol. 2013, Article ID 578942, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Mohammadi and R. Mokhtari, “Solving the generalized regularized long wave equation on the basis of a reproducing kernel space,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4003–4014, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. G. Akram and H. U. Rehman, “Numerical solution of eighth order boundary value problems in reproducing kernel space,” Numerical Algorithms, vol. 62, no. 3, pp. 527–540, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. K. Kadalbajoo and P. Arora, “B-spline collocation method for the singular-perturbation problem using artificial viscosity,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 650–663, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. A. Kaushik, V. Kumar, and A. K. Vashishth, “An efficient mixed asymptotic-numerical scheme for singularly perturbed convection diffusion problems,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8645–8658, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. A. Andargie and Y. N. Reddy, “Fitted fourth-order tridiagonal finite difference method for singular perturbation problems,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 90–100, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. M. M. Chawla, “A fourth-order tridiagonal finite difference method for general non-linear two-point boundary value problems with mixed boundary conditions,” IMA Journal of Applied Mathematics, vol. 21, no. 1, pp. 83–93, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. M. M. Chawla and P. N. Shivakumar, “An efficient finite difference method for two-point boundary value problems,” Neural, Parallel & Scientific Computations, vol. 4, no. 3, pp. 387–395, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet