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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 325364, 9 pages
http://dx.doi.org/10.1155/2015/325364
Research Article

Stability Analysis of One-Leg Methods for Nonlinear Neutral Delay Integrodifferential Equations

Department of Mathematics, Xiangtan University, Xiangtan 411105, China

Received 8 April 2015; Revised 16 June 2015; Accepted 24 June 2015

Academic Editor: Antonia Vecchio

Copyright © 2015 Yuexin Yu and Liping Wen. 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. A. Bocharov and F. A. Rihan, “Numerical modelling in biosciences using delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 183–199, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springe, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. J. Zhao, Y. Xu, and M. Z. Liu, “Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1062–1079, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. Y. Xu and J. J. Zhao, “Stability of Runge-Kutta methods for neutral delay-integro-differential-algebraic system,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 571–583, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. C. Zhang and S. Vandewalle, “Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations,” Advances in Computational Mathematics, vol. 28, no. 4, pp. 383–399, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. S. F. Wu and S. Q. Gan, “Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2426–2443, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. Y.-X. Yu and S.-F. Li, “Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations,” Science in China—Series A: Mathematics, vol. 50, no. 4, pp. 464–474, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. P. Hu and C. M. Huang, “Analytical and numerical stability of nonlinear neutral delay integro-differential equations,” Journal of the Franklin Institute, vol. 348, no. 6, pp. 1082–1100, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. Y. X. Yu, L. P. Wen, and S. F. Li, “Nonlinear stability of Runge-Kutta methods for neutral delay integro-differential equations,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 543–549, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. C. J. Zhang, T. T. Qin, and J. Jin, “An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 548–556, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. C. J. Zhang and Y. Y. He, “The extended one-leg methods for nonlinear neutral delay-integro-differential equations,” Applied Numerical Mathematics, vol. 59, no. 6, pp. 1409–1418, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. T. Koto, “Stability of Runge-Kutta methods for delay integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 145, no. 2, pp. 483–492, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Brunner and R. Vermiglio, “Stability of solutions of delay functional integro-differential equations and their discretizations,” Computing. Archives for Scientific Computing, vol. 71, no. 3, pp. 229–245, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. C. J. Zhang and S. Vandewalle, “Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations,” IMA Journal of Numerical Analysis, vol. 24, no. 2, pp. 193–214, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. J. Zhang and S. Vandewalle, “General linear methods for Volterra integro-differential equations with memory,” SIAM Journal on Scientific Computing, vol. 27, no. 6, pp. 2010–2031, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. C. M. Huang, “Stability of linear multistep methods for delay integro-differential equations,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2830–2838, 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. G.-D. Hu and T. Mitsui, “Stability analysis of numerical methods for systems of neutral delay-differential equations,” BIT Numerical Mathematics, vol. 35, no. 4, pp. 504–515, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Y. Liu, “Numerical solution of implicit neutral functional-differential equations,” SIAM Journal on Numerical Analysis, vol. 36, no. 2, pp. 516–528, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. A. Bellen, N. Guglielmi, and M. Zennaro, “Numerical stability of nonlinear delay differential equations of neutral type,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 251–263, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. R. Vermiglio and L. Torelli, “A stable numerical approach for implicit non-linear neutral delay differential equations,” BIT Numerical Mathematics, vol. 43, no. 1, pp. 195–215, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. W.-S. Wang, S.-F. Li, and K. Su, “Nonlinear stability of Runge-Kutta methods for neutral delay differential equations,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 175–185, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. L. Torelli, “Stability of numerical methods for delay differential equations,” Journal of Computational and Applied Mathematics, vol. 25, no. 1, pp. 15–26, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. A. Bellen and M. Zennaro, “Strong contractivity properties of numerical methods for ordinary and delay differential equations,” Applied Numerical Mathematics, vol. 9, no. 3–5, pp. 321–346, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. K. J. In 't Hout, “Stability analysis of Runge-Kutta methods for systems of delay differential equations,” IMA Journal of Numerical Analysis, vol. 17, no. 1, pp. 17–27, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. Zennaro, “Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations,” Numerische Mathematik, vol. 77, no. 4, pp. 549–563, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. C. M. Huang, S. F. Li, H. Y. Fu, and G. N. Chen, “Stability and error analysis of one-leg methods for nonlinear delay differential equations,” Journal of Computational and Applied Mathematics, vol. 103, no. 2, pp. 263–279, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Clarendon Press, Oxford, UK, 2003.
  29. G. Dahlquist, “G-stability is equivalent to A-stability,” BIT Numerical Mathematics, vol. 18, no. 4, pp. 384–401, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus