Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 607827, 9 pages
http://dx.doi.org/10.1155/2014/607827
Research Article

Stability Analysis of Runge-Kutta Methods for Nonlinear Functional Differential and Functional Equations

Department of Mathematics, Xiangtan University, Xiangtan 411105, China

Received 5 September 2013; Revised 15 April 2014; Accepted 18 April 2014; Published 6 May 2014

Academic Editor: Mehmet Sezer

Copyright © 2014 Yuexin Yu 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. V. Kolmanovskii and A. Myshkis, Applied Theory of Functional-Differential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  2. P. Pepe, Z.-P. Jiang, and E. Fridman, “A new Lyapunov-Krasovskii methodology for coupled delay differential and difference equations,” International Journal of Control, vol. 81, no. 1, pp. 107–115, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. P. Pepe and E. I. Verriest, “On the stability of coupled delay differential and continuous time differences equations,” IEEE Transactions on Automatic Control, vol. 48, no. 8, pp. 1422–1427, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. I. Karafyllis, P. Pepe, and Z.-P. Jiang, “Stability results for systems described by coupled retarded functional differential equations and functional difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3339–3362, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. H. Li and K. Gu, “Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels,” Automatica, vol. 46, no. 5, pp. 902–909, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. Y.-K. Liu, “Runge-Kutta-collocation methods for systems of functional-differential and functional equations,” Advances in Computational Mathematics, vol. 11, no. 4, pp. 315–329, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. A. Bellen, Z. Jackiewicz, and M. Zennaro, “Stability analysis of one-step methods for neutral delay-differential equations,” Numerische Mathematik, vol. 52, no. 6, pp. 605–619, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. T. Koto, “A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations,” BIT Numerical Mathematics, vol. 36, no. 4, pp. 855–859, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. C.-J. Zhang and S.-Z. Zhou, “The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations,” Science in China A: Mathematics, vol. 41, no. 11, pp. 1151–1157, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. L. Qiu, B. Yang, and J.-X. Kuang, “The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations,” Numerische Mathematik, vol. 81, no. 3, pp. 451–459, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. C.-M. Huang and Q.-S. Chang, “Linear stability of general linear methods for systems of neutral delay differential equations,” Applied Mathematics Letters, vol. 14, no. 8, pp. 1017–1021, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. Y.-K. 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
  14. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. C.-J. Zhang, “Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations,” Journal of Computational Mathematics, vol. 20, no. 6, pp. 583–590, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. Y.-X. Yu, L.-P. Wen, and S.-F. Li, “Stability analysis of general linear methods for nonlinear neutral delay differential equations,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1389–1398, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. W.-S. Wang, Y. Zhang, and S.-F. Li, “Nonlinear stability of one-leg methods for delay differential equations of neutral type,” Applied Numerical Mathematics, vol. 58, no. 2, pp. 122–130, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. W.-S. Wang, S.-F. Li, and K. Su, “Nonlinear stability of general linear methods for neutral delay differential equations,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 592–601, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. C.-M. Huang and Q.-S. Chang, “Stability analysis of numerical methods for systems of functional-differential and functional equations,” Computers & Mathematics with Applications, vol. 44, no. 5-6, pp. 717–729, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. S.-Q. Gan and W.-M. Zheng, “Stability of multistep Runge-Kutta methods for systems of functional-differential and functional equations,” Applied Mathematics Letters, vol. 17, no. 5, pp. 585–590, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. S.-Q. Gan, “Asymptotic stability of Rosenbrock methods for systems of functional differential and functional equations,” Mathematical and Computer Modelling, vol. 44, no. 1-2, pp. 144–150, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. Y.-X. Yu and S.-F. Li, “Stability analysis of nonlinear functional differential and functional equations,” Applied Mathematics Letters, vol. 22, no. 5, pp. 787–791, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. Y.-X. Yu and L.-P. Wen, “Stability analysis of one-leg methods for nonlinear functional differential and functional equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 817–824, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, The Netherlands, 1984. View at MathSciNet
  27. S. F. Li, Theory of Computational Methods for Stiff Differential Equations, Hunan Science and Technology Press, Changsha, China, 1997, (Chinese).
  28. K. Burrage and J. C. Butcher, “Non-linear stability of a general class of differential equation methods,” BIT Numerical Mathematics, vol. 20, no. 2, pp. 185–203, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus