Computing

, Volume 90, Issue 1–2, pp 57–71 | Cite as

The extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations

Article

Abstract

This paper deals with the extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. Nonlinear stability and numerical implementation of the methods are investigated. It is proven under the suitable conditions that the extended Pouzet–Runge–Kutta methods are globally and asymptotically stable for problems of the class \({\mathbb{NRI}{(\alpha,\beta,\gamma,\nu)}}\). Numerical examples further illustrate the theoretical results and the methods’ effectiveness.

Keywords

Neutral delay-integro-differential equations Nonlinear stability Pouzet–Runge–Kutta methods 

Mathematics Subject Classification (2000)

65L20 65L06 65R20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bellen A, Guglielmi N, Zennaro M (1999) On the contractivity and asymptotic stability of systems of delay differential equations of neutral type. Numer Math 39: 1–24MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brunner H, Vermiglio R (2003) Stability of solutions of delay functional integro-differential equations and their discretizations. Computing 71: 229–245MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brunner H (2004) Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  4. 4.
    Brunner H, van der Houwen PJ (1986) The numerical solution of Volterra equations. North-Holland, AmsterdamMATHGoogle Scholar
  5. 5.
    Burrage K, Butcher JC (1980) Nonlinear stability of a general class of differential equations methods. BIT 20: 185–203MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hairer E, Wanner G (1996) Solving ordinary differential equations II: stiff and differential-algebraic problems. Springer, BerlinMATHGoogle Scholar
  7. 7.
    Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  8. 8.
    Huang C, Fu H, Li S, Chen G (1999) Stability analysis of Runge–Kutta methods for non-linear delay differential equations. BIT 39: 270–280MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jin J (2007) A numerical solution for neutral delay integro-differential equations with Volterra type. Math Appl 20(supplement):31–33Google Scholar
  10. 10.
    Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations. Kluwer, DordrechtMATHGoogle Scholar
  11. 11.
    Vermiglio R, Torelli L (2003) A stable numerical approach for implicit non-linear neutral delay differential equations. BIT 43: 195–215MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Yuexin Y, Wen L, Li S (2007) Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations. Appl Math Comput 191: 543–549MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yuexin Y, Li S (2007) Stability analysis of Runge–Kutta methods for nonlinear neutral delay integro-differential equations. Sci China (Ser A) 50: 464–474CrossRefMathSciNetGoogle Scholar
  14. 14.
    Zennaro M (1997) Asymptotic stability analysis of Runge–Kutta methods for nonlinear systems of delay differential equations. Numer Math 77: 549–563MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zhang C, He Y (2009) The extended one-leg methods for nonlinear neutral delay-integro-differential equations. Appl Numer Math 59: 1409–1418MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Zhang C, Vandewalle S (2004) Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal 24: 193–214CrossRefMathSciNetGoogle Scholar
  17. 17.
    Zhang C, Vandewalle S (2006) General linear methods for Volterra integro-differential equations with memory. SIAM J Sci Comput 27: 2010–2031MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhang C, Qin T, Jin J (2009) An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations. Appl Math Comput 215: 548–556MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations