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Dissipativity of Multistep Runge–Kutta Methods for Nonlinear Neutral Delay Integro Differential Equations with Constrained Grid

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Cybernetics and Mathematics Applications in Intelligent Systems (CSOC 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 574))

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Abstract

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations. We investigate the dissipativity properties of \( (k,l) \)-algebraically stable multistep Runge-Kutta methods with constrained grid. The finite-dimensional and infinite-dimensional dissipativity results of \( (k,l) \)-algebraically stable multistep Runge-Kutta methods are obtained.

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References

  1. Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences with delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Xiao, A.G.: On the solvability of general linear methods for dissipative dynamical systems. J. Comput. Math. 18, 633–638 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Humphries, A.R., Stuart, A.M.: Model problems in numerical stability theory for initial value problems. SIAM Rev. 36, 226–257 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Humphries, A.R., Stuart, A.M.: Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal. 31, 1452–1485 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hill, A.T.: Global dissipativity for A-stable methods. SIAM J. Numer. Anal. 34, 119–142 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hill, A.T.: Dissipativity of Runge-Kutta methods in Hilbert spaces. BIT 37, 37–42 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xiao, A.G.: Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces. Math. Numer. Sin. 22, 429–436 (2000). (in Chinese)

    MathSciNet  Google Scholar 

  8. Huang, C.M.: Dissipativity of Runge-Kutta methods for dissipative systems with delays. IMA J. Numer. Anal. 20, 153–166 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, C.M., Chen, G.: Dissipativity of linear-methods for delay dynamical systems. Math. Numer. Sin. 22, 501–506 (2000). (in Chinese)

    Google Scholar 

  10. Huang, C.M.: Dissipativity of one-leg methods for dissipative systems with delays. Appl. Numer. Math. 35, 11–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, C.M.: Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays. Math. Comput. Model. 40, 1285–1296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Tian, H.J.: Numerical and analytic dissipativity of the θ-method for delay differential equations with a bounded lag. Int. J. Bifurcation Chaos, 1839–1845 (2004)

    Google Scholar 

  13. Wen, L.P.: Numerical stability analysis for nonlinear Volterra functional differential equations in abstract spaces, Ph.D. thesis, Xiangtan University (2005). (in Chinese)

    Google Scholar 

  14. Gan, S.Q.: Dissipativity of linear θ-methods for integro-differential equations. Comput. Math Appl. 52, 449–458 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gan, S.Q.: Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J. Comput. Appl. Math. 206, 898–907 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gan, S.Q.: Exact and discretized dissipativity of the pantograph equation. J. Comput. Math. 25(1), 81–88 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Burrage, K., Butcher, J.C.: Nonlinear stability of a general class of differential equation methods. BIT 20, 185–203 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, S.F.: Theory of computational methods for stiff differential equation. Huan Science and Technology Publisher, Changsha (1997)

    Google Scholar 

  19. Cheng, Z., Huang, C.M.: Dissipativity for nonlinear neutral delay differential equations. J. Syst. Simul. 19(14), 3184–3187 (2007)

    Google Scholar 

  20. Wen, L.P., Wang, W.S., Yu, Y.X.: Dissipativity of θ-methods for a class of nonlinear neutral delay differential equations. Appl. Math. Comput. 202(2), 780–786 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Wang, W.S., Li, S.F.: Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay. Appl. Math. Lett. 21(9), 983–991 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, S.F., Gan, S.Q.: Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput. Math Appl. 55, 2426–2443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qi, R., Zhang, C.J., Zhang, Y.J.: Dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. Acta Math. Applicatae Sinica 28(2), 225–236 (2012). English Series

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work were supported by the Creative Talent Project Foundation of Heilongjiang Province Education Department (UNPYSCT-2015102).

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Authors and Affiliations

  1. Department of Mathematics, Heilongjiang Institute of Technology, Harbin, 150050, China

    Haiyan Yuan

  2. Harbin Institute of Technology, College of Management, Harbin, 150001, China

    Cheng Song

Authors
  1. Haiyan Yuan
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  2. Cheng Song
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Corresponding author

Correspondence to Haiyan Yuan .

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Editors and Affiliations

  1. Faculty of Applied Informatics, Tomas Bata University in Zlín Faculty of Applied Informatics, Zlín, Czech Republic

    Radek Silhavy

  2. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Roman Senkerik

  3. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zuzana Kominkova Oplatkova

  4. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zdenka Prokopova

  5. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Petr Silhavy

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Yuan, H., Song, C. (2017). Dissipativity of Multistep Runge–Kutta Methods for Nonlinear Neutral Delay Integro Differential Equations with Constrained Grid. In: Silhavy, R., Senkerik, R., Kominkova Oplatkova, Z., Prokopova, Z., Silhavy, P. (eds) Cybernetics and Mathematics Applications in Intelligent Systems. CSOC 2017. Advances in Intelligent Systems and Computing, vol 574. Springer, Cham. https://doi.org/10.1007/978-3-319-57264-2_4

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  • DOI: https://doi.org/10.1007/978-3-319-57264-2_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57263-5

  • Online ISBN: 978-3-319-57264-2

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