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Numerical Stability of the Runge-Kutta Methods for Equations \(u'(t)=au(t)+bu([\frac{K}{N}t])\) in Science Computation

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Social Computing (ICYCSEE 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 623))

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Abstract

Differential equation has widely applied in science and engineering calculation. Runge Kutta method is a main method for solving differential equations. In this paper, the numerical properties of Runge-Kutta methods for the equation \(u'(t)=au(t)+bu([\frac{K}{N}t])\) is dealed with, where K and N is relatively prime and \(K<N, K, N\in \mathbb {Z}^{+}.\) The conditions are obtained under which the numerical solutions preserve the analytical stability properties of the analytic ones and some numerical experiments are given.

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References

  1. Liu, M.Z., Song, M.H., Yang, Z.W.: Stability of Runge-Kutta methods in the numerical solution of equation \(u^{\prime }(t)=au(t)+bu([t])\). J. Comput. Appl. Math. 166, 361–370 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cooke, K.L., Wiener, J.: Retarded differential equation with piecewise constant delays. J. Math. Anal. Appl. 99, 265–297 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Shah, S.M., Wiener, J.: Advanced differetial equations with piecewise constant argument deviations. Int. J. Math. Sci. 6, 671–703 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Liu, M.Z., Ma, S.F., Yang, Z.W.: Stability analysis of Runge-Kutta methods for unbounded retarded differetial equations with piecewise continuous arguments. Comput. Math. Appl. 191, 57–66 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Lv, W.J., Yang, Z.W., Liu, M.Z.: Stability of the Euler-Maclaurin methods for neutral differential equations with piecewise continuous. Comput. Math. Appl. 186, 1480–1487 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Lv, W.J., Yang, Z.W., Liu, M.Z.: Stability of Runge-Kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments. Comput. Math. Appl. 54, 326–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liang, H., Liu, M.Z., Lv, W.J.: Stability of \(\theta \)-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett. 23, 198–206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang, Z.W., Liu, M.Z., Nieto, J.: Runge-Kutta methods for first-order periodic boundary value differetial equations with piecewise constant argument. J. Comput. Appl. Math. 233, 990–1004 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Song, M.H., Yang, Z.W., Liu, M.Z.: Stability of \(\theta \)-methods advanced differetial equations with piecewise continuous argument. Comput. Math. Appl. 49, 1205–1301 (2005)

    MathSciNet  Google Scholar 

  10. Wanner, G., Hairer, E., Nørsett, S.P.: Order stars and stability therems. BIT 18, 475–489 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wiener, J.: Differetial equations with piecewise continuous delays. In: Lakshmikantham, V. (ed.) Trends in the Theory and Practice of Nonlinear Differetial Equations, pp. 547–552. Marcel Dekker, New York (1983)

    Google Scholar 

  12. Wiener, J.: Generalised Solutions of Differetial Equations. World Scientific, Singapore (1993)

    Google Scholar 

  13. Iserles, A., Nørsett, S.P.: Order stars and rational approximations to EXP(Z). Appl. Numer. Math. 5, 63–70 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bucher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. Wiley, New York (1987)

    Google Scholar 

  15. Dekker, K., Verwer, J.G.: Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, Amsterdam (1984)

    Google Scholar 

  16. Busenberg, S., Cooke, K.L.: Models of vertically transmitted diseases with sequential-continuous dynamics. In: Lakshmikantham, V. (ed.) Nonlinear phenomena in Mathematical Sciences. Academic Press, New York (1982)

    Google Scholar 

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Acknowledgments

This work is supported by the Research Fund of the Natural Science Foundation of Heilongjiang Province (No. A201214) and the National Natural Science Foundation of China(61501148).

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Authors and Affiliations

  1. School of Applied Science, Harbin University of Science and Technology, Harbin, 150080, China

    Yingchun Song & Xianhua Song

Authors
  1. Yingchun Song
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  2. Xianhua Song
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Corresponding author

Correspondence to Yingchun Song .

Editor information

Editors and Affiliations

  1. Harbin Institute of Technology , Harbin, China

    Wanxiang Che

  2. Harbin Engineering University , Harbin, China

    Qilong Han

  3. Harbin Institute of Technology , Harbin, China

    Hongzhi Wang

  4. Northeast Forestry University , Harbin, China

    Weipeng Jing

  5. National University of Defense Technology , Changsha, China

    Shaoliang Peng

  6. Harbin Engineering University , Harbin, China

    Junyu Lin

  7. Harbin Univ. of Science and Technology , Harbin, China

    Guanglu Sun

  8. Harbin Univ. of Science and Technology , Harbin, China

    Xianhua Song

  9. Harbin Engineering University , Harbin, China

    Hongtao Song

  10. Harbin Sea of Clouds & Computer Tech. , Harbin, China

    Zeguang Lu

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Song, Y., Song, X. (2016). Numerical Stability of the Runge-Kutta Methods for Equations \(u'(t)=au(t)+bu([\frac{K}{N}t])\) in Science Computation. In: Che, W., et al. Social Computing. ICYCSEE 2016. Communications in Computer and Information Science, vol 623. Springer, Singapore. https://doi.org/10.1007/978-981-10-2053-7_45

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  • DOI: https://doi.org/10.1007/978-981-10-2053-7_45

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2052-0

  • Online ISBN: 978-981-10-2053-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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