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Numerical Method to Solve the Cauchy Problem with Previous History

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Cybernetics and Systems Analysis Aims and scope

Abstract

The paper analyzes the theoretical aspects of constructing a family of single-stage multi-step methods for solving the Cauchy problem with prehistory for ordinary differential equations. The authors consider general issues related to discretization, approximation, convergence, and stability. The problem of improving the accuracy of numerical solutions is analyzed in detail. The results presented in the paper are also applicable for the numerical solution of partial differential equations.

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References

  1. V. A. Prusov, A. E. Doroshenko, S. V. Prihod’ko, Yu. M. Tyrchak, and R. I. Chernysh, “Methods for efficient modeling and forecasting of regional atmospheric processes,” Problemy Programmir., No. 2, 274–287 (2004).

  2. V. A. Prusov and A. Yu. Doroshenko, Modeling of Natural and Technogenic Processes in the Atmosphere [in Ukrainian], Naukova Dumka, Kyiv (2006).

    Google Scholar 

  3. V. A. Prusov and A. Yu. Doroshenko, “Multistep method of the numerical solution of the problem of modeling the circulation of atmosphere in the Cauchy problem,” Cybern. Syst. Analysis, Vol. 51, No. 4, 547–555 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Prusov and A. Doroshenko, “Modeling and forecasting atmospheric pollution over region,” Annales Univ. Sci. Budapest, Vol. 46, 71–79 (2003).

    MathSciNet  MATH  Google Scholar 

  5. V. Prusov, A. Doroshenko, and Yu. Tyrchak, “Highly efficient methods for regional weather forecasting,” System Research and Inform. Technologies, No. 4, 312–319 (2005).

  6. V. A. Prusov and A. Yu. Doroshenko, “Methods of efficient modeling and forecasting regional atmospheric processes,” in: Advances in Air Pollution Modeling for Environmental Security. NATO Science Series, Vol. 54, Springer, Printed in the Netherlands (2005), pp. 1012–1023.

  7. J. A. Young, “Comparative properties of some time differencing schimes for linear and nonlinear oscillations,” Mon. Wea. Rev., Vol. 96, No. 6, 118–127 (1968).

    Article  Google Scholar 

  8. N. N. Kalitkin, Numerical Methods [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  9. L. Collatz, The Numerical Treatment of Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 60, Springer-Verlag, Berlin–Heidelberg (1960).

    Book  Google Scholar 

  10. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

  11. I. S. Berezin and N. P. Zhidkov, Computing Methods [in Russian], Vol. 2, Fizmatgiz, Moscow (1962).

    MATH  Google Scholar 

  12. A. D. Gorbunov, Finite Difference Methods to Solve the Cauchy Problem for the System of Ordinary Differential Equations [in Russian], Izd. MGU, Moscow (1973).

    Google Scholar 

  13. S. F. Zaletkin, “Numerical solution of the Cauchy problem for ordinary linear homogeneous differential equations on large integration intervals,” Vych. Metody i Programmirovanie, Vol. XXVI, 27–41 (1977).

  14. V. I. Krylov, V. V. Bobkov, and P. I. Monastyrskii, Computing Methods [in Russian], Vol. 2, Nauka, Moscow (1976).

    Google Scholar 

  15. W. E. Milne, Numerical Solution of Differential Equations, Dover Pubns. (1970).

  16. J. M. Ortega and W. G Pool, An Introduction to Numerical Methods for Differential Equations, Pitman Publ. (1981).

  17. G. Hall and J.M. Watt (eds.), Modern Numerical Methods for Ordinary Differential Equations, Oxford University Press (1976).

  18. R.W. Hamming, Numerical Methods for Scientists and Engineers, Dover Books on Mathematics, Dover Publ. (1987).

  19. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971), pp. 146–156.

    MATH  Google Scholar 

  20. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, J. Wiley & Sons, New York–London (1962).

    MATH  Google Scholar 

  21. L. F. Shampine and M.K. Gordon, Computer Solution of Ordinary Differential Equations, W. H. Freeman, San-Francisco (1975).

    MATH  Google Scholar 

  22. H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Ser. Springer Tracts in Natural Philosophy, Vol. 23, Springer-Verlag, Berlin–Heidelberg (1973).

  23. A. G. Kurosh, A Course of Higher Algebra [in Russian], Nauka, Moscow (1975).

    Google Scholar 

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    V. A. Prusov

  2. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    A. Yu. Doroshenko

Authors
  1. V. A. Prusov
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  2. A. Yu. Doroshenko
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Correspondence to V. A. Prusov.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2017, pp. 42–67.

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Prusov, V.A., Doroshenko, A.Y. Numerical Method to Solve the Cauchy Problem with Previous History. Cybern Syst Anal 53, 34–56 (2017). https://doi.org/10.1007/s10559-017-9905-y

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  • DOI: https://doi.org/10.1007/s10559-017-9905-y

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