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Analysis of systems of singular ordinary differential equations

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Abstract

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones.

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References

  1. E. P. Doolan, J. Miller, and W. S. Childers, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980).

    MATH  Google Scholar 

  2. N. S. Bakhvalov,“Optimization of methods of solution of boundary-value problems in the presence of a boundary layer,” Zh. Vychisl. Mat. Mat. Fiz.,9, No. 4, 841–859 (1969).

    MATH  Google Scholar 

  3. Yu. P. Boglayev,“Numerical methods of solution of singularly perturbed problems,” Diff. Uravn..21, No. 10, 1804–1806 (1985).

    Google Scholar 

  4. A. H. Nayfeh, Perturbation Methods [Russian translation], Mir, Moscow (1976).

    Google Scholar 

  5. A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  6. A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations [in Russian], Vysshaya Shkola, Moscow (1990).

    MATH  Google Scholar 

  7. D. Greenspan,“A new explicit discrete mechanics with applications,” J. Franklin Inst.,294, No. 4, 231–240 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  9. V. A. Pliss, Nonlocal Problems of Vibration Theory [in Russian], Nauka, Moscow (1964).

    Google Scholar 

  10. N. N. Krasovskii, Some Problems of the Theory of Stability of Motion [in Russian], Gos. Izd. Fiz. Mat. Lit. (1959).

  11. I. M. Romanishin and L.A. Sinitskii,“Analysis of systems with slow change of the parameters of the input actions,” Teor. Elektrotekh.,51, 72–78 (1992).

    Google Scholar 

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Authors and Affiliations

  1. I. Franko State University, Lvov, Ukraine

    V. V. Grigorenko & L. A. Sinitskii

  2. Physicomechanical Institute of the National Academy of Sciences of Ukraine, Lvov, Ukraine

    I. M. Romanishin

Authors
  1. V. V. Grigorenko
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  2. I. M. Romanishin
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  3. L. A. Sinitskii
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Additional information

Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.

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Grigorenko, V.V., Romanishin, I.M. & Sinitskii, L.A. Analysis of systems of singular ordinary differential equations. Cybern Syst Anal 35, 769–776 (1999). https://doi.org/10.1007/BF02733411

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  • DOI: https://doi.org/10.1007/BF02733411

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