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Construction of asymptotic solutions of linear singularly perturbed systems of second order with degeneracy

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Ukrainian Mathematical Journal Aims and scope

Abstract

The paper discusses the asymptotic behavior of the general solution of a linear singularly perturbed system

$$\varepsilon ^{2h} A(t,\varepsilon )\frac{{d^2 x}}{{dt^2 }} + \varepsilon ^h B(t,\varepsilon )\frac{{dx}}{{dt^2 }} + C(t,\varepsilon )x = 0,$$

, where x∈Rn, t∈t0; T], h∈N, ε∈ (0, ε0], ε≪1 and det A(t, 0) ≡ 0. It is assumed that the quadratic pencil of matrices C(t, 0) + λB(t, 0) + λ2A(t, 0) is regular and has either simple “finite” and “infinite” elementary divisors or just one multiple elementary divisor.

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Literature cited

  1. S. F. Feshchenko, “Small oscillations of a system with finitely many degrees of freedom,” Nauk. Zap. Kiiv. Ped. Inst., Fiz.-Mat. Ser.,9, No. 4, 99–155 (1949).

    Google Scholar 

  2. S. F. Feshchenko, N. I. Shkil', and L. D. Nikolenko, Asymptotic Methods in the Theory of Linear Differential Equations [in Russian], Naukova Dumka, Moscow (1966).

    Google Scholar 

  3. N. I. Shkil' and Z. Shamanov, “On the asymptotic solution of a system of linear differential equations of second order,” in: Approximate Methods of Mathematical Analysis [in Russian], Kiev. Ped. Inst., Kiev (1978), pp. 137–146.

    Google Scholar 

  4. N. I. Shkil' and T. K. Meiliev, “On asymptotic representation of solutions of a system of linear differential equations of second order with a small parameter as coefficient of a derivative of fractional rank,” Dokl. Akad. Nauk UkrSSR, Ser. A, No. 4, 264–267 (1979).

    Google Scholar 

  5. N. I. Shkil' and I. M. Konet, “Asymptotic properties of formal fundamental matrices of systems of linear differential equations of second order containing a parameter,” Ukr. Mat. Zh.,35, No. 1, 124–130 (1983).

    Google Scholar 

  6. V. P. Yakovets, “Asymptotic integration of linear systems of partial differential equations of second order with slowly varying coefficients,” Author's Abstract of Dissertation, Candidate of Phys.-Math. Sciences, Kiev (1983).

  7. N. I. Shkil', I. I. Starun, and V. P. Yakovets, Asymptotic Integration of Linear Systems of Ordinary Differential Equations [in Russian], Vishcha Shkola, Kiev (1989).

    Google Scholar 

  8. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  9. P. Lancaster and J. Maroulas, “Inverse eigenvalue problems for damped vibrating systems,” J. Math. Anal. Appl.,123, No. 1, 238–261 (1987).

    Google Scholar 

  10. Yu. E. Boyarintsev, V. A. Danilov, A. A. Loginov, and V. F. Chistyakov, Numerical Methods for the Solution of Singular Systems [in Russian], Nauka, Novosibirsk (1989).

    Google Scholar 

  11. M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations [in Russian], Nauka, Moscow (1969).

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute, Academy of Sciences of the Ukraine, Kiev

    V. P. Yakovets

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  1. V. P. Yakovets
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1102–1112, August, 1992.

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Yakovets, V.P. Construction of asymptotic solutions of linear singularly perturbed systems of second order with degeneracy. Ukr Math J 44, 1003–1013 (1992). https://doi.org/10.1007/BF01057121

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  • DOI: https://doi.org/10.1007/BF01057121

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