Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Asymptotic expansions of the solutions of linear, singularly perturbed differential equations

Abstract

Asymptotic expansions are constructed for the general solution of a finitedimensional singularly perturbed linear differential equation in cases of strong degeneracy of the structure matrix. For additional restrictions on the elements of the structure matrix these cases can be reduced to the case when the new structure matrix is nondegenerate.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    N. N. Moiseev, “Asymptotic representation of the solutions of linear differential equations in the case of multiple elementary divisors of the characteristic equation,” Dokl. Akad. Nauk SSSR,170, No. 4, 780–782 (1966).

  2. 2.

    N. N. Moiseev, Asymptotic Methods in Nonlinear Mechanics [in Russian], Moscow (1981).

  3. 3.

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

  4. 4.

    A. G. Eliseev, “Theory of singular perturbations for systems of differential equations in the case of a multiple spectrum of the limit operator, I,” Izv. Akad. Nauk SSSR, Ser. Mat.,48, No. 5, 999–1020 (1984).

  5. 5.

    A. G. Eliseev, “Theory of singular perturbations for systems of differential equations in the case of a multiple spectrum of the limit operator. III,” Izv. Akad. Nauk SSSR, Ser. Mat.,48, No. 6, 1175–1195 (1984).

  6. 6.

    A. G. Eliseev and S. A. Lomov, “Asymptotic integration of singularly perturbed problems,” Usp. Mat. Nauk,43, No. 3, 3–53 (1988).

  7. 7.

    D. A. Moshinskii, “Construction of asymptotic solutions of systems of linear differential equations with slowlyvarying coefficients,” in: Asymptotic Methods in the Theory of Nonlinear Vibrations [in Russian], Trans. All-Union Conf., Kiev (1979), pp. 125–133.

  8. 8.

    K. I. Chernyshov, “Stability and asymptotic expansions of the solutions of an operator differential equation not solved for the derivative,” in: Method of Lyapunov Functions in the Analysis of the Dynamics of Systems [in Russian], Novosibirsk (19870; pp. 129–143.

  9. 9.

    K. I. Chernyshov, “Method of standard decomposition of singularly perturbed differential equations,” Dokl. Akad. Nauk SSSR,311, No. 6, 1311–1316 (1990).

  10. 10.

    K. I. Chernyshov, “Asymptotic expansions of solutions of an equation with a Fredholm operator in front of the derivative in the critical case,” in: Colloq. Math. Soc. J. Bolyai. 53, Qualitative Theory of Differential Equations, Szeged (1988).

  11. 11.

    V. I. Arnol'd, “On Matrices dependent upon parameters,” Usp. Mat. Nauk,26, No. 2, 101–114 (1971).

  12. 12.

    K. L. Territin, “Asymptotic expansion of the solutions of systems of ordinary differential equations containing parameters,” Matematika [Russian translation],1, No. 2, 29–59 (1957).

  13. 13.

    V. N. Bogaevskii and A Ya. Povzner, Algebraic Methods in Nonlinear Perturbation Theory [in Russian], Moscow (1987).

  14. 14.

    N. D. Kopachevskii, S. G. Krein, and Ngo Zui Kan, Operator Methods in Linear Fluid Mechanics [in Russian], Moscow (1989).

Download references

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 195, pp. 161–178, 1991.

The author deeply thanks S. G. Krein, A. P. Oskolov, and V. M. Babich for useful discussions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chernyshov, K.I. Asymptotic expansions of the solutions of linear, singularly perturbed differential equations. J Math Sci 62, 3153–3164 (1992). https://doi.org/10.1007/BF01095689

Download citation

Keywords

  • Differential Equation
  • General Solution
  • Asymptotic Expansion
  • Additional Restriction
  • Linear Differential Equation