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Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

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Abstract

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.

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References

  1. Bogolyubov N. N., Some Statistical Methods in Mathematical Physics [in Russian], Izdat. Akad. Nauk USSR, Kiev (1945).

    Google Scholar 

  2. Bogolyubov N. N. and Mitropol’skii Yu. A., Asymptotic Methods in the Theory on Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  3. Simonenko I. B., The Averaging Method in the Theory of Nonlinear Equations of Parabolic Type with Applications to the Problems of Hydrodynamic Stability [in Russian], Rostovsk. Univ., Rostov-on-Don (1989).

    Google Scholar 

  4. Levenshtam V. B., “Asymptotic integration of quasilinear parabolic equations with coefficients rapidly oscillating with respect to time,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 36–43 (2002).

  5. Levenshtam V. B., “Higher-order approximations of the averaging method for parabolic initial-boundary value problems with rapidly oscillating coefficients,” Differentsial’nye Uravneniya, 39, No.10, 1395–1403 (2003).

    Google Scholar 

  6. Bogolyubov N. N., “Perturbation theory in nonlinear mechanics,” Sb. Inst. Stroit. Mekh. Akad. Nauk USSR, No. 4, 9–34 (1950).

  7. Kapitsa P. L., “Dynamic stability of a pendulum with oscillating suspension point,” Prikl. Mat. i Mekh., 59, No.6, 922–929 (1995).

    Google Scholar 

  8. Chelomei V. N., “On the possibility of raising the stability of elastic systems by means of vibrations,” Dokl. Akad. Nauk SSSR, 110, No.3, 345–347 (1956).

    Google Scholar 

  9. Zen’kovskaya S. M. and Simonenko I. B., “On influence of high-frequency vibrations on the origin of convection,” Izv. Akad. Nauk SSSR Mekh. Zhidk. i Gaza, No. 5, 51–55 (1966).

  10. Yudovich V. I., “Vibrodynamics of systems with constraints,” Dokl. Ross. Akad. Nauk, 354, No.5, 622–624 (1997).

    Google Scholar 

  11. Yudovich V. I., “The dynamics of a material particle on a smooth vibrating surface,” Prikl. Mat. i Mekh., 62, No.6, 968–976 (1998).

    Google Scholar 

  12. Yudovich V. I., “Vibrodynamics and vibrogeometry of systems with constraints,” submitted to VINITI on July 17, 2003, No. 1407-B2003.

  13. Yudovich V. I., “Vibrodynamics and vibrogeometry of systems with constraints. II,” submitted to VINITI on July 17, 2003, No. 1408-B2003.

  14. Vishik M. I. and Lyusternik L. A., “Regular degeneration and a boundary layer for linear differential equations with a small parameter,” Uspekhi Mat. Nauk, 12, No.5, 3–122 (1957).

    Google Scholar 

  15. Solonnikov V. A., “On boundary value problems for linear parabolic systems of general differential equations,” Trudy Mat. Inst. Steklov., 83, 3–162 (1965).

    Google Scholar 

  16. Sobolevskii P. E., “On parabolic type equations in Banach spaces,” Trudy Moskov. Mat. Obshch., 10, 207–350 (1961).

    Google Scholar 

  17. Krasnosel’skii M. A., et al., Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

    Google Scholar 

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Authors and Affiliations

  1. Rostov State University, Rostov-on-Don, Russia

    V. B. Levenshtam

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  1. V. B. Levenshtam
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Original Russian Text Copyright © 2005 Levenshtam V. B.

The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.

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Levenshtam, V.B. Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands. Sib Math J 46, 637–651 (2005). https://doi.org/10.1007/s11202-005-0064-4

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  • DOI: https://doi.org/10.1007/s11202-005-0064-4

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