Skip to main content
Log in

Neumann Boundary-Value Problem for a Singularly Perturbed Heat Equation with Pulse Action

  • Published:
Nonlinear Oscillations

Abstract

We propose an algorithm for the construction of an asymptotic solution of the Neumann boundary-value problem for a singularly perturbed heat equation with pulse action at fixed times. A theorem on the order with which the asymptotic solution satisfies the original problem is proved.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinematics [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  2. V. F. Butuzov and T. A. Urazgil’dina, “Asymptotic solution of the problem of propagation of heat in thin bodies,” Ukr. Mat. Zh., 39, No.1, 13–21 (1987).

    Google Scholar 

  3. K. A. Velizhanina, E. A. Vozhukova, and N. N. Nefedov, “On the influence of viscosity and thermal conductivity on characteristics of a cylindrical resonator,” Akust. Zh., 32, Issue 1, 114–116 (1986).

    Google Scholar 

  4. B. S. Pol’skii, Numerical Simulation of Semiconductor Devices [in Russian], Zinatne, Riga (1986).

    Google Scholar 

  5. S. Yu. Dobrokhotov and V. P. Maslov, “Finite-gap almost periodic solutions in WKB approximations,” in: VINITI Series in Contemporary Problems of Mathematics [in Russian], Vol. 15, VINITI, Moscow (1980), pp. 3–94.

    Google Scholar 

  6. V. P. Maslov and G. A. Omel’yanov, “Asymptotic soliton-like solutions of equations with small dispersion,” Usp. Mat. Nauk, 36, Issue 3 (219), 63–123 (1981).

    Google Scholar 

  7. O. Aksel’son, E. V. Glushkov, and N. V. Glushkova, “Method of local Green functions for a singularly perturbed convection-diffusion problem,” Dokl. Ros. Akad. Nauk, 388, No.2, 166–167 (2003).

    Google Scholar 

  8. A. A. Bobodzhanov and V. F. Safonov, “Singularly perturbed nonlinear integro-differential systems with rapidly varying kernels,” Mat. Zametki, 72, No.5, 654–664 (2002).

    Google Scholar 

  9. J. Mo, “A class of nonlinear singularly perturbed problems for reaction diffusion equations,” Acta Math. Sci., Ser. B, 23, No.3, 377–385 (2003).

    Google Scholar 

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

    Google Scholar 

  11. A. B. Vasil’eva, “Asymptotics of solutions of some problems for ordinary differential equations with small parameter in the coefficient of the leading derivative,” Usp. Mat. Nauk, 18, No.3, 15–86 (1963).

    Google Scholar 

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

    Google Scholar 

  13. V. D. Mil’man and A. D. Myshkis, “On the stability of motion in the presence of impulses,” Sib. Mat. Zh., 1, No.2, 233–237 (1960).

    Google Scholar 

  14. Yu. A. Mitropol’skii, A. M. Samoilenko, and N. A. Perestyuk, “On the justification of the averaging method for second-order equations with pulse action,” Ukr. Mat. Zh., 29, No.6, 750–762 (1977).

    Google Scholar 

  15. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).

    Google Scholar 

  16. A. M. Samoilenko, Yu. I. Kaplun, and V. H. Samoilenko, “Singularly perturbed differential equations with pulse action,” Ukr. Mat. Zh., 54, No.8, 1089–1099 (2002).

    Google Scholar 

  17. L. V. Khomchenko, “Singularly perturbed heat equation with pulse action,” Visn. Kyiv. Nats. Univ., Issues 11–12, 101–105 (2004).

  18. A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations [in Ukrainian], Lybid’, Kyiv (2003).

    Google Scholar 

  19. A. M. Samoilenko, S. A. Kryvosheya, and M. O. Perestyuk, Differential Equations in Problems [in Ukrainian], Lybid’, Kyiv (2003).

    Google Scholar 

  20. N. I. Ivanchov, “Boundary-value problems for a parabolic equation with integral conditions,” Differents. Uravn., 40, No.4, 547–564 (2004).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 89–122, January–March, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samoilenko, V.H., Khomchenko, L.V. Neumann Boundary-Value Problem for a Singularly Perturbed Heat Equation with Pulse Action. Nonlinear Oscill 8, 87–121 (2005). https://doi.org/10.1007/s11072-005-0040-8

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11072-005-0040-8

Keywords

Navigation