Skip to main content
SpringerLink
Log in

Corner boundary layer in nonlinear elliptic problems containing first-order derivatives

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

In a rectangular domain, the first boundary-value problem is considered for the following singularly perturbed elliptic equation:

$$\varepsilon ^2 \Delta u - \varepsilon ^\alpha A(x,y)\frac{{\partial u}} {{\partial y}} = F(u,x,y,\varepsilon )$$

with the function F, which is nonlinear in u. The complete asymptotic solution expansion, which is uniform in a closed rectangle, is constructed for α > 1. If 0 < α < 1, the uniform asymptotic approximation is constructed as a zero and first approximation. The features of the asymptotic behavior are noted at α = 1.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Butuzov, V.F., The asymptotic properties of solutions of the equation μ2Δu − k 2(x, y)u = f(x, y) in a rectangle, Diff. Eq., 1973, vol. 9, pp. 1274–1279.

    MathSciNet  Google Scholar 

  2. Butuzov, V.F., On asymptotics of solutions of singularly perturbed equations of elliptic type in the rectangle, Diff. Eq., 1975, vol. 11, pp. 780–790.

    MathSciNet  Google Scholar 

  3. Butuzov, V.F., A singularly perturbed elliptic equation with two small parameters, Diff. Eq., 1976, vol. 12, pp. 1261–1272.

    Google Scholar 

  4. Denisov, I.V., Quasilinear singularly perturbed elliptic equations in a rectangle, Compt. Mathem. Mathem. Phys., 1995, vol. 35, pp. 1341–1350.

    MATH  MathSciNet  Google Scholar 

  5. Denisov, I.V., The problem of finding the dominant term of the corner part of the asymptotics of the solution to a singularly perturbed elliptic equation with a nonlinearity, Compt. Mathem. Mathem. Phys., 1999, vol. 39, pp. 747–759.

    MATH  MathSciNet  Google Scholar 

  6. Denisov, I.V., The corner boundary layer in nonlinear singularly perturbed elliptic equations, Compt. Mathem. Mathem. Phys., 2001, vol. 41, pp. 362–378.

    MATH  MathSciNet  Google Scholar 

  7. Denisov, I.V., The corner boundary layer in nonmonotone singularly perturbed boundary value problems with nonlinearities, Compt. Mathem. Mathem. Phys., 2004, vol. 44, pp. 1592–1610.

    MathSciNet  Google Scholar 

  8. Denisov, I.V., Corner boundary layer in nonlinear singularly perturbed elliptic problems, Compt. Mathem. Mathem. Phys., 2008, vol. 48, pp. 59–75.

    Article  MathSciNet  Google Scholar 

  9. Denisov, I.V., On some classes of functions, in Chebyshevskiy sbornik, Tom 10, vyp. 2 (Chebyshev’s Collection, vol. 10, no. 2), Tula: Tul. Gos. Ped. Univ., 2009, pp. 79–108.

    Google Scholar 

  10. Vasilieva, A.B. and Butuzov, V.F., Asimptoticheskie metody v teorii singulyarnyh vozmusheniy (Asymptotic Methods in Theory of Singular Perturbations), Moscow: Vysshaya Shkola, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Vorob’evy Gory 1, str. 2, Moscow, 119991, Russia

    V. F. Butuzov

  2. Tula State Pedagogical University, pr. Lenina 125, Tula, 300026, Russia

    I. V. Denisov

Authors
  1. V. F. Butuzov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. V. Denisov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. F. Butuzov.

Additional information

Original Russian Text © V.F. Butuzov, I.V. Denisov, 2014, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2014, No. 1, pp. 7–31.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Butuzov, V.F., Denisov, I.V. Corner boundary layer in nonlinear elliptic problems containing first-order derivatives. Aut. Control Comp. Sci. 48, 458–476 (2014). https://doi.org/10.3103/S0146411614070050

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411614070050

Keywords

Search

Navigation