Skip to main content
SpringerLink
Log in

Galerkin method for a third-order differential-operator equation

  • Numerical Methods
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.

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. Dezin, A.A., Operator-Differential Equations, Tr. Mat. Inst. Steklova, 2000, vol. 229.

  2. Vinogradova, P. and Zarubin, A., Projection Method for Cauchy Problem for an Operator-Differential Equation, Numer. Funct. Anal. Optim., 2009, vol. 30, nos. 1–2, pp. 148–167.

    Article  MATH  MathSciNet  Google Scholar 

  3. Vinogradova, P., Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations, Numer. Funct. Anal. Optim., 2010, vol. 31, no. 3, pp. 339–365.

    Article  MATH  MathSciNet  Google Scholar 

  4. Lyakhov, D.A. and Lomovtsev, F.E., The Method ofWeak Solutions of the Auxiliary Cauchy Problem for Investigating the Smoothness of Solutions of Second-Order Hyperbolic Operator-Differential Equations with Variable Domains, Vestn. Belarus. Gos. Univ. Ser. 1, 2010, no. 2, pp. 75–82.

    Google Scholar 

  5. Khodos, S.P., The Euler-Poisson-Darboux Equation with Variable Domains of Discontinuous Operators, Vestn. Belarus. Gos. Univ. Ser. 1, 2010, no. 1, pp. 81–87.

    Google Scholar 

  6. Aliev, A.R., Boundary Value Problems for a Class of Higher-Order Operator-Differential Equations with Variable Coefficients, Mat. Zametki, 2003, vol. 74, no. 6, pp. 803–814.

    Article  Google Scholar 

  7. Dubinskii, Yu.A., Certain Differential-Operator Equations of Arbitrary Order, Mat. Sb., 1973, vol. 90 (132), no. 1, pp. 3–22.

    MathSciNet  Google Scholar 

  8. Yurchuk, N.I., Boundary Value Problems for Differential Equations with Operator Coefficients That Depend on a Parameter. I. A Priori Estimates, Differ. Uravn., 1976, vol. 12, no. 9, pp. 1645–1661.

    MATH  Google Scholar 

  9. Yurchuk, N.I., Solvability of Boundary Value Problems for Certain Operator-Differential Equations, Differ. Uravn., 1977, vol. 13, no. 4, pp. 626–636.

    MATH  Google Scholar 

  10. Vasilevskii, K.V., A Boundary Value Problem for a Third-Order Binomial Operator-Differential Equation with Variable Domains of Unbounded Operators, Vestn. Belarus. Gos. Univ. Ser. 1, 2010, no. 3, pp. 110–114.

    Google Scholar 

  11. Aliev, A.R., On the Solvability of Initial-Boundary Value Problems for a Class of Third-Order Operator-Differential Equations, Mat. Zametki, 2011, vol. 90, no. 3, pp. 323–339.

    Article  MathSciNet  Google Scholar 

  12. Mamedov, A.M., A Boundary Value Problem for a Class of Third-Order Operator-Differential Equations, Mat. Zametki, 2010, vol. 87, no. 4, pp. 632–635.

    Article  Google Scholar 

  13. Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., et al., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969.

    Google Scholar 

  14. Glushko, V.P. and Krein, S.G., Inequalities for Norms of Derivatives in Weighted L p Spaces, Sibirsk. Mat. Zh., 1960, vol. 1, no. 3, pp. 343–382.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Far Eastern State Transport University, Khabarovsk, Russia

    P. V. Vinogradova & A. G. Zarubin

  2. Pacific National University, Khabarovsk, Russia

    P. V. Vinogradova & A. G. Zarubin

Authors
  1. P. V. Vinogradova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. Zarubin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. V. Vinogradova.

Additional information

Original Russian Text © P.V. Vinogradova, A.G. Zarubin, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 2, pp. 242–249.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vinogradova, P.V., Zarubin, A.G. Galerkin method for a third-order differential-operator equation. Diff Equat 50, 246–253 (2014). https://doi.org/10.1134/S0012266114020128

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266114020128

Keywords

Search

Navigation