Skip to main content
SpringerLink
Log in

On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions

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

Abstract

We consider the stability of an explicit finite-difference scheme for a linear hyperbolic equation with nonlocal integral boundary conditions. By studying the spectrum of the transition matrix of the explicit three-layer difference scheme, we obtain a sufficient condition for stability in a special norm.

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. Mu, C., Liu, D., and Zhou, S., Properties of Positive Solutions for a Nonlocal Reaction-Diffusion Equation with Nonlocal Nonlinear Boundary Condition, J. Korean Math. Soc., 2010, vol. 47, no. 6, pp. 1317–1328.

    Article  MathSciNet  MATH  Google Scholar 

  2. Cannon, J.R., The Solution of the Heat Equation Subject to the Specification of Energy, Quart. Appl. Math., 1963, vol. 21, pp. 155–160.

    MathSciNet  Google Scholar 

  3. Nakhushev, A.M., An Approximate Method for Solving Boundary Value Problems for Differential Equations and Its Application to the Dynamics of Ground Moisture and Ground Water, Differ. Uravn., 1982, vol. 18, no. 1, pp. 72–81.

    MathSciNet  Google Scholar 

  4. Chudnovskii, A.F., Teplofizika pochv (Thermophysics of Soils), Moscow, 1976.

    Google Scholar 

  5. Day, W.A., A Decreasing Property of Solutions of Parabolic Equations with Applications to Thermoelasticity, Quart. Appl. Math., 1983, vol. 40, no. 4, pp. 468–475.

    MATH  Google Scholar 

  6. Ashyralyev, A. and Aggez, N., Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition, Discrete Dyn. Nat. Soc., 2011, vol. 2011, Article ID 562385, p. 15.

    Article  MathSciNet  Google Scholar 

  7. Kozhanov, A.I. and Pul’kina, L.S., On the Solvability of Boundary Value Problems with a Nonlocal Boundary Condition of Integral Form for Multidimensional Hyperbolic Equations, Differ. Uravn., 2006, vol. 42, no. 9, pp. 1166–1179.

    MathSciNet  Google Scholar 

  8. Pul’kina, L.S., A Nonlocal Problem with Integral Conditions for a Hyperbolic Equation, Differ. Uravn., 2004, vol. 40, no. 7, pp. 887–892.

    MathSciNet  Google Scholar 

  9. Ramezani, M., Dehghan, M., and Razzaghi, M., Combined Finite Difference and Spectral Methods for the Numerical Solution of Hyperbolic Equation with an Integral Condition, Numer. Methods Partial Differential Equations, 2008, vol. 24, no. 1, pp. 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  10. Sapagovas, M., On Spectral Properties of Three-Layer Difference Schemes for Parabolic Equations with Nonlocal Conditions, Differ. Uravn., 2012, vol. 48, no. 7, pp. 1033–1041.

    Google Scholar 

  11. Samarskii, A.A. and Gulin, A.V., Chislennye metody (Numerical Methods), Moscow, 1989.

    Google Scholar 

  12. Sapagovas, M., Meškauskas, T., and Ivanauskas, F., Numerical Spectral Analysis of a Difference Operator with Non-Local Boundary Conditions, Appl. Math. Comput., 2012, vol. 218, no. 14, pp. 7515–7527.

    Article  MathSciNet  MATH  Google Scholar 

  13. Sapagovas, M.P. and Štikonas, A.D., On the Structure of the Spectrum of a Differential Operator with a Nonlocal Condition, Differ. Uravn., 2005, vol. 41, no. 7, pp. 961–969.

    MathSciNet  Google Scholar 

  14. Sapagovas, M., On the Stability of a Finite-Difference Scheme for Nonlocal Parabolic Boundary-Value Problems, Lithuanian Math. J., 2008, vol. 48, no. 3, pp. 339–356.

    Article  MathSciNet  MATH  Google Scholar 

  15. Gulin, A.V., Ionkin, N.I., and Morozova, V.A., Investigation of the Norm in Problems of the Stability of Nonlocal Difference Schemes, Differ. Uravn., 2006, vol. 42, no. 7, pp. 914–923.

    MathSciNet  Google Scholar 

  16. Collatz, L., Funktionalanalysis und numerische Mathematik, Berlin, 1964.

    Book  MATH  Google Scholar 

  17. Lancaster, P., Lambda-Matrices and Vibrating Systems, Oxford, 1966.

    MATH  Google Scholar 

  18. Gulin, A.V., Morozova, V.A., and Udovichenko, N.S., Stability of a Nonlocal Difference Problem with a Complex Parameter, Differ. Uravn., 2011, vol. 47, no. 8, pp. 1105–1108.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vilnius University, Vilnius, Lithuania

    F. F. Ivanauskas, Yu. A. Novitski & M. P. Sapagovas

Authors
  1. F. F. Ivanauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. A. Novitski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. P. Sapagovas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © F.F. Ivanauskas, Yu.A. Novitski, M.P. Sapagovas, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 7, pp. 877–884.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivanauskas, F.F., Novitski, Y.A. & Sapagovas, M.P. On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions. Diff Equat 49, 849–856 (2013). https://doi.org/10.1134/S0012266113070070

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266113070070

Keywords

Search

Navigation