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On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions

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Abstract

We consider three-layer difference schemes for a one-dimensional linear parabolic equation with nonlocal integral conditions. A three-layer scheme is written out in an equivalent form of a two-layer scheme. We analyze the dependence of the spectrum of the difference operator on the parameters occurring in the integral conditions. We derive stability conditions for the original three-layer scheme in a specially defined energy norm.

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References

  1. Goolin, A.V., Ionkin, N.I., and Morozova, V.A., Difference Schemes with Nonlocal Boundary Conditions, Comput. Methods Appl. Math., 2001, vol. 1, no. 1, pp. 62–71.

    MathSciNet  MATH  Google Scholar 

  2. Gulin, A.V. and Morozova, V.A., On a Family of Nonlocal Difference Schemes, Differ. Uravn., 2009, vol. 45, no. 7, pp. 1001–1013.

    MathSciNet  Google Scholar 

  3. Gulin, A.V., Ionkin, N.I., and Morozova, V.A., On the Stability of a Nonlocal Two-Dimensional Difference Problem, Differ. Uravn., 2001, vol. 37, no. 7, pp. 926–932.

    MathSciNet  Google Scholar 

  4. Cahlon, B., Kulkarni, D.M., and Shi, P., Stepwise Stability for the Heat Equation with a Nonlocal Constraint, SIAM J. Numer. Anal., 1995, vol. 32, no. 2, pp. 571–593.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Jachimavičienė, J., Jesevičiūtė, Ž., and Sapagovas, M., The Stability of Finite-Difference Schemes for a Pseudoparabolic Equation with Nonlocal Conditions, Numer. Funct. Anal. Optim., 2009, vol. 30, no. 9, pp. 988–1001.

    Article  MathSciNet  MATH  Google Scholar 

  7. Ivanauskas, F., Meškauskas, T., and Sapagovas, M., Stability of Difference Schemes for Two-Dimensional Parabolic Equations with Non-Local Boundary Conditions, Appl. Math. Comput., 2009, vol. 215, no. 7, pp. 2716–2732.

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Gulin, A.V., Morozova, V.A., and Udovichenko, N.S., A Criterion for the Stability of a Family of Nonlocal Difference Schemes, Differ. Uravn., 2010, vol. 46, no. 7, pp. 966–982.

    MathSciNet  Google Scholar 

  10. Ekolin, G., Finite Difference Methods for a Nonlocal Boundary Value Problem for Heat Equation, BIT, 1991, vol. 31, pp. 245–261.

    Article  MathSciNet  MATH  Google Scholar 

  11. Fairweather, G. and Lopez-Marcos, J.C., Galerkin Methods for a Semilinear Parabolic Problem with Nonlocal Conditions, Adv. Comput. Math., 1996, vol. 6, no. 1, pp. 243–262.

    Article  MathSciNet  Google Scholar 

  12. Lin, Y., Xu, S., and Yin, H.M., Finite Difference Approximations for a Class of Non-Local Parabolic Equations, Int. J. Math. Math. Sci., 1997, vol. 20, no. 1, pp. 147–164.

    Article  MathSciNet  MATH  Google Scholar 

  13. Čiegis, R., Štikonas, A., Štikonienė, O., and Suboch, O., A Monotone Difference Scheme for a Parabolic Problem with Nonlocal Boundary Conditions, Differ. Uravn., 2002, vol. 38, no. 7, pp. 968–975.

    MathSciNet  Google Scholar 

  14. Čiegis, R. and Tumanova, N., Numerical Solution of Parabolic Problems with Nonlocal Boundary Conditions, Numer. Funct. Anal. Optim., 2010, vol. 31, no. 12, pp. 1318–1329.

    Article  MathSciNet  MATH  Google Scholar 

  15. Noye, B.J. and Dehghan, M., New Explicit Finite Difference Schemes for Two-Dimensional Diffusion Subject to Specification of Mass, Numer. Methods Partial Differential Equations, 1999, vol. 15, no. 4, pp. 521–534.

    Article  MathSciNet  MATH  Google Scholar 

  16. Čiegis, R.Yu., Efficient Difference Schemes for Solving a Two-Dimensional Parabolic Problem with an Integral Condition, Differ. Uravn., 2005, vol. 41, no. 7, pp. 975–979.

    MathSciNet  Google Scholar 

  17. Sapagovas, M., Kairytė, G., Štikonienė, O., and Štikonas, A., Alternating Direction Method for a Two-Dimensional Parabolic Equation with a Nonlocal Boundary Condition, Math. Model. Anal., 2007, vol. 12, no. 1, pp. 131–142.

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, J. and Sun, Z.Z., Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions, Numer. Math. J. Chinese Univ. (English Ser.), 2007, vol. 16, no. 2, pp. 97–111.

    MATH  Google Scholar 

  19. Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1977.

    Google Scholar 

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

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

  22. Sapagovas, M., On the Stability of Finite-Difference Schemes for One-Dimensional Parabolic Equations Subject to Integral Conditions, Zh. Vychisl. Mat. Mat. Fiz., 2005, vol. 92, pp. 77–90.

    Google Scholar 

  23. Collatz, L., Funktionalanalysis und Numerische Mathematik, Berlin, 1964.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

    M. Sapagovas

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  1. M. Sapagovas
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Original Russian Text © M. Sapagovas, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 7, pp. 1033–1041.

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Sapagovas, M. On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions. Diff Equat 48, 1018–1027 (2012). https://doi.org/10.1134/S0012266112070129

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