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Difference Schemes for Delay Parabolic Equations with Periodic Boundary Conditions

  • Allaberen Ashyralyev
  • Deniz AgirsevenEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

The initial-boundary value problem for the delay parabolic partial differential equation with nonlocal conditions is studied. The convergence estimates for solutions of first and second order of accuracy difference schemes in Hölder norms are obtained. The theoretical statements are supported by a numerical example.

Keywords

Difference schemes Delay parabolic equation Hölder spaces Convergence 

References

  1. 1.
    Al-Mutib, A.N.: Stability properties of numerical methods for solving delay differential equations. J. Comput. and Appl. Math. 10(1), 71–79 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bellen, A., Jackiewicz, Z., Zennaro, M.: Stability analysis of one-step methods for neutral delay-differential equations. Numer. Math. 52(6), 605–619 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cooke, K.L., Györi, I.: Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Comput. Math. Appl. 28, 81–92 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Torelli, L.: Stability of numerical methods for delay differential equations. J. Comput. and Appl. Math. 25, 15–26 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Yenicerioglu, A.F.: The behavior of solutions of second order delay differential equations. J. Math. Anal. Appl. 332(2), 1278–1290 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ashyralyev, A., Akca, H.: Stability estimates of difference schemes for neutral delay differential equations. Nonlinear Anal.: Theory Methods Appl. 44(4), 443–452 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ashyralyev, A., Akca, H., Yenicerioglu, A.F.: Stability properties of difference schemes for neutral differential equations. Differ. Equ. Appl. 3, 57–66 (2003)Google Scholar
  8. 8.
    Liu J., Dong P., Shang G.: Sufficient conditions for inverse anticipating synchronization of unidirectional coupled chaotic systems with multiple time delays. In: Proceedings of the Chinese Control and Decision Conference, pp. 751–756. IEEE (2010)Google Scholar
  9. 9.
    Mohamad, S., Akca, H., Covachev, V.: Discrete-time cohen-grossberg neural networks with transmission delays and impulses. Differ. Differ. Equ. Appl. Book Ser.: Tatra Mountains Math. Publ. 43, 145–161 (2009)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Ashyralyev, A., Sobolevskii, P.E.: On the stability of the delay differential and difference equations. Abstr. Appl. Anal. 6(5), 267–297 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ashyralyev, A., Sobolevskii, P.E.: New Difference Schemes for Partial Differential Equations. Birkhäuser Verlag, Boston (2004)zbMATHCrossRefGoogle Scholar
  12. 12.
    Gabriella, D.B.: Delay differential equations with unbounded operators acting on delay terms. Nonlinear Anal.: Theory Methods Appl. 52(1), 1–18 (2003)zbMATHCrossRefGoogle Scholar
  13. 13.
    Ashyralyev, A.: Fractional spaces generated by the positive differential and difference operator in a banach space. In: Tas, K., Tenreiro Machado, J.A., Baleanu, D. (eds.) Mathematical Methods in Engineering, pp. 13–22. Springer, Netherlands (2007)CrossRefGoogle Scholar
  14. 14.
    Ashyralyev, A., Agirseven, D.: On convergence of difference schemes for delay parabolic equations. Comput. Math. Appl. 66(7), 1232–1244 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Akca, H., Shakhmurov, V.B., Arslan, G.: Differential-operator equations with bounded delay. Nonlinear Times Dig. 2, 179–190 (1995)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Akca, H., Covachev, V.: Spatial discretization of an impulsive Cohen-Grossberg neural network with time-varying and distributed delays and reaction-diffusion terms. Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Mat. 17(3), 15–16 (2009)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Ashyralyev A., Agirseven D.: Finite difference method for delay parabolic equations. In: AIP Conference Proceedings of 1389 Numerical Analysis And Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, pp. 573–576 (2011)Google Scholar
  18. 18.
    Agirseven, D.: Approximate solutions of delay parabolic equations with the Dirichlet condition. Abstr. Appl. Anal. 2012, 1–31 (2012). doi: 10.1155/2012/682752. Article ID 682752MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ashyralyev, A., Agirseven, D.: Well-posedness of delay parabolic difference equations. Adv. Differ. Equ. 2014, 1–18 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ashyralyev, A., Agirseven, D.: Approximate solutions of delay parabolic equations with the Neumann condition, In: AIP Conference Proceedings, ICNAAM 2012, vol. 1479, pp. 555–558 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey
  2. 2.ITTUAshgabatTurkmenistan
  3. 3.Department of MathematicsTrakya UniversityEdirneTurkey

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