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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 230190, 13 pages
http://dx.doi.org/10.1155/2012/230190
Research Article

On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

Received 7 April 2012; Accepted 24 April 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Allaberen Ashyralyev and Okan Gercek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi University Press, Tbilisi, Georgia, 1981.
  2. A. M. Nakhushev, Equations of Mathematical Biology, Textbook for Universities, Vysshaya Shkola, Moscow, Russia, 1995.
  3. M. S. Salakhitdinov and A. K. Urinov, Boundary Value Problems for Equations of Mixed Type with a Spectral Parameter, Fan, Tashkent, Uzbekistan, 1997.
  4. A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semi-linear hyperbolic-parabolic equations,” Nonlinear Analysis. Theory, Methods and Applications, vol. 47, no. 5, pp. 3585–3592, 2001.
  5. J. I. Diaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, “An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator,” Journal of Differential Equations, vol. 198, no. 2, pp. 321–355, 2004. View at Publisher · View at Google Scholar
  6. D. Guidetti, B. Karasozen, and S. Piskarev, “Approximation of abstract differential equations,” Journal of Mathematical Sciences, vol. 122, no. 2, pp. 3013–3054, 2004. View at Publisher · View at Google Scholar
  7. R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,” Boundary Value Problems, no. 1, pp. 9–42, 2005.
  8. A. Ashyralyev, G. Judakova, and P. E. Sobolevskii, “A note on the difference schemes for hyperbolic-elliptic equations,” Abstract and Applied Analysis, vol. 2006, Article ID 14816, 13 pages, 2006. View at Publisher · View at Google Scholar
  9. A. S. Berdyshev and E. T. Karimov, “Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type,” Central European Journal of Mathematics, vol. 4, no. 2, pp. 183–193, 2006. View at Publisher · View at Google Scholar
  10. A. Ashyralyev, C. Cuevas, and S. Piskarev, “On well-posedness of difference schemes for abstract elliptic problems in Lp([0, T]; E) spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 1-2, pp. 43–65, 2008. View at Publisher · View at Google Scholar
  11. A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems for elliptic-parabolic differential and difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 904824, 16 pages, 2008. View at Publisher · View at Google Scholar
  12. M. De la Sen, “About the stability and positivity of a class of discrete nonlinear systems of difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 595367, 18 pages, 2008. View at Publisher · View at Google Scholar
  13. P. V. Vinogradova, “Error estimates for a projection-difference method for a linear differential-operator equation,” Differential Equations, vol. 44, no. 7, pp. 970–979, 2008. View at Publisher · View at Google Scholar
  14. A. Ashyralyev and Y. Ozdemir, “On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1100–1118, 2009. View at Publisher · View at Google Scholar
  15. A. Castro, C. Cuevas, and C. Lizama, “Well-posedness of second order evolution equation on discrete time,” Journal of Difference Equations and Applications, vol. 16, no. 10, pp. 1165–1178, 2010. View at Publisher · View at Google Scholar
  16. A. Ashyralyev and O. Gercek, “On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems,” Abstract and Applied Analysis, vol. 2010, Article ID 705172, 17 pages, 2010. View at Publisher · View at Google Scholar
  17. A. Ashyralyev, “On the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 49, pp. 1–16, 2011.
  18. V. B. Shakhmurov, “Regular degenerate separable differential operators and applications,” Potential Analysis, vol. 35, no. 3, pp. 201–222, 2011. View at Publisher · View at Google Scholar
  19. J. Martin-Vaquero, A. Queiruga-Dios, and A. H. Encinas, “Numerical algorithms for diffusion-reaction problems with non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5487–5495, 2012.
  20. O. Gercek, Difference Schemes of Nonlocal Boundary Value Problems for Elliptic-Parabolic Differential Equations [Ph.D. thesis], Yildiz Teknik University, Istanbul, Turkey, 2010.
  21. A. Ashyralyev and O. Gercek, “On multipoint nonlocal elliptic-parabolic difference problems,” Vestnik of Odessa National University, vol. 15, no. 10, pp. 135–156, 2010.
  22. P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971.
  23. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkhäuser, Basle, Switzerland, 2004.
  24. A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basle, Switzerland, 1994.
  25. P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.