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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 237657, 12 pages
http://dx.doi.org/10.1155/2012/237657
Research Article

Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

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

Received 30 March 2012; Accepted 17 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 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. M. S. Salakhitdinov, Equations of Mixed-Composite Type, Fan, Tashkent, Uzbekistan, 1974.
  2. T. D. Dzhuraev, Boundary Value Problems for Equations of Mixed and Mixed Composite Types, Fan, Tashkent, Uzbekistan, 1979.
  3. D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi University Press, Tbilisi, Georgia, 1981.
  4. V. N. Vragov, Boundary Value Problems for Nonclassical Equations of Mathematical Physics, Textbook for Universities, NGU, Novosibirsk, Russia, 1983.
  5. D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.
  6. A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, Moskow, Russia, 1995.
  7. S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.
  8. A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations,” Nonlinear Analysis-Theory, Methods and Applications, vol. 47, no. 5, pp. 3585–3592, 2001.
  9. D. Guidetti, B. Karasözen, 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 · View at Zentralblatt MATH
  10. J. I. Díaz, 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 · View at Zentralblatt MATH
  11. A. Ashyralyev, “A note on the nonlocal boundary value problem for elliptic-parabolic equations,” Nonlinear Studies, vol. 13, no. 4, pp. 327–333, 2006. View at Zentralblatt MATH
  12. 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 · View at Zentralblatt MATH
  13. 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 · View at Zentralblatt MATH
  14. 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 · View at Zentralblatt MATH
  15. J. Martín-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.
  16. A. Ashyralyev and H. Soltanov, “On elliptic-parabolic equations in a Hilbert space,” in Proceedings of the IMM of CS of Turkmenistan, pp. 101–104, Ashgabat, Turkmenistan, 1995.
  17. 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 · View at Zentralblatt MATH
  18. A. Ashyralyev and O. Gercek, “Finite difference method for multipoint nonlocal elliptic-parabolic problems,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 2043–2052, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. 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 · View at Zentralblatt MATH
  20. 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.
  21. P. E. Sobolevskii, “The theory of semigroups and the stability of difference schemes,” in Operator Theory in Function Spaces, pp. 304–337, Nauka, Novosibirsk, Russia, 1977.
  22. A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994.
  23. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
  24. A. Ashyralyev, Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations [Doctor of Sciences Thesis], Kiev, Ukraine, 1992.
  25. P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971.
  26. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004.
  27. P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.