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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 603018, 14 pages
http://dx.doi.org/10.1155/2012/603018
Research Article

The Difference Problem of Obtaining the Parameter of a Parabolic Equation

1Department of Computer Technology, Turkmen Agricultural University, Gerology Street,74400 Asgabat, Turkmenistan
2Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey
3Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

Received 4 April 2012; Accepted 23 April 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Charyyar Ashyralyyev and Oznur Demirdag. 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. Dehghan, “Determination of a control parameter in the two-dimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. T. Kimura and T. Suzuki, “A parabolic inverse problem arising in a mathematical model for chromatography,” SIAM Journal on Applied Mathematics, vol. 53, no. 6, pp. 1747–1761, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography,” Inverse Problems, vol. 2, no. 5, pp. 373–397, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Y. S. Eidelman, Boundary value problems for di¤erential equations with parameters [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1984.
  5. Y. S. Eidelman, “Conditions for the solvability of inverse problems for evolution equations,” Doklady Akademii Nauk Ukrainskoj SSR Serija A, no. 7, pp. 28–31, 1990 (Russian).
  6. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Di¤erential Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 2004.
  7. R. P. Agarwal and V. B. Shakhmurov, “Multipoint problems for degenerate abstract differential equations,” Acta Mathematica Hungarica, vol. 123, no. 1-2, pp. 65–89, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. O. Ashyralyev and P. E. Sobolevskiĭ, “The linear operator interpolation theory and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289–1291, 1984 (Russian).
  9. A. Ashyralyev, I. Karatay, and P. E. Sobolevskii, “Well-posedness of the nonlocal boundary value problem for parabolic difference equations,” Discrete Dynamics in Nature and Society, no. 2, pp. 273–286, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of the nonlocal boundary value problem for parabolic differential equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53–61, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Ashyralyev, “High-accuracy stable difference schemes for well-posed NBVP,” in Modern Analysis and Applications, vol. 191 of Operator Theory: Advances and Applications, pp. 229–252, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Ashyralyev, “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557–573, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. S. Eidelman, “Two-point boundary value problem for a differential equation with a parameter,” Dopovidi Akademii Nauk Ukrainskoi RSR Seriya A-Fiziko-Matematichni ta Technichni Nauki, no. 4, pp. 15–18, 1983 (Russian).
  14. A. I. Prilepko, “Inverse problems of potential theory,” Matematicheskie Zametki, vol. 14, pp. 755–767, 1973, English translation Mathematical Notes, vol. 14, 1973.
  15. A. D. Iskenderov and R. G. Tagiev, “ The inverse problem of determining the right-hand sides of evolution equations in Banach space,” Nauchnyye Trudy Azerbaidzhanskogo Gosudarstvennogo Universiteta, no. 1, pp. 51–56, 1979 (Russian).
  16. W. Rundell, “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,” Applicable Analysis, vol. 10, no. 3, pp. 231–242, 1980. View at Publisher · View at Google Scholar
  17. A. I. Prilepko and I. A. Vasin, “Some time-dependent inverse problems of hydrodynamics with final observation,” Doklady Akademii Nauk SSSR, vol. 314, no. 5, pp. 1075–1078, 1990, English translation Soviet Mathematics. Doklady, vol. 42, 1991.
  18. A. I. Prilepko and A. B. Kostin, “On certain inverse problems for parabolic equations with final and integral observation,” Matematicheskii Sbornik, vol. 183, no. 4, pp. 49–68, 1992, English translation Russian Academy of Sciences. Sbornik Mathematics, vol. 75, 1993. View at Publisher · View at Google Scholar
  19. A. I. Prilepko and I. V. Tikhonov, “Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation,” Matematicheskie Zametki, vol. 51, no. 2, pp. 77–87, 1992, English translation Mathematical Notes, vol. 2, pp. 77–87, 1992. View at Publisher · View at Google Scholar
  20. D. G. Orlovskii, “On a problem of determining the parameter of an evolution equation,” Differentsial'nye Uravneniya, vol. 26, no. 9, pp. 1614–1621, 1990, English translation Differential Equations, vol. 26, 1990.
  21. J. R. Cannon, Y. L. Lin, and S. Xu, “Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations,” Inverse Problems, vol. 10, no. 2, pp. 227–243, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. M. Dehghan, “Finding a control parameter in one-dimensional parabolic equations,” Applied Mathematics and Computation, vol. 135, no. 1-2, pp. 491–503, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Ashyralyev and O. Demirdag, “A note on boundary value parabolic problems,” Vestnik of Odessa National University. Mathematics and Mechanics, vol. 16, no. 16, pp. 131–143, 2010.
  25. A. Ashyralyev and O. Demirdağ, “A note on a problem of obtaining the parameter of a parabolic equation,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 10–20, 2011.
  26. C.-R. Ye and Z.-Z. Sun, “On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 214–225, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. A. Ashyralyev and O. Demirdağ, “On the numerical solution of parabolic equation with the Neumann condition arising in determination of a control parameter,” AIP Conference Proceedings, vol. 1389, pp. 613–616, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Ashyralyev and A. S. Erdoğan, “On the numerical solution of a parabolic inverse problem with the Dirichlet condition,” International Journal of Mathematics and Computation, vol. 11, no. 11, pp. 73–81, 2011.
  29. A. Ashyralyev and A. S. Erdogan, “Well-posedness of the inverse problem of a multidimensional parabolic equation,” Vestnik of Odessa National University. Mathematicsand Mechanics, vol. 15, no. 18, pp. 129–135, 2010.
  30. A. Ashyralyev, “Fractional spaces generated by the positivite di¤erential and difference operator in a Banach space,” in Proceedings of the Mathematical Methods and Engineering, K. Taş, Ed., pp. 13–22, Springer, The Netherlands, 2007. View at Publisher · View at Google Scholar
  31. A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Operator Theory Advances and Applications, Birkhäuser, Boston, Berlin, 1994.
  32. Y. A. Smirnitskii and P. E. Sobolevskii, “Positivity of multidimensional di¤erence operators in the C-norm,” Uspekhi Matematicheskikh Nauk, vol. 36, no. 4, pp. 202–203, 1981 (Russian).
  33. Y. A. Smirnitskii, Fractional powers of elliptic di¤erence operators [Ph.D. thesis], Voronezh State University, Voronezh, Russia, 1983.
  34. P. E. Sobolevskiĭ, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971 (Russian).