About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 548017, 11 pages
http://dx.doi.org/10.1155/2013/548017
Research Article

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

1Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey
2TAU, Gerogly Street 143, 74400 Ashgabat, Turkmenistan

Received 5 July 2013; Revised 14 August 2013; Accepted 20 August 2013

Academic Editor: Abdullah Said Erdogan

Copyright © 2013 Charyyar Ashyralyyev and Mutlu Dedeturk. 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. A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, vol. 231 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet
  2. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, Walter de Gruyter GmbH & Company, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. V. V. Soloviev, “Inverse problems of source determination for the Poisson equation on the plane,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 44, no. 5, pp. 862–871, 2004 (Russian). View at MathSciNet
  4. D. G. Orlovskiĭ, “Inverse Dirichlet problem for an equation of elliptic type,” Differential Equations, vol. 44, no. 1, pp. 124–134, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. Orlovsky and S. Piskarev, “On approximation of inverse problems for abstract elliptic problems,” Journal of Inverse and Ill-Posed Problems, vol. 17, no. 8, pp. 765–782, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. Ashyralyev and A. S. Erdogan, “Well-posedness of the inverse problem of a multidimensional parabolic equation,” Vestnik of Odessa National University, Mathematics and Mechanics, vol. 15, no. 18, pp. 129–135, 2010.
  7. 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 MathSciNet
  8. 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. J11, pp. 73–81, 2011. View at MathSciNet
  9. C. Ashyralyyev, A. Dural, and Y. Sozen, “Finite difference method for the reverse parabolic problem,” Abstract and Applied Analysis, vol. 2012, Article ID 294154, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. C. Ashyralyyev and O. Demirdag, “The difference problem of obtaining the parameter of a parabolic equation,” Abstract and Applied Analysis, vol. 2012, Article ID 603018, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. Ch. Ashyralyyev and M. Dedeturk, “A finite difference method for the inverse elliptic problem with the Dirichlet condition,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 132–155, 2013.
  12. A. Ashyralyev and M. Urun, “Determination of a control parameter for the Schrodinger equation,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 156–166, 2013.
  13. 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 MathSciNet
  14. V. G. Romanov, “A three-dimensional inverse problem of viscoelasticity,” Doklady Mathematics, vol. 84, no. 3, pp. 833–836, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. S. I. Kabanikhin, “Definitions and examples of inverse and ill-posed problems,” Journal of Inverse and Ill-Posed Problems, vol. 16, no. 4, pp. 317–357, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Y. S. Éidel'man, “An inverse problem for an evolution equation,” Mathematical Notes, vol. 49, no. 5, pp. 535–540, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  17. K. Sakamoto and M. Yamamoto, “Inverse heat source problem from time distributing overdetermination,” Applicable Analysis, vol. 88, no. 5, pp. 735–748, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. V. Gulin, N. I. Ionkin, and V. A. Morozova, “On the stability of a nonlocal two-dimensional finite-difference problem,” Differential Equations, vol. 37, no. 7, pp. 970–978, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  19. G. Berikelashvili, “On a nonlocal boundary-value problem for two-dimensional elliptic equation,” Computational Methods in Applied Mathematics, vol. 3, no. 1, pp. 35–44, 2003. View at MathSciNet
  20. M. P. Sapagovas, “Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions,” Differential Equations, vol. 44, no. 7, pp. 1018–1028, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  21. 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 MathSciNet
  22. D. Orlovsky and S. Piskarev, “The approximation of Bitzadze-Samarsky type inverse problem for elliptic equations with Neumann conditions,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 118–131, 2013.
  23. A. Ashyralyev and F. S. O. Tetikoglu, “FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 454831, 22 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. A. Ashyralyev and E. Ozturk, “The numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition,” Abstract and Applied Analysis, vol. 2013, Article ID 730804, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. A. Ashyralyev, “Well-posedness of the difference schemes for elliptic equations in Cτβ,γ(E) spaces,” Applied Mathematics Letters, vol. 22, no. 3, pp. 390–395, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  26. S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, Russia, 1966.
  27. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  28. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.
  29. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2, Birkhäuser, Basel, Switzerland, 1989.