Abstract and Applied Analysis

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

Charyyar Ashyralyyev and Mutlu Dedeturk

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Abstract

A finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the first and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 548017, 11 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450133

Digital Object Identifier
doi:10.1155/2013/548017

Mathematical Reviews number (MathSciNet)
MR3111818

Zentralblatt MATH identifier
1291.65328

Citation

Ashyralyyev, Charyyar; Dedeturk, Mutlu. Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 548017, 11 pages. doi:10.1155/2013/548017. https://projecteuclid.org/euclid.aaa/1393450133


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References

  • 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.
  • 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.
  • 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).
  • D. G. Orlovskiĭ, “Inverse Dirichlet problem for an equation of elliptic type,” Differential Equations, vol. 44, no. 1, pp. 124–134, 2008.
  • 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.
  • 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.
  • A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. Dehghan, “Determination of a control parameter in the two-dimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001.
  • V. G. Romanov, “A three-dimensional inverse problem of viscoelasticity,” Doklady Mathematics, vol. 84, no. 3, pp. 833–836, 2011.
  • 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.
  • Y. S. Éidel'man, “An inverse problem for an evolution equation,” Mathematical Notes, vol. 49, no. 5, pp. 535–540, 1991.
  • K. Sakamoto and M. Yamamoto, “Inverse heat source problem from time distributing overdetermination,” Applicable Analysis, vol. 88, no. 5, pp. 735–748, 2009.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • A. Ashyralyev, “Well-posedness of the difference schemes for elliptic equations in ${C}_{\tau }^{\beta ,\gamma }(E)$ spaces,” Applied Mathematics Letters, vol. 22, no. 3, pp. 390–395, 2009.
  • S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, Russia, 1966.
  • 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.
  • P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.
  • A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2, Birkhäuser, Basel, Switzerland, 1989.