Abstract and Applied Analysis

FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions

Allaberen Ashyralyev and Fatma Songul Ozesenli Tetikoglu

Full-text: Open access


A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. The first and second-orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.

Article information

Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 454831, 22 pages.

First available in Project Euclid: 5 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ashyralyev, Allaberen; Ozesenli Tetikoglu, Fatma Songul. FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 454831, 22 pages. doi:10.1155/2012/454831. https://projecteuclid.org/euclid.aaa/1365174075

Export citation


  • O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, Russia, 1973.
  • M. L. Vishik, A. D. Myshkis, and O. A. Oleinik, “Partial differential equations,” in Mathematics in USSR in the Last 40 Years, 1917–1957, vol. 1, pp. 563–599, Fizmatgiz, Moscow, Russia, 1959.
  • S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, Russia, 1966.
  • V. L. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Differential-Operator Equations, Naukova Dumka, Kiev, Russia, 1984.
  • 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.
  • F. Criado-Aldeanueva, F. Criado, N. Odishelidze, and J. M. Sanchez, “On a control problem governed by a linear partial differential equation with a smooth functional,” Optimal Control Applications & Methods, vol. 31, no. 6, pp. 497–503, 2010.
  • A. Ashyralyev, “Nonlocal boundary-value problems for elliptic equations: well-posedness in Bochner spaces,” in Proceedings of the ICMS International Conference on Mathematical Science, vol. 1309 of AIP Conference Proceedings, pp. 66–84, Bolu, Turkey, November 2010.
  • A. Ashyralyev, “On well-posedness of the nonlocal boundary value problems for elliptic equations,” Numerical Functional Analysis and Optimization, vol. 24, no. 1-2, pp. 1–15, 2003.
  • I. A. Gurbanov and A. A. Dosiev, “On the numerical solution of nonlocal boundary problems for quasilinear elliptic equations,” in Approximate Methods for Operator Equations, pp. 64–74, Baku State University, Baku, Azerbaijan, 1984.
  • A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, 2 Iterative Methods, Birkhäuser, Basel, Switzerland, 1989.
  • A. Ashyralyev, C. Cuevas, and S. Piskarev, “On well-posedness of difference schemes for abstract elliptic problems in ${\text{L}}_{p}$$([0,1],E)$ spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 1-2, pp. 43–65, 2008.
  • A. V. Bitsadze and A. A. Samarskii, “On some simplest generalizations of linear elliptic problems,” Doklady Akademii Nauk SSSR, vol. 185, pp. 739–740, 1969.
  • A. Ashyralyev and E. Ozturk, “Numerical solutions of Bitsadze-Samarskii problem for elliptic equations,” in Further Progress in Analysis: Proceedings of the 6th International ISAAC Congress Ankara, Turkey 13–18 August 2007, pp. 698–707, World Scientific, 2009.
  • A. P. Soldatov, “A problem of Bitsadze-Samarskii type for second-order elliptic systems on the plane,” Russian in Doklady Akademii Nauk, vol. 410, no. 5, pp. 607–611, 2006.
  • D. G. Gordeziani, “On a method of resolution of Bitsadze-Samarskii boundary value problem,” Abstracts of Reports of Institute of Applied Mathematics, vol. 2, pp. 38–40, 1970.
  • D. V. Kapanadze, “On the Bitsadze-Samarskii nonlocal boundary value problem,” Differential Equations, vol. 23, no. 3, pp. 543–545, 1987.
  • 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.
  • 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.
  • V. Shakhmurov and R. Shahmurov, “Maximal B-regular integro-differential equation,” Chinese Annals of Mathematics B, vol. 30, no. 1, pp. 39–50, 2009.
  • 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.
  • P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University, Voronezh, Russia, 1975.