On the Bitsadze-Samarskii Type Nonlocal Boundary Value Problem with the Integral Condition for an Elliptic Equation

Abstract

In the present paper, the Bitsadze-Samarskii type nonlocal boundary value problem with the integral condition for an abstract elliptic differential equation in a Hilbert space is studied. Theorem on well-posedness of this problem in H\"{o}lder spaces with a weight is established. The nonlocal boundary value problem for multidimensional elliptic equations with the Dirichlet condition is studied. The first order of accuracy difference scheme for the approximate solution of the Bitsadze-Samarskii type nonlocal boundary value problem is investigated. Theorem on well-posedness of this difference scheme in difference analogue of H\"{o}lder spaces with a weight is established.

Keywords

• Bitsadze-Samarskii type nonlocal boundary value problem,Difference scheme,Elliptic Equation,Wellposedness

References

1. [1] A. V. Bitsadze, A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Doklady Akademii Nauk SSSR, 85(4) (1969), 739-740.
2. [2] D. V. Kapanadze, On a nonlocal Bitsadze-Samarskii boundary value problem, Differ. Uravn., 23 (1987), 543-545.
3. [3] V. A. Il’in, E. I. Moiseev, A two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference interpretation, Mat. Model.,2 (1990), 139-156.
4. [4] A. Ashyralyev, A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space, J. Math. Anal. Appl., 344 (2008), 557-573.
5. [5] A. Ashyralyev, E. Ozturk,On Bitsadze-Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: Well-Posedness, Appl. Math. Comput., 219 (2012), 1093-1107.
6. [6] A. Ashyralyev, E. Ozturk, On a difference scheme of fourth-order of accuracy for the Bitsadze-Samarskii type nonlocal boundary value problem, Math. Methods Appl. Sci., 36 (2013), 936-955.
7. [7] A. Ashyralyev, E. Ozturk, On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary value problem, Bound. Value Probl., 14 (2014), 1687-2770.
8. [8] A. Ashyralyev, E. Ozturk, Stability of difference schemes for Bitsadze-Samarskii type nonlocal boundary value problem involving integral condition, Filomat,28 (2014), 1027- 1047.

9. [9] E. A. Volkova, A. A. Dosiyev, S. C. Buranay, On the solution of a nonlocal problem, Comput. Math. Appl., 66 (2013), 330-338.
10. [10] Z. V. Kiguradze, Domain decomposition and parallel algorithm for Bitsadze- Samarskii boundary value problem, Rep. Enlarged Sess. Semin. I.Vekua Appl. Math., 10 (1995), 49-51.
11. [11] T. Jangveladze, Z. Kiguradze, G. Lobjanidze, On variational formulation and decomposition methods for Bitsadze-Samarskii nonlocal boundary value problem for two-dimensional second order elliptic equations, Proc. I.Vekua Inst. Appl. Math., 57 (2007), 66-77.
12. [12] P. E. Sobolevskii, On elliptic equations in a Banach space, Differ. Uravn.,4 (1969), 1346-1348.
13. [13] S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, 1966.
14. [14] A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, Basel, 1997.
15. [15] P. E. Sobolevskii, Well-posedness of difference elliptic equations, Rendiconti dell’Istituto di Matematica dell’Università di Trieste, 28 (1996), 337-382.
16. [16] A. Ashyralyev, Well-posed solvability of the boundary value problem for difference equations of elliptic type, Nonlinear Analy-Theor, 24 (1995), 251-256.
17. [17] A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Operator Theory: Advances and Applications, Birkhäuser, Basel, 2004.
18. [18] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions-II, Comm. Pure Appl. Math., 17 (1964), 35-92.
19. [19] G. Berikelashvili, On a nonlocal boundary value problem for a two-dimensional elliptic equation, Comput. Methods Appl. Math., 3 (2003), 35-44.
20. [20] A. Ashyralyev, C. Cuevas, S. Piskarev, On well-posedness of difference schemes for abstract elliptic problems in Lp ([0; 1] ;E) spaces, Numer. Func. Anal. Opt., 29 (2008), 43-65.
21. [21] A. Ashyralyev, Well-posedness of the difference schemes for elliptic equations in C; (E) spaces, Appl. Math. Lett., 22 (2009), 390-395.
22. [22] P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, 1975.
23. [23] L. M. Gershteyn, P. E. Sobolevskii, Well-posedness of the general boundary value problem for the second order elliptic equations in a Banach space, Differ. Uravn., 11 (1975), 1335-1337.
24. [24] A. Ashyralyev, Ch. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Anal.-Model., 19 (2014), 350-366.
25. [25] A. Ashyralyev, Well-posedness of the elliptic equations in a space of smooth functions, Boundary value problems for nonclassic equations of mathematical physics, Sibirsk, Otdel, SSSR- Novosibirsk,2 (1989), 82-86.