A Note On The Stability of Solution for Elliptic-Schrödinger Type Nonlocal Boundary Value Problem

Abstract

In the present article, a problem for a Elliptic-Schrödinger equation with nonlocal boundary value problem is considered. The stability estimates are established for the solution of Elliptic-Schrödinger problem for nonlocal boundary problem . A theorem, with proof, for stability of the solution of this problem for differential equations of elliptic-Schrödinger type in a Hilberts space H with self-adjoint positive definite operator A is considered. On the other hand, conclusion section is presented.

Keywords

• Partila differential equation
• Nonlocal boundary value problem
• stability

References

1. [1] M. S. Salahitdinov, Equations of Mixed-Composite Type, FAN, Tashkent, Uzbekistan, 1974.
2. [2] T. D. Drjuev, Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, FAN, Tashkent, Uzbekistan, 1979.
3. [3] M. G. Karatopraklieva, “A nonlocal boundary value problem for an equation of mixed type”, Differensial'nye Uravneniya, vol. 27, pp. 68-79, 1991, in Russian.
4. [4] D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat, Turkmenistan, 1995.
5. [5] S. N. Glazatov, Sobolev Inst. of Math. SB RAS, Preprint no: 46, 26p (1998).
6. [6] A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary valur problems for hyperbolic-parabolic equations”, Taiwan. J. Math., vol. 11, pp. 1075-1089, 2007.
7. [7] A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems of elliptic-parabolic differential and difference equations”, Discrete. Dyn. Nat. Soc. vol. 2008, pp. 1-16, 2008.
8. [8] A. Ashyralyev and A. Sirma, “Nonlocal boundary value problems for Shrödinger equations”, Comput. Math. Appl., vol. 55, pp. 392-407, 2008.

9. [9] A. Ashyralyev and O. Yıldırım, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations”, Taiwanese Journal of Mathematics, vol 14, no. 1, pp. 165-194, 2010.
10. [10] A. Ashyralyev, and B. Hicdurmaz, “A note on the fractional Schrödinger differential equatios”, Kybernetes, vol. 40, pp. 736-750, 2011.
11. [11] Y. Ozdemir, and M. Kucukunal, “A note on nonlocal boundary value problems for hyperbolic Schrödinger equations”, Abstr. Appl. Anal., vol. 2012, pp. 1-12, 2012.
12. [12] A. Ashyralyev and O. Yıldırım, “A note on the second order accuracy stable difference schemes for the nonlocal boundary value hyperbolic problem”, Abstr. Appl. Anal., vol. 2012, pp. 1-29, 2012.
13. [13] P. Quittner and P. Souplet, “Optimal Lioville-type theorems for noncooperative elliptic Schrödinger systems and equations”, Comm. Math. Phys., vol. 311, pp. 1-19, 2012.
14. [14] B. Liu and L. Ma, “Symmetry results for eliiptic Schrödinger systems on half spaces”, J. Math. Anal. Appl., vol. 401, pp. 259-268, 2013.
15. [15] N. Godet and N. Tzvetkov, “Strichartz estimates for the periodic non-elliptic Schrödinger equation”, vol. 350, no. 21-22, pp. 995-958, 2012.
16. [16] P. Souplet, “Lioville-type theorems for ellitic Schrödinger systems associated with copositive matrices”, Networks & Heterogeneous Media, vol. 7, no. 4, pp. 697-988, 2012.
17. [17] Y. Ozdemir and M. Eser, “Numerical solution of the elliptic-Schrödinger equation with the Dirichlet and Neumann condition”, AIP Conference Proceedings, vol. 1611, pp. 410-414, 2014.
18. [18] Y. Ozdemir and M. Eser, “On nonlocal boundary value problems for elliptic-Schrödinger equations”, AIP Conference Proceedings, vol. 1676, 2015.