Abstract and Applied Analysis

A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

Yildirim Ozdemir and Mehmet Kucukunal

Full-text: Open access

Abstract

The nonlocal boundary value problem d 2 u ( t ) / d t 2 + A u ( t ) = f ( t )    ( 0 t 1 ) , i ( d u ( t ) / d t ) + A u ( t ) = g ( t )    ( - 1 t 0 ) , u ( 0 + ) = u ( 0 - ) , u t ( 0 + ) = u t ( 0 - ) , A u ( - 1 ) = α u ( μ ) + φ , 0 < μ 1 , for hyperbolic Schrödinger equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 687321, 12 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174045

Digital Object Identifier
doi:10.1155/2012/687321

Mathematical Reviews number (MathSciNet)
MR2935138

Zentralblatt MATH identifier
1246.35035

Citation

Ozdemir, Yildirim; Kucukunal, Mehmet. A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 687321, 12 pages. doi:10.1155/2012/687321. https://projecteuclid.org/euclid.aaa/1365174045


Export citation

References

  • M. S. Salakhitdinov, Equations of Mixed-Composite Type, FAN, Tashkent, Uzbekistan, 1974.
  • T. D. Djuraev, Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, FAN, Tashkent, Uzbekistan, 1979.
  • M. G. Karatopraklieva, “A nonlocal boundary value problem for an equation of mixed type,” Differensial'nye Uravneniya, vol. 27, no. 1, p. 68, 1991 (Russian).
  • D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat, Turkmenistan, 1995.
  • S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.
  • A. Ashyralyev and N. Aggez, “A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations,” Numerical Functional, vol. 25, no. 5-6, pp. 439–462, 2004.
  • A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary value problems for hyperbolic-parabolic equations,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1075–1089, 2007.
  • A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems for elliptic-parabolic differential and difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 904824, 16 pages, 2008.
  • A. Ashyralyev and A. Sirma, “Nonlocal boundary value problems for the Schrödinger equation,” Computers & Mathematics with Applications, vol. 55, no. 3, pp. 392–407, 2008.
  • A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,” Taiwanese Journal of Mathematics, vol. 14, no. 1, pp. 165–194, 2010.
  • A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011.
  • A. Ashyralyev and F. Ozger, “The hyperbolic-elliptic equation with the nonlocal condition,” AIP Conference Proceedings, vol. 1389, pp. 581–584, 2011.
  • Z. Zhao and X. Yu, “Hyperbolic Schrödinger equation,” Advances in Applied Clifford Algebras, vol. 14, no. 2, pp. 207–213, 2004.
  • A. A. Oblomkov and A. V. Penskoi, “Laplace transformations and spectral theory of two-dimensional semidiscrete and discrete hyperbolic Schrödinger operators,” International Mathematics Research Notices, no. 18, pp. 1089–1126, 2005.
  • A. Avila and R. Krikorian, “Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles,” Annals of Mathematics, vol. 164, no. 3, pp. 911–940, 2006.
  • M. Kozlowski and J. M. Kozlowska, “Development on the Schrodinger equation for attosecond laser pulse interaction with planck gas,” Laser in Engineering, vol. 20, no. 3-4, pp. 157–166, 2010.
  • K. Tselios and T. E. Simos, “Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 173–181, 2005.
  • D. P. Sakas and T. E. Simos, “Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 161–172, 2005.
  • G. Psihoyios and T. E. Simos, “A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 137–147, 2005.
  • Z. A. Anastassi and T. E. Simos, “An optimized Runge-Kutta method for the solution of orbital problems,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 1–9, 2005.
  • T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616–1621, 2009.
  • S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase-lag order of nonlinear explicit two-step $P$-stable method for linear periodic IVPs,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2467–2474, 2009.
  • T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331–1352, 2010.
  • M. E. Koksal, “Recent developments on operator-difference schemes for solving nonlocal BVPs for the wave equation,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 210261, 14 pages, 2011.
  • A. Ashyralyev and M. E. Koksal, “On the numerical solution of hyperbolic PDEs with variable space operator,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1086–1099, 2009.
  • H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.
  • S. Piskarev and S.-Y. Shaw, “On certain operator families related to cosine operator functions,” Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 527–546, 1997.
  • P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdat, Voronezh Gosud University, Voronezh, Russia, 1975.