Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2022, Cilt: 7 Sayı: 2, 107 - 115, 30.09.2022

Öz

Kaynakça

  • [1] Adams RA. Sobolev spaces. Academic Press, New York, 1975.
  • [2] Alomari AK, Noorani MSM, Nazar R. Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation. 14(4), 2009, 1196-1207.
  • [3] Becerril R, Guzman FS, Rendon-Romero A, Valdez-Alvarado S. Solving the time-dependent Schrödinger equation using finite difference methods. Revista Mexicana de Fisica E. 54(2), 2008, 120-132.
  • [4] Biazar J, Ghazvini H. Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method. Physics Letters A. 366(1-2), 2007, 79-84.
  • [5] Cavalcanti MM, Correa WJ, Sepulveda CMA, Asem RV. Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping. Studia Universitatis Babes¸-Bolyai Mathematica. 64(2), 2019, 161-172.
  • [6] Chagas CQ, Diehl NML, Guidolin PL. Some properties for the Steklov averages. 2017; Available at arXiv:1707.06368.
  • [7] Chan TF, Lee D, Shen L. Stable explicit schemes for equations of the Schrödinger type. SIAM Journal on Numerical Analysis. 23(2), 1986, 274-281.
  • [8] Chan TF, Shen L. Stability analysis of difference schemes for variable coefficient Schrödinger type equations. SIAM Journal on Numerical Analysis. 24(2), 1987, 336-349.
  • [9] Chen J, Zhang LM. Numerical approximation of solution for the coupled nonlinear Schrödinger equations. Acta Mathematicae Applicatae Sinica. English Series 33(2), 2017, 435-450.
  • [10] Dai W. An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient. SIAM Journal on Numerical Analysis. 29(1), 1992, 174-181.
  • [11] Ghanbari B. An analytical study for (2+1)-dimensional Schrödinger equation. The Scientific World Journal . Vol.2014, 2014, 1-5.
  • [12] He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons and Fractals. 26(3), 2005, 695-700.
  • [13] Hosseinzadeh Kh. An analytic approximation to the solution of Schrödinger equation by VIM. Applied Mathematical Sciences. 11(16), 2017, 813-818.
  • [14] Hu H, Hu H. Maximum norm error estimates of fourth-order compact difference scheme for the nonlinear Schrödinger equation involving a quintic term. Journal of inequalities and applications. 2018:180, 2018.
  • [15] Iskenderov AD, Yagubov GY. Optimal control problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation. Proceedings of the Lankaran State University, Natural Sciences Series, 2007, 3-56.
  • [16] Iskenderov AD, Yagubov GY, Musayeva MA. Identification of Quantum potentials. Casıoglu, Baku, Azerbaijan, 2012.
  • [17] Ivanauskas F, Radziunas M. On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations. SIAM Journal on Numerical Analysis. 36(5), 1999, 1466-1481.
  • [18] Khuri SA. A new approach to the cubic Schrödinger eguation: An application of the decomposition technique. Applied Mathematics and Computation. 97(2-3), 1998, 251-254.
  • [19] Kücük GD, Yagub G, Celik E. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete Contin Dyn Syst-S. 12(3), 2019, 503-512.
  • [20] Ladyzhenskaya OA, Solonnikov VA, Ural’tseva NN. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. (Enligsh transl.), Providence, R.I.,1968.
  • [21] Mahmudov NM. Solvability of boundary value problems for a Schrödinger equation with pure ımaginary coefficient in the nonlinear part of this equation. Proceedings of IMM of NAS of Azerbaijan. 27(35), 2007, 25-36.
  • [22] Mousaa MM, Ragab SF. Application of the homotopy perturbation method to linear and nonlinear Schrödinger equations. Zeitschrift für Naturforschung A. 63a, 2008, 140-144.
  • [23] Potapov MM, Razgulin AV. Difference methods in problems of the optimal control problems of the steady selfaction of light beams. USSR Comput. Math. and Math. Phys. 30(4), 1990, 134-142.
  • [24] Radziünas M. On convergence and stability of difference schemes for nonlinear Schrödinger type equations. Lithuanian Mathematical Journal. 36(2), 1996, 178-194.
  • [25] Sadighi A, Ganji DD. Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy-perturbation and Adomian decomposition methods. Physics Letters A. 372(4), 2008, 465-469.

On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator

Yıl 2022, Cilt: 7 Sayı: 2, 107 - 115, 30.09.2022

Öz

In this paper, we apply the finite difference method to a Schrödinger equation which contains a momentum operator. For this, we constitute a difference scheme. A priori estimate for the solution of difference scheme is obtained. By using this estimate, we prove that the difference scheme is unconditionally stable.

Kaynakça

  • [1] Adams RA. Sobolev spaces. Academic Press, New York, 1975.
  • [2] Alomari AK, Noorani MSM, Nazar R. Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation. 14(4), 2009, 1196-1207.
  • [3] Becerril R, Guzman FS, Rendon-Romero A, Valdez-Alvarado S. Solving the time-dependent Schrödinger equation using finite difference methods. Revista Mexicana de Fisica E. 54(2), 2008, 120-132.
  • [4] Biazar J, Ghazvini H. Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method. Physics Letters A. 366(1-2), 2007, 79-84.
  • [5] Cavalcanti MM, Correa WJ, Sepulveda CMA, Asem RV. Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping. Studia Universitatis Babes¸-Bolyai Mathematica. 64(2), 2019, 161-172.
  • [6] Chagas CQ, Diehl NML, Guidolin PL. Some properties for the Steklov averages. 2017; Available at arXiv:1707.06368.
  • [7] Chan TF, Lee D, Shen L. Stable explicit schemes for equations of the Schrödinger type. SIAM Journal on Numerical Analysis. 23(2), 1986, 274-281.
  • [8] Chan TF, Shen L. Stability analysis of difference schemes for variable coefficient Schrödinger type equations. SIAM Journal on Numerical Analysis. 24(2), 1987, 336-349.
  • [9] Chen J, Zhang LM. Numerical approximation of solution for the coupled nonlinear Schrödinger equations. Acta Mathematicae Applicatae Sinica. English Series 33(2), 2017, 435-450.
  • [10] Dai W. An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient. SIAM Journal on Numerical Analysis. 29(1), 1992, 174-181.
  • [11] Ghanbari B. An analytical study for (2+1)-dimensional Schrödinger equation. The Scientific World Journal . Vol.2014, 2014, 1-5.
  • [12] He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons and Fractals. 26(3), 2005, 695-700.
  • [13] Hosseinzadeh Kh. An analytic approximation to the solution of Schrödinger equation by VIM. Applied Mathematical Sciences. 11(16), 2017, 813-818.
  • [14] Hu H, Hu H. Maximum norm error estimates of fourth-order compact difference scheme for the nonlinear Schrödinger equation involving a quintic term. Journal of inequalities and applications. 2018:180, 2018.
  • [15] Iskenderov AD, Yagubov GY. Optimal control problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation. Proceedings of the Lankaran State University, Natural Sciences Series, 2007, 3-56.
  • [16] Iskenderov AD, Yagubov GY, Musayeva MA. Identification of Quantum potentials. Casıoglu, Baku, Azerbaijan, 2012.
  • [17] Ivanauskas F, Radziunas M. On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations. SIAM Journal on Numerical Analysis. 36(5), 1999, 1466-1481.
  • [18] Khuri SA. A new approach to the cubic Schrödinger eguation: An application of the decomposition technique. Applied Mathematics and Computation. 97(2-3), 1998, 251-254.
  • [19] Kücük GD, Yagub G, Celik E. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete Contin Dyn Syst-S. 12(3), 2019, 503-512.
  • [20] Ladyzhenskaya OA, Solonnikov VA, Ural’tseva NN. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. (Enligsh transl.), Providence, R.I.,1968.
  • [21] Mahmudov NM. Solvability of boundary value problems for a Schrödinger equation with pure ımaginary coefficient in the nonlinear part of this equation. Proceedings of IMM of NAS of Azerbaijan. 27(35), 2007, 25-36.
  • [22] Mousaa MM, Ragab SF. Application of the homotopy perturbation method to linear and nonlinear Schrödinger equations. Zeitschrift für Naturforschung A. 63a, 2008, 140-144.
  • [23] Potapov MM, Razgulin AV. Difference methods in problems of the optimal control problems of the steady selfaction of light beams. USSR Comput. Math. and Math. Phys. 30(4), 1990, 134-142.
  • [24] Radziünas M. On convergence and stability of difference schemes for nonlinear Schrödinger type equations. Lithuanian Mathematical Journal. 36(2), 1996, 178-194.
  • [25] Sadighi A, Ganji DD. Analytic treatment of linear and nonlinear Schrödinger equations: A study with homotopy-perturbation and Adomian decomposition methods. Physics Letters A. 372(4), 2008, 465-469.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Volume VII Issue II
Yazarlar

Nigar Yıldırım Aksoy 0000-0001-6767-3363

Emel Sarıahmet Bu kişi benim 0000-0003-0844-3810

Yayımlanma Tarihi 30 Eylül 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 7 Sayı: 2

Kaynak Göster

APA Yıldırım Aksoy, N., & Sarıahmet, E. (2022). On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator. Turkish Journal of Science, 7(2), 107-115.
AMA Yıldırım Aksoy N, Sarıahmet E. On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator. TJOS. Eylül 2022;7(2):107-115.
Chicago Yıldırım Aksoy, Nigar, ve Emel Sarıahmet. “On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator”. Turkish Journal of Science 7, sy. 2 (Eylül 2022): 107-15.
EndNote Yıldırım Aksoy N, Sarıahmet E (01 Eylül 2022) On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator. Turkish Journal of Science 7 2 107–115.
IEEE N. Yıldırım Aksoy ve E. Sarıahmet, “On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator”, TJOS, c. 7, sy. 2, ss. 107–115, 2022.
ISNAD Yıldırım Aksoy, Nigar - Sarıahmet, Emel. “On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator”. Turkish Journal of Science 7/2 (Eylül 2022), 107-115.
JAMA Yıldırım Aksoy N, Sarıahmet E. On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator. TJOS. 2022;7:107–115.
MLA Yıldırım Aksoy, Nigar ve Emel Sarıahmet. “On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator”. Turkish Journal of Science, c. 7, sy. 2, 2022, ss. 107-15.
Vancouver Yıldırım Aksoy N, Sarıahmet E. On the Stability of Finite Difference Scheme for the Schrödinger Equation Including Momentum Operator. TJOS. 2022;7(2):107-15.