Abstract

In this study, the initial value stochastic Schrödinger type problem in an abstract Hilbert space with the self-adjoint operator is investigated.

Rothe-Maruyama method for the numerical solution of this problem is presented. Theorem on the convergence of this difference scheme is established. A numerical example is given.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 2
Year: 2021

DOI: 10.12732/ijam.v34i2.3

Download Section

Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

1. [1] A. Ashyralyev, On modified Crank-Nicholson difference schemes for stochastic parabolic equation, Numer. Funct. Anal. Optim., 29, No 3-4 (2008), 268-282.
2. [2] A. Ashyralyev, A. Sirma, Nonlocal boundary value problems for the Schr¨odinger equation, Comput. Math. Appl., 55 (2008), 392-407.
3. [3] A. Kharab, H. Eleuch, Analytical solution of the position dependent mass Schr¨odinger equation with a hyperbolic tangent potential, International Journal of Applied Mathematics, 32, No 2 (2019), 357-367; doi:10.12732/ijam.v32i2.14.
4. [4] A. Ashyralyev, M.E. San, An approximation of semigroups method for stochastic parabolic equations, Abstr. Appl. Anal., 2012 (2012), 1-24; doi:10.1155/2012/684248.
5. [5] C. Privot, M. R¨ochner, A Concise Course on Stochastic Partial Differential Equations, Springer-Verlag, Berlin (2007).
6. [6] L. Bouten, M. Guta, H. Maassen, Stochastic Schr¨odinger equations, J. Phys. A. Math. Gen., 37 (2004), 3189-3209.
7. [7] C. Ashyralyyev, Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition, Math. Meth. Appl. Sci., 43 (2020), 5369–5379.
8. [8] H.O. Fattorini, Second Order Linear Differential Equations in Banach Space, Notas de Matematica, North-Holland (1985).
9. [9] A. Ashyralyev, H.O. Fattorini, On uniform difference-schemes for 2ndorder singular pertubation problems in Banach spaces, SIAM J. Math. Anal., 23, No 1 (1992), 29–54.
10. [10] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications Book 152, Cambridge University Press (2014). 258 A. Sirma
11. [11] A. Ashyralyev, D. Agirseven, On the stable difference schemes for the Schr¨odinger equation with time delay, Computational Methods in Applied Mathematics, 20, No 1 (2020), 27-38.