TIME-DEPENDENT SOURCE IDENTIFICATION
SCHRÖDINGER TYPE PROBLEM
Allaberen Ashyralyev1,2,3, Mesut Urun1,4 1 Department of Mathematics, Near East University
Nicosia, TRNC Mersin – 10, TURKEY
2 Peoples' Friendship University of Russia
(RUDN University), Ul. Miklukho Maklaya 6
Moscow – 117198, RUSSIA
3 Institute of Mathematics and Mathematical Modeling
Almaty, KAZAKHSTAN
4 Maritime Vocational School, Galatasaray University
Istanbul – 34349, TURKEY

Abstract

In this study, the source identification problem with non-local boundary conditions for the time-dependent, one-dimensional Schrödinger equation is studied. Stability estimates are constructed for the solution of source identification problem. A first order of accuracy difference scheme is investigated for the numerical solution of this problem.

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.v34i2.7

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.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker (1987).
  2. [2] S.I. Kabanikhin, Methods for solving dynamic inverse problems for hyperbolic equations, J. of Inverse and Ill-Posed Problems, 12 (2014), 493-517.
  3. [3] Y.Y. Belov, Inverse Problems for Partial Differential Equations, In: Inverse and Ill-Posed Problems Series, 32, De Grayter (2002). 308 A. Ashyralyev, M. Urun
  4. [4] Y.A. Gryazin, M.V. Klibanov, T.R. Lucas, Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography, Inverse Problems, 15, No 2 (1999), 373-397.
  5. [5] C. Ashyralyyev, High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition, Boundary Value Problems, 2014, No 5 (2014); doi: 10.1186/1687-2770.
  6. [6] C. Ashyralyyev, High order approximation of the inverse elliptic problem with Dirichlet-Neumann conditions, Filomat, 28, No 5 (2014), 947-962.
  7. [7] A. Ashyralyev, C. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Analysis: Modelling and Control, 19, No 3 (2014), 350-366.
  8. [8] A. Ashyralyev, M. Urun, Determination of a control parameter for the difference Schr¨odinger equation, Abstract and Applied Analysis, 2013 (2013), ID 548201.
  9. [9] A.B. Kostin, The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation, Matematicheskii Sbornik, 204, No 10 (2013), 1391-1434 (In Russian).
  10. [10] A. Ashyralyev, On a problem of determining the parameter of a parabolic equation, Ukrainian Mathematical J., 62, No 9 (2010), 1200-1210.
  11. [11] M. Choulli, M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameter, J. of Inverse and III-posed Problems, 7, No 3 (1999), 241-254.
  12. [12] A. Ashyralyev, F. Emharab, Source identification problems for hyperbolic differential and difference equations. J. of Inverse and Ill-posed Problems, 27, No 3 (2019), 301-315.
  13. [13] A. Ashyralyev A.U. Sazaklioglu, Investigation of a time-dependent source identification inverse problem with integral overdetermination, Numerical Functional Analysis and Optimization, 38, No 10 (2017), 1276-1294.
  14. [14] S. Saitoh, V.K. Tuan, M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. of Inequalities in Pure and Applied Mathematics, 3, No 5 (2002), Art. 80. TIME-DEPENDENT SOURCE IDENTIFICATION... 309
  15. [15] N.I. Ivanchov, On the determination of unknown source in the heat equation with nonlocal boundary conditions, Ukrainian Mathematical J., 47, No 10 (1995), 1647-1652.
  16. [16] V.T. Borukhov, P.N. Vabishchevich, Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation, Computer Physics Communications, 126 (2000), 32-36.
  17. [17] A. Ashyralyev, A.S. Erdogan, Well-posedness of the right-hand side identification problem for a parabolic equation, Ukrainian Mathematical J., 66, No 2 (2014), 165-177.
  18. [18] A.A. Samarskii, P.N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, Walter de Gruyter, Berlin-New York (2007).
  19. [19] G. Di Blasio, A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, J. of Inverse and Ill-Posed Problems, 15, No 7 (2007), 709-734.
  20. [20] A. Ashyralyev, D. Agirseven, On source identification problem for a delay parabolic equation, Nonlinear Analysis: Modelling and Control, 19, No 3 (2014), 335-349.
  21. [21] I. Orazov and M.A. Sadybekov, On a class of problems of determining the temperature and density of heat sources given initial and final temperature, Siberian Mathematical J., 53 (2012), 146-151.
  22. [22] A. Ashyralyev, A.S. Erdogan, A.U. Sazaklioglu, Numerical solution of a source identification problem, Journal of Inverse and Ill-posed Problems, 27, No 4 (2019), 457-468.
  23. [23] F.A. Aliev, N.A. Ismailov, A.A. Namazov, M.F. Rajabov, Algorithm for calculating the parameters of formation of gas-liquid mixture in the shoe of gas lift well, Applied and Computational Mathematics, 15, No 3 (2016), 370-376.
  24. [24] B. Hicdurmaz, Finite difference schemes for time-fractional Schr¨odinger equations via fractional linear multi step method, International J. of Computer Mathematics (2020); doi:10.1080/00207160/1834088.
  25. [25] A. Ashyralyev, M. Urun, Time-dependent source identification problem for the Schr¨odinger equation with nonlocal boundary conditions, In: AIP 310 A. Ashyralyev, M. Urun Conf. Proc., 2183 (Third Int. Conf. of Math. Sci., Maltepe Univ, Istanbul, Sep. 04-08, 2019) (Ed. by H. Cakalli and others), Amer. Inst. Phys. (2019), Art. 070016.
  26. [26] B. Hicdurmaz, Initial Boundary Value Problems for Fractional Schr¨odinger Differential Equations, PhD Thesis, Gebze Technical University, Kocaeli (2015).
  27. [27] A. Sirma, Nonlocal Boundary Value Problems for Schr¨odinger Equation, PhD Thesis, Gebze Institute of Technology, Kocaeli (2007).