Abstract
In the present paper, the first and second order of accuracy difference schemes for the approximate solutions of the initial value problem for Schrödinger equation with time delay in a Hilbert space are presented. The theorem on stability estimates for the solutions of these difference schemes is established. The application of theorems on stability of difference schemes for the approximate solutions of the initial boundary value problems for Schrödinger partial differential equation is provided. Additionally, some illustrative numerical results are presented.
Funding source: Ministry of Education and Science of the Russian Federation
Award Identifier / Grant number: 02.A03.21.0008
Funding statement: The publication has been prepared with the support of the “RUDN University Program 5-100” and published under target program BR05236656 of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan.
Acknowledgements
We would like to thank the referees for their helpful suggestions to the improvement of our paper.
References
[1] D. Agirseven, On the stability of the Schrödinger equation with time delay, Filomat 32 (2018), no. 3, 759–766. 10.2298/FIL1803759ASearch in Google Scholar
[2] X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math. Comp. 73 (2004), no. 248, 1779–1799. 10.1090/S0025-5718-04-01631-XSearch in Google Scholar
[3] A. Ashyralyev and D. Agirseven, On convergence of difference schemes for delay parabolic equations, Comput. Math. Appl. 66 (2013), no. 7, 1232–1244. 10.1016/j.camwa.2013.07.018Search in Google Scholar
[4] A. Ashyralyev and D. Agirseven, Bounded solutions of nonlinear hyperbolic equations with time delay, Electron. J. Differential Equations 2018 (2018), Paper No. 21. Search in Google Scholar
[5] A. Ashyralyev and B. Hicdurmaz, A note on the fractional Schrödinger differential equations, Kybernetes 40 (2011), no. 5–6, 736–750. 10.1108/03684921111142287Search in Google Scholar
[6] A. Ashyralyev and B. Hicdurmaz, On the numerical solution of fractional Schrödinger differential equations with the Dirichlet condition, Int. J. Comput. Math. 89 (2012), no. 13–14, 1927–1936. 10.1080/00207160.2012.698841Search in Google Scholar
[7] A. Ashyralyev and B. Hicdurmaz, A stable second order of accuracy difference scheme for a fractional Schrödinger differential equation, Appl. Comput. Math. 17 (2018), no. 1, 10–21. Search in Google Scholar
[8] A. Ashyralyev and A. Sarsenbi, Well-posedness of an elliptic equation with involution, Electron. J. Differential Equations 2015 (2015), Paper No. 284. 10.1186/s13661-015-0297-5Search in Google Scholar
[9] A. Ashyralyev and A. Sirma, Nonlocal boundary value problems for the Schrödinger equation, Comput. Math. Appl. 55 (2008), no. 3, 392–407. 10.1016/j.camwa.2007.04.021Search in Google Scholar
[10] A. Ashyralyev and A. Sirma, A note on the numerical solution of the semilinear Schrödinger equation, Nonlinear Anal. 71 (2009), no. 12, e2507–e2516. 10.1016/j.na.2009.05.048Search in Google Scholar
[11] H. Bereketlioglu and M. Lafci, Behavior of the solutions of a partial differential equation with a piecewise constant argument, Filomat 31 (2017), no. 19, 5931–5943. 10.2298/FIL1719931BSearch in Google Scholar
[12] S. Bhalekar and J. Patade, Analytical solutions of nonlinear equations with proportional delays, Appl. Comput. Math. 15 (2016), no. 3, 331–345. Search in Google Scholar
[13] T. Chen, S.-f. Zhou and C.-d. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser. 26 (2010), no. 4, 633–642. 10.1007/s10255-007-7101-ySearch in Google Scholar
[14] K. L. Cooke and J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), 265–297. 10.1016/0022-247X(84)90248-8Search in Google Scholar
[15] K. L. Cooke and J. Wiener, A survey of differential equations with piecewise continuous arguments, Delay Differential Equations and Dynamical Systems, Lecture Notes in Math. 1475, Springer, Berlin (1991), 1–15. 10.1007/BFb0083475Search in Google Scholar
[16] H.-Y. Cui, Z.-J. Han and G.-Q. Xu, Stabilization for Schrödinger equation with a time delay in the boundary input, Appl. Anal. 95 (2016), no. 5, 963–977. 10.1080/00036811.2015.1047830Search in Google Scholar
[17] G. Eskin and J. Ralston, Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Comm. Math. Phys. 173 (1995), no. 1, 199–224. 10.1007/BF02100187Search in Google Scholar
[18] D. G. Gordeziani and G. A. Avalishvili, Time-nonlocal problems for Schrödinger-type equations. I: Problems in abstract spaces, Differ. Equ. 41 (2005), no. 5, 703–711. 10.1007/s10625-005-0205-3Search in Google Scholar
[19] D. G. Gordeziani and G. A. Avalishvili, Time-nonlocal problems for Schrödinger-type equations. II: Results for specific problems, Differ. Equ. 41 (2005), no. 6, 852–859. 10.1007/s10625-005-0224-0Search in Google Scholar
[20] B.-Z. Guo and Z.-C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems Control Lett. 54 (2005), no. 11, 1135–1142. 10.1016/j.sysconle.2005.04.008Search in Google Scholar
[21] B.-Z. Guo and K.-Y. Yang, Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay, IEEE Trans. Automat. Control 55 (2010), no. 5, 1226–1232. 10.1109/TAC.2010.2042363Search in Google Scholar
[22] H. Han, J. Jin and X. Wu, A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain, Comput. Math. Appl. 50 (2005), no. 8–9, 1345–1362. 10.1016/j.camwa.2005.05.006Search in Google Scholar
[23] W. Kang and E. Fridman, Boundary constrained control of delayed nonlinear Schrödinger equation, IEEE Trans. Automat. Control 63 (2018), no. 11, 3873–3880. 10.1109/TAC.2018.2800526Search in Google Scholar
[24] G. Kuralay and H. Özbay, Design of first order controllers for a flexible robot arm with time delay, Appl. Comput. Math. 16 (2017), no. 1, 48–58. Search in Google Scholar
[25] M. E. Mayfield, Nonreflective boundary conditions for Schrodinger’s equation, ProQuest LLC, Ann Arbor, MI, 1989; Ph.D. thesis, University of Rhode Island, 1989. Search in Google Scholar
[26] H. Nakatsuji, Inverse Schrödinger equation and the exact wave function, Phys. Rev. A 65 (2002), Article ID 052122. 10.1103/PhysRevA.65.052122Search in Google Scholar
[27] S. Nicaise and S.-E. Rebiai, Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback, Port. Math. 68 (2011), no. 1, 19–39. 10.4171/PM/1879Search in Google Scholar
[28] V. Serov and L. Päivärinta, Inverse scattering problem for two-dimensional Schrödinger operator, J. Inverse Ill-Posed Probl. 14 (2006), no. 3, 295–305. 10.1515/156939406777340946Search in Google Scholar
[29] A. L. Skubachevskiĭ, On the problem of damping a control system with aftereffect, Dokl. Akad. Nauk 335 (1994), no. 2, 157–160. Search in Google Scholar
[30] V. V. Smagin and E. V. Shepilova, Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time, Differ. Equ. 44 (2008), no. 4, 580–592. 10.1134/S0012266108040113Search in Google Scholar
[31] P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations (in Russian), Izdat. Voronezh. Gosud. Univ., Voronezh, 1975. Search in Google Scholar
[32] K. Sriram and M. S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa, J. Theoret. Biol. 231 (2004), no. 1, 23–38. 10.1016/j.jtbi.2004.04.006Search in Google Scholar PubMed
[33] J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle, J. Theoret. Biol. 241 (2006), no. 3, 617–627. 10.1016/j.jtbi.2005.12.020Search in Google Scholar PubMed
[34] J. Sun, L. Kou, G. Guo, G. Zhao and Y. Wang, Existence of weak solutions of stochastic delay differential systems with Schrödinger–Brownian motions, J. Inequal. Appl. (2018), Paper No. 100. 10.1186/s13660-018-1691-1Search in Google Scholar PubMed PubMed Central
[35] V. V. Vlasov and N. A. Rautian, Spectral Analysis of Functional Differential Equations, MAKS Press, Moscow, 2016. Search in Google Scholar
[36] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific Publishing, Singapore, 1993. 10.1142/1860Search in Google Scholar
[37] K.-Y. Yang and C.-Z. Yao, Stabilization of one-dimensional Schrödinger equation with variable coefficient under delayed boundary output feedback, Asian J. Control 15 (2013), no. 5, 1531–1537. 10.1002/asjc.667Search in Google Scholar
[38] Z. Zhao and W. Ge, Traveling wave solutions for Schrödinger equation with distributed delay, Appl. Math. Model. 35 (2011), no. 2, 675–687. 10.1016/j.apm.2010.07.025Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston