This paper includes a MAPLE® code giving numerical solution of two
dimensional Schrödinger equation in a functional space. The Galerkin
method has been used to get the approximate solution. The results have
been examined with numerical examples.
This study is one of very few studies which have investigated to obtain
an efficient computation tool for numerical examinations of two
dimensional Schrödinger Equation.
F. D. Tappert, "The parabolic approximation method. Wave propagation and underwater acoustic," Lecture Notes in Physics Springer, Berlin, vol. 70, pp. 224-287, 1977. View at Google Scholar | View at Publisher
M. Levy, Parabolic equation methods for electromagnetic wave propagation. London, United Kingdom: The Institution of Electrical Engineers, 2000.
Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Applications of the parabolic wave equations to X-ray diffraction optics," Optics Communications, vol. 118, pp. 619-636, 1995. View at Google Scholar | View at Publisher
W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: Analysis and assessment," Journal of Lightwave Technology, vol. 10, pp. 295-305, 1992. View at Google Scholar | View at Publisher
M. Dehghan and A. Taleei, "A chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrodinger equations," Computer Physics Communications, vol. 182, pp. 2519-2529, 2011. View at Google Scholar | View at Publisher
M. Dehghan and V. Mohammadi, "Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein-Gordon-Schrodinger (KGS) equations," Computer and Mathematics with Applications, vol. 71, pp. 892-921, 2016.View at Google Scholar | View at Publisher
M. Dehghan, "Finite difference procedures for solving a problem arising in modeling and design of certain opto-electronic devices," Mathematics and Computers in Simulation, vol. 71, pp. 16-30, 2006. View at Google Scholar | View at Publisher
R. J. Cheng and Y. M. Cheng, "Solving unsteady Schrodinger equation using the improved element-free Galerkin method," Chinese Physics B, vol. 25, pp. 1-9, 2016.
A. Xavier, B. Christophe, and K. Pauline, "Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and numerical results," Numerische Mathematik, vol. 125, pp. 191-223, 2013. View at Publisher
J. Kalita, P. Chhabra, and S. Kumar, "A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation," Journal of Computational and Applied Mathematics, vol. 197, pp. 141-149, 2006. View at Google Scholar | View at Publisher
M. Subaşı, "On the finite-difference schemes for the numerical solution of two dimensional Schrödinger equation," Numerical Methods of Partial Differential Equations, vol. 18, pp. 752-758, 2002. View at Google Scholar | View at Publisher
A. Taleei and M. Dehghan, "Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrodinger equations," Computer Physics Communications, vol. 185, pp. 1515-1528, 2014. View at Google Scholar | View at Publisher
M. Dehghan and F. Emami-Naeini, "The sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrodinger equation with nonhomogeneous boundary conditions," Applied Mathematical Modelling, vol. 37, pp. 9379-9397, 2013. View at Google Scholar | View at Publisher
G. Yagubov, F. Toyoglu, and M. Subaşı, "An optimal control problem for two-dimensional Schrodinger equation," Applied Mathematics and Computation, vol. 218, pp. 6177–6187, 2012. View at Google Scholar | View at Publisher
This study received no specific financial support.
The authors declare that they have no competing interests.
We are very grateful to the referee for his/her appropriate and
constructive suggestions and for his/her proposed corrections to improve