# Numerical solution of Schrodinger equation using compact finite differences method and the cubic spline functions

## DOI:

https://doi.org/10.14419/ijamr.v3i4.3743## Published:

2014-11-22## Keywords:

Compact finite difference, Cubic Spline functions, Numerical solution, Schrodinger equation.## Abstract

In this paper, a high-order method for solving the Schrodinger equation is introduced. We apply a compact finite difference approximation for discretizing spatial derivatives and we use the C^{1}-cubic spline collocation method for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. We can obtain both pointwise approximations at the all mesh points and, a cubic spline solution in each space step by the method. Numerical results show that the method is an efficient technique for solving the one-dimensional Schrodinger equation.Â

## References

R. Abdur, A.I.B.MD. Ismail, Numerical studies on two-dimensional schrodinger equation by chebyshev spectral collocation method, U.P.B. Sci. Bull. (2011) 1223-1227.

M. Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simul. 71 (2006) 16-30.

M. Dehghan, D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrodinger equation using mesh less boundary integral equation method, Int. J. Numer. Math. Engng. 76 (2008) 501-520.

M. Dehghan, A. Shokri, A numerical method for two-dimensional Schrodinger equation using collocation and radial basis functions, Comput. Math. Appl. 54 (2007) 136-146.

F.Y. Hajj, Solution of the Schrodinger equation in two and three dimensions, J. Phys. B. 18 (1985) 1-11.

L.Gr. Ixaru, Operations on oscillatory functions, Comput. Phys. Comm. 105 (1997) 1-9.

J.C. Kalita, P. Chhabra, S. Kumar, A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrodinger equation, J. Comput. Appl. Math. 197 (2006) 141-149.

S. Kim, Compact schemes for acoustics in the frequency domain, Math. Comput. Modeling, 37 (2003) 1335-1341.

Y.V. Kopylov, A.V. Popov, A.V. Vinogradov, Applications of the parabolic wave equations to X-ray diffraction optics, Optics Comm. 118 (1995) 619-636.

A. Mohebbi, M. Dehghan, High-order compact solution of the one-dimensional heat and advection-diffusion equations, Applied Mathematical Modeling, 34 (2010) 3071-3084.

A. Mohebbi, M. Dehghan, The use of compact boundary value method for the solution of two-dimensional Schrodinger equation, Journal of Computational and Applied Mathematics, 225 (2009) 124-134.

S. Sallam, M. Naim Anwar, M. R. Abdel-Aziz, Unconditionally stable C1-cubic spline collocation method for solving parabolic equations, International Journal of Computer Mathematics, (2004) 813-821.

S. Sallam and M. Naim Anwar, Stabilized C1-cubic spline collocation method for solving first-order ordinary initial value problems, International Journal of Computer Mathematics, (2000) 87-96.

J.S. Shang, High-order compact difference schemes for time-dependent Maxwell equations, J. Comput. Phys. 153 (1999) 312-333.

W.F. Spotz, High-order compact finite difference schemes for computational mechanics, Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1995.

M. Subasi, On the finite difference schemes for the numerical solution of two dimensional Schrodinger equation, Numer. Methods Partial Differential Equations, 18 (2002) 752-758.