Numerical solution to the unsteady two‐dimensional Schrödinger equation using meshless local boundary integral equation method

A meshless local boundary integral equation (LBIE) method is proposed to solve the unsteady two-dimensional Schrodinger equation. The method is based on the LBIE with moving least-squares (MLS) approximation. For the MLS approximation, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time-stepping method is employed to deal with the time derivative. An efficient method for dealing with singular domain integrations that appear in the discretized equations is presented. Finally, numerical results are considered for some examples to demonstrate the accuracy, efficiency and high rate of convergence of this method. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  Anton Arnold,et al.  Numerically Absorbing Boundary Conditions for Quantum Evolution Equations , 1998, VLSI Design.

[2]  S. Long,et al.  Analysis of thin plates by the local boundary integral equation (LBIE) method , 2002 .

[3]  Satya N. Atluri,et al.  The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity , 2000 .

[4]  Satya N. Atluri,et al.  New concepts in meshless methods , 2000 .

[5]  J. Sládek,et al.  Meshless local boundary integral equation method for 2D elastodynamic problems , 2003 .

[6]  V. Vavourakis,et al.  A MLPG(LBIE) numerical method for solving 2D incompressible and nearly incompressible elastostatic problems , 2006 .

[7]  Mehdi Dehghan,et al.  Parameter determination in a partial differential equation from the overspecified data , 2005, Math. Comput. Model..

[8]  Christophe Besse,et al.  Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions , 2004, Math. Comput..

[9]  Wei-Ping Huang,et al.  The finite-difference vector beam propagation method: analysis and assessment , 1992 .

[10]  Mehdi Dehghan,et al.  Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices , 2006, Math. Comput. Simul..

[11]  Demosthenes Polyzos,et al.  A MLPG (LBIE) METHOD FOR SOLVING FREQUENCY DOMAIN ELASTIC PROBLEMS , 2003 .

[12]  Ch. Zhang,et al.  Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids , 2003 .

[13]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[14]  Murat Subaşi,et al.  On the finite‐differences schemes for the numerical solution of two dimensional Schrödinger equation , 2002 .

[15]  A. V. Popov,et al.  Application of the parabolic wave equation to X-ray diffraction optics , 1995 .

[16]  Chuanzeng Zhang,et al.  A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials , 2005 .

[17]  Fred D. Tappert,et al.  The parabolic approximation method , 1977 .

[18]  Hyun Gyu Kim,et al.  A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods , 1999 .

[19]  V. Sladek,et al.  Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties , 2000 .

[20]  Vladimir Sladek,et al.  Application of local boundary integral equation method into micropolar elasticity , 2003 .

[21]  D. Polyzos,et al.  A meshless local boundary integral equation method for solving transient elastodynamic problems , 2005 .

[22]  Mehdi Dehghan,et al.  A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions , 2007, Comput. Math. Appl..

[23]  Mehdi Dehghan.  Determination of a control parameter in the two-dimensional diffusion equation , 2001 .

[24]  S. Atluri,et al.  A meshless local boundary integral equation (LBIE) method for solving nonlinear problems , 1998 .

[25]  Mehdi Dehghan,et al.  Fourth-order techniques for identifying a control parameter in the parabolic equations , 2002 .

[26]  E. J. Sellountos,et al.  A MLPG (LBIE) approach in combination with BEM , 2005 .

[27]  S. N. Atluri,et al.  Numerical integration of singularities in meshless implementation of local boundary integral equations , 2000 .

[28]  S. Atluri,et al.  A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems , 1999 .

[29]  Jiten C. Kalita,et al.  A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation , 2006 .

[30]  Vladimir Sladek,et al.  Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method , 2003 .

[31]  W. Ang,et al.  A dual‐reciprocity boundary element solution of a generalized nonlinear Schrödinger equation , 2004 .