Numerical Methods for Schrödinger-Type Problems

Abstract

Several physical phenomena are modeled by initial-boundary value problems which can be formulated as SchrOdinger -type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrodinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrodinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving SchrOdinger-type systems are also outlined.Keywords: Schrodinger systems, linear and nonlinear Schrodinger equations, parabolic wave equation, vibrating beams and plates, fourth order parabolic equations, finite difference methods, finite element Galerkin methods, orthogonal spline collocation methods, spectral methods IntroductionWe consider the numerical solution of problems that can be written as initialboundary value problems (IBVPs) for real systems of partial differential equations of the form

Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning

“…|Eu| ij x [3] 1.1301e-8 4.8987e-9 4.5232e-9 4.4073e-9 1.3425e-10 x [7] 3.9975e-10 1.6408e-9 2.9687e-9 6.0980e-10 5.2440e-9 x [11] 5.8580e-9 9.4360e-10 1.3082e-9 2.1858e-10 2.2288e-9 x [14] 3.9906e-9 5.2506e-9 1.5004e-9 8.5276e-10 4.4032e-9…”

“…Chang et al studied eight different schemes for the generalized onedimensional NLS equation in [8]. For more about the numerical methods for the NLS equation, see the review paper by Fairweather et al [11].…”

Dynamical behavior of a competition model of Telcom and Cable Television industry

“…8 The first step in analyzing the model (4) is to determine the steady state [2][3][4][5][6], which are as follows:…”

“…As a result, the correct competitive strategy is instituted. 5,6 In [1], we investigate the interaction of China Telecom and Cable Television using a discrete model. However, in this paper, we present theoretical research on the competition strategy of the CATV industry under the trend of the three networks integration using an ordinary differential equation.…”

A modified numerical scheme for the cubic Schrödinger equation

“…Numerical methods for solving the CNLS equation were found among others in [20][21][22][23][24][25] using finite-difference methods, in [26][27][28][29] with finite-elements, in [30] with pseudospectral methods, in [31] with Lanczos' Tau method, in [32,33] with the Adomian decomposition method, in [34] with the variation iteration method, etc. For more about the numerical methods for the CNLS equation, see the review article [35].…”