@article {5033,
title = {A spectral solution of the Sturm-Liouville equation: comparison of classical and nonclassical basis sets},
journal = {Journal of Computational and Applied Mathematics},
volume = {136},
number = {1-2},
year = {2001},
note = {ISI Document Delivery No.: 475METimes Cited: 16Cited Reference Count: 28},
month = {Nov},
pages = {17-35},
type = {Article},
abstract = {This paper considers a spectral method of solution of the Stunn-Liouville equation and the associated Schroedinger equation. The main objective is to develop a collocation method based on quadrature (collocation) points generated from nonclassical polynomials. The polynomials and associated quadrature points are calculated with Gautschi{\textquoteright}s Stieltjes procedure from some specified weight function. The particular spectral method used here with nonclassical basis sets is referred to as the Quadrature Discretization Method (QDM). The QDM and a related weighted QDM are applied to several Sturm-Liouville and Schroedinger equations and the results are compared with the traditional spectral methods based on Chebyshev and Legendre quadrature points. The results are also compared with the results of other workers wherever available. The QDM was found to give the most rapid convergence relative to other methods for the problems studied. (C) 2001 Elsevier Science B.V. All rights reserved.},
keywords = {LAGRANGE, MESHES, METHOD QDM, nonclassical polynomials, NUMERICAL-INTEGRATION, ORTHOGONAL POLYNOMIALS, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, Schroedinger equation, spectral methods, Sturm-Liouville equation},
isbn = {0377-0427},
url = {://000171167400002},
author = {Chen, H. and Shizgal, B. D.}
}
@article {4131,
title = {The quadrature discretization method in the solution of the Fokker-Planck equation with nonclassical basis functions},
journal = {Journal of Chemical Physics},
volume = {107},
number = {19},
year = {1997},
note = {ISI Document Delivery No.: YG655Times Cited: 12Cited Reference Count: 122},
month = {Nov},
pages = {8051-8063},
type = {Review},
abstract = {Fokker-Planck equations are used extensively to study a variety of problems in nonequilibrium statistical mechanics. A discretization method referred to as the quadrature discretization method (QDM) is introduced for the time-dependent solution of Fokker-Planck equations. The QDM is based on the discretization of the probability density function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to some weight function. For the Fokker-Planck equation, the weight functions that have often provided rapid convergence of the eigenvalues of the Fokker-Planck operator are the steady distributions at infinite time. Calculations are carried out for several systems with bistable potentials that arise in the study of optical bistability. reactive systems and climate models. The rate of convergence of the eigenvalues and the eigenfunctions of the Fokker-Planck equation is very rapid with this approach. The time evolution is determined in terms of the expansion of the distribution function in the eigenfunctions. (C) 1997 American Institute of Physics.},
keywords = {CLIMATIC TRANSITIONS, COLORED NOISE, DISCRETE-ORDINATE METHOD, INITIAL-VALUE PROBLEM, ONE-DIMENSIONAL, OPTICAL BISTABILITY, ORTHOGONAL POLYNOMIALS, SCHRODINGER-EQUATION, STOCHASTIC RESONANCE, SYSTEMS, TIME-DEPENDENT NUCLEATION},
isbn = {0021-9606},
url = {://A1997YG65500053},
author = {Shizgal, B. D. and Chen, H. L.}
}
@article {3811,
title = {The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions},
journal = {Journal of Chemical Physics},
volume = {104},
number = {11},
year = {1996},
note = {ISI Document Delivery No.: TY728Times Cited: 34Cited Reference Count: 66},
month = {Mar},
pages = {4137-4150},
type = {Article},
abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker-Planck equations. The Fokker-Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics, For this class of potentials, the groundstate wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker-Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrodinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O-2 and Ar-Xe with realistic pair potentials. For O-2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrodinger equation with the Henon-Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrodinger equation is very rapid with this approach. (C) 1996 American Institute of Physics.},
keywords = {BOUND-STATES, COEFFICIENTS, DISCRETE-ORDINATE METHOD, EIGENVALUE PROBLEMS, FOKKER-PLANCK EQUATION, ORTHOGONAL POLYNOMIALS, POTENTIALS, QUANTUM-MECHANICS, RECURRENCE, ROVIBRATIONAL STATES},
isbn = {0021-9606},
url = {://A1996TY72800030},
author = {Shizgal, B. D. and Chen, H.}
}